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topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#TopologicalProperties'>Topological properties</a></li> <li><a href='#products_of_cwcomplexes'>Products of CW-complexes</a></li> <li><a href='#up_to_homotopy_equivalence'>Up to homotopy equivalence</a></li> <li><a href='#Subcomplexes'>Subcomplexes</a></li> <li><a href='#CellularApproximationTheorem'>Cellular approximation theorem</a></li> <li><a href='#fibrations'>Fibrations</a></li> <li><a href='#SingularHomology'>Singular homology</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>CW-complex</strong> is a <a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> which is, or can be, built up inductively, by a process of <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> along their <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> <a class="existingWikiWord" href="/nlab/show/n-spheres">(n-1)-spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a> built from the basic topological cells <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^{n-1} \hookrightarrow D^n</annotation></semantics></math>.</p> <p>Being, therefore, essentially combinatorial objects, CW complexes are the principal objects of interest in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>; in fact, most spaces of interest to algebraic topologists are <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a> to CW-complexes. Notably the <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of every <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, hence also of every <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, etc., is a CW complex.</p> <p>Milnor has argued that the category of spaces which are homotopy equivalent to CW-complexes, also called <a class="existingWikiWord" href="/nlab/show/m-cofibrant+spaces">m-cofibrant spaces</a>, is a <a class="existingWikiWord" href="/nlab/show/nice+category+of+spaces">convenient category of spaces</a> for <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>.</p> <p>Also, CW complexes are among the <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant objects</a> in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>. In fact, <em>every</em> topological space is <em><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weakly homotopy equivalent</a></em> to a CW-complex (but need not be <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">strongly homotopy equivalent</a> to one). See also at <em><a class="existingWikiWord" href="/nlab/show/CW-approximation">CW-approximation</a></em>. Since every topological space is a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> in this <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure, this means that the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> on the CW-complexes is a category of “homotopically very good representatives” of <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>. See at <em><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em> and <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a></em> for more on this.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(origin of the “CW” terminology)</strong></p> <p>The terminology “CW-complex” goes back to <a class="existingWikiWord" href="/nlab/show/John+Henry+Constantine+Whitehead">John Henry Constantine Whitehead</a> (and see the discussion in <a href="#HatcherTopologyOfCellComplexes">Hatcher, “Topology of cell complexes”, p. 520</a>).</p> <p>To quote from the original paper, which was “an address delivered before the Princeton Meeting of the (American Mathematical) Society on November 2, 1946”, Whitehead states:</p> <blockquote> <p>In this presentation we abandon <a class="existingWikiWord" href="/nlab/show/simplicial+complexes">simplicial complexes</a> in favor of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a>. This first part consists of geometrical preliminaries, including some elementary propositions concerning what we call closure finite complexes with weak topology, abbreviated to CW-complexes, …</p> </blockquote> <p>Thus the “CW” stands for the following two properties shared by any CW-complex:</p> <ul> <li> <p><strong>C</strong> = “closure finiteness”: a <a class="existingWikiWord" href="/nlab/show/compact+subset">compact subset</a> of a CW-complex intersects the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of only finitely many cells (<a href="classical+model+structure+on+topological+spaces#CompactSubsetsAreSmallInCellComplexes">prop.</a>), hence in particular so does the closure of any cell.</p> </li> <li> <p><strong>W</strong> = “weak topology”: Since a CW-complex is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> over its cells, and as such equipped with the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> of the cell inclusion maps, a subset of a CW-complex is open or closed precisely if its restriction to (the closure of) each cell is open or closed, respectively.</p> </li> </ul> <p>(Whitehead called the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of the <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> the “cells”, so that their closure of each cell is the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disk.)</p> </div> <p><br /></p> <h2 id="Definition">Definition</h2> <p>In the following let <a class="existingWikiWord" href="/nlab/show/Top">Top</a> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, or any of its variants, <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>.</p> <div class="num_defn" id="SpheresAndDisks"> <h6 id="definition_2">Definition</h6> <p><strong>(spheres and disks)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">D^n \in Top</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a>, for instance realized (up to <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>) as the <a class="existingWikiWord" href="/nlab/show/closed+ball">closed unit ball</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and equipped with the induced <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">S^{n-1} \in Top</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/n-sphere">(n-1)-sphere</a>, for instance realized as the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of the <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a>, also equipped with the corresponding <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">i_n \;\colon\; S^{n-1} \hookrightarrow D^n</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> that exhibits this <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> inclusion.</p> <p>We also call these functions the <em>generating cofibrations</em> (of the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>).</p> </li> </ul> <p>Notice that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">S^{-1} = \emptyset</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">S^0 = \ast \sqcup \ast</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_defn" id="SingleCellAttachment"> <h6 id="definition_3">Definition</h6> <p><strong>(single cell attachment)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, then an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cell <a class="existingWikiWord" href="/nlab/show/attaching+space">attachment</a></strong> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the result of gluing an <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, along a prescribed image of its bounding <a class="existingWikiWord" href="/nlab/show/n-sphere">(n-1)-sphere</a> (def. <a class="maruku-ref" href="#SpheresAndDisks"></a>):</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; S^{n-1} \longrightarrow X </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, then the <em><a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>∪</mo> <mi>ϕ</mi></msub><msup><mi>D</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> X \cup_\phi D^n \,\in Top </annotation></semantics></math></div> <p>is the topological space which is the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of the boundary inclusion of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, hence the universal space that makes the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mi>ϕ</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>∪</mo> <mi>ϕ</mi></msub><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \cup_\phi D^n } \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/GluingHemispheres.jpg" width="400" /></div> <p>If we take the defining boundary inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\iota_n \colon S^{n-1} \to D^n</annotation></semantics></math> itself as an attaching map, then we are gluing two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks to each other along their common boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math>. The result is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. </annotation></semantics></math></div> <p>(graphics from Ueno-Shiga-Morita 95)</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>A single cell <a class="existingWikiWord" href="/nlab/show/attaching+space">attachment</a> of a 0-cell, according to def. <a class="maruku-ref" href="#SingleCellAttachment"></a> is the same as forming the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \sqcup \ast</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/point">point</a> space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>∅</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msup><mi>D</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><mo>⊔</mo><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (S^{-1} = \emptyset) &\overset{\exists !}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ (D^0 = \ast) &\longrightarrow& X \sqcup \ast } \,. </annotation></semantics></math></div> <p>In particular if we start with the <a class="existingWikiWord" href="/nlab/show/empty+topological+space">empty topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">X = \emptyset</annotation></semantics></math> itself, then by <a class="existingWikiWord" href="/nlab/show/attaching+space">attaching</a> 0-cells we obtain a <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a>. To this then we may attach higher dimensional cells.</p> </div> <div class="num_defn" id="CellAttachments"> <h6 id="definition_4">Definition</h6> <p><strong>(attaching many cells at once)</strong></p> <p>If we have a <a class="existingWikiWord" href="/nlab/show/set">set</a> of attaching maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S^{n_i-1} \overset{\phi_i}{\longrightarrow} X\}_{i \in I}</annotation></semantics></math> (as in def. <a class="maruku-ref" href="#SingleCellAttachment"></a>), all to the same space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we may think of these as one single continuous function out of the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> of their <a class="existingWikiWord" href="/nlab/show/domain">domain</a> spheres</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\phi_i)_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} S^{n_i-1} \longrightarrow X \,. </annotation></semantics></math></div> <p>Then the result of attaching <em>all</em> the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the pushout of the corresponding <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of boundary inclusions:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>∪</mo> <mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></msub><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X \cup_{(\phi_i)_{i \in I}} \left(\underset{i \in I}{\sqcup} D^{n_i}\right) } \,. </annotation></semantics></math></div></div> <p>Apart from attaching a set of cells all at once to a fixed base space, we may “attach cells to cells” in that after forming a given cell attachment, then we further attach cells to the resulting attaching space, and ever so on:</p> <div class="num_defn" id="RelativeCellComplexes"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/relative+cell+complexes">relative cell complexes</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a topological space, then a <em>topological <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a></em> of countable height based on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \colon X \longrightarrow Y </annotation></semantics></math></div> <p>and a <a class="existingWikiWord" href="/nlab/show/sequential+diagram">sequential diagram</a> of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>↪</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> X = X_{-1} \hookrightarrow X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots </annotation></semantics></math></div> <p>such that</p> <ol> <li> <p>each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_k \hookrightarrow X_{k+1}</annotation></semantics></math> is exhibited as a cell attachment according to def. <a class="maruku-ref" href="#CellAttachments"></a>, hence presented by a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>S</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} S^{n_i - 1} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_{k+1} } \,. </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><munder><mo>∪</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Y = \underset{k\in \mathbb{N}}{\cup} X_k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of all these cell attachments, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> is the canonical inclusion; or stated more abstractly: the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> is the inclusion of the first component of the diagram into its <a class="existingWikiWord" href="/nlab/show/colimit">colimiting cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>k</mi></msub><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\underset{\longrightarrow}{\lim}_k X_k</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi><mo>=</mo><munder><mi>lim</mi><mo>⟶</mo></munder><msub><mi>X</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet } </annotation></semantics></math></div></li> </ol> <p>If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">X = \emptyset</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a> then the result is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\emptyset \hookrightarrow Y</annotation></semantics></math>, which is equivalently just a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> built form “attaching cells to nothing”. This is then called just a <em>topological <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></em> of countable hight.</p> <p>Finally, a topological (relative) cell complex of countable hight is called a <strong>CW-complex</strong> if the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-st cell attachment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_k \to X_{k+1}</annotation></semantics></math> is entirely by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k+1)</annotation></semantics></math>-cells, hence exhibited specifically by a pushout of the following form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>S</mi> <mi>k</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mi>D</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i \in I}{\sqcup} S^{k} &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& X_k \\ \downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{k+1} &\longrightarrow& X_{k+1} } \,. </annotation></semantics></math></div> <p>A <em><a class="existingWikiWord" href="/nlab/show/finite+CW-complex">finite CW-complex</a></em> is one which admits a presentation in which there are only finitely many attaching maps, and similarly a <em>countable CW-complex</em> is one which admits a presentation with countably many attaching maps.</p> <p>Given a CW-complex, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> is also called its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/skeleton">skeleton</a>.</p> </div> <p>A <strong>cellular map</strong> between CW-complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>Y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f(X_n) \subset Y_n</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="TopologicalProperties">Topological properties</h3> <p> <div class='num_prop' id='CWComplexesAreLocallyContractible'> <h6>Proposition</h6> <p>Every CW-complex is a <a class="existingWikiWord" href="/nlab/show/locally+contractible+topological+space">locally contractible topological space</a>.</p> </div> (e.g. <a href="#Hatcher02">Hatcher 2002, prop. A.4</a>).</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a>)</strong> <br /> Every CW-complex is a</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/normal+topological+space">normal topological space</a>, in particular a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact topological space</a>.</p> </li> </ol> <p></p> </div> </p> <p> <div class='num_prop' id='CWComplexIsCompactlyGenerated'> <h6>Proposition</h6> <p>Every CW-complex is a <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a>.</p> </div> </p> <p> <div class='proof'> <h6>Proof</h6> <p>Since a CW-complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> over attachments of standard <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_i}</annotation></semantics></math> (its cells), by the characterization of colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> (<a href="Top#DescriptionOfLimitsAndColimitsInTop">prop.</a>) a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks are compact, this implies one direction: if a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> intersected with all compact subsets is closed, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is closed.</p> <p>For the converse direction, since <a class="existingWikiWord" href="/nlab/show/a+CW-complex+is+a+Hausdorff+space">a CW-complex is a Hausdorff space</a> and since <a class="existingWikiWord" href="/nlab/show/compact+subspaces+of+Hausdorff+spaces+are+closed">compact subspaces of Hausdorff spaces are closed</a>, the intersection of a closed subset with a compact subset is closed.</p> </div> </p> <p>In fact:</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Every CW-complex is a <a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+space">Delta-generated topological space</a>.</p> </div> See there, <a href="Delta-generated+topological+space#CWComplexesAreDTopologicalSpaces">this Prop.</a></p> <h3 id="products_of_cwcomplexes">Products of CW-complexes</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> is an inclusion of CW-complexes, then the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math> is naturally itself a CW-complex, such that the quotient map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to X/A</annotation></semantics></math> is cellular.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a CW-complex and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+CW-complex">finite CW-complex</a>, then the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">X \times K</annotation></semantics></math> is naturally itself a CW-complex (see <a href="#BrookeTaylor17">Brooke-Taylor 2017</a> for more and more generality, and see Prop. <a class="maruku-ref" href="#ClosureOfCWComplexesUnderCartesianProduct"></a> below).</p> <p>For example the <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> of a CW-complex itself carries the structure of a CW-complex.</p> <p>Similarly for <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> CW-complexes: the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of a pointed CW-complex with a finite pointed CW-complex is a pointed CW-complex. For example the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S^1 \wedge X</annotation></semantics></math> of a pointed CW-complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is itself a CW-complex.</p> <div class="num_prop" id="ClosureOfCWComplexesUnderCartesianProduct"> <h6 id="proposition">Proposition</h6> <p><strong>(product preserves CW-complexes in <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> with attaching maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ϕ</mi> <mi>α</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\phi_\alpha\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Ψ</mi> <mi>β</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\Psi_\beta\}</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/k-ification">k-ification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">(X \times Y)_c</annotation></semantics></math> of their <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math> (hence their Cartesian product in the category of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>) is again a CW-complex with attaching maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Φ</mi> <mi>α</mi></msub><mo>×</mo><msub><mi>Ψ</mi> <mi>β</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\Phi_\alpha \times \Psi_\beta\}</annotation></semantics></math>.</p> <p>If either of the two CW-complexes is a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological space</a> or if both are countable CW-complexes (have a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of cells) then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>c</mi></msub><mo>≃</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> (X\times Y)_c \simeq X \times Y </annotation></semantics></math></div> <p>and so then the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math> itself has CW-complex structure.</p> </div> <p>(e.g. <a href="#Hatcher02">Hatcher 2002, theorem A.6</a>)</p> <h3 id="up_to_homotopy_equivalence">Up to homotopy equivalence</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Every CW complex is homotopy equivalent to (the <a href="simplicial+complex#geometric_realisations_and_polyhedra">realization</a> of) a <a class="existingWikiWord" href="/nlab/show/simplicial+complex">simplicial complex</a>.</p> </div> <p>See <a href="#Gray">Gray</a>, Corollary 16.44 (p. 149) and Corollary 21.15 (p. 206). For more see at <em><a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a></em>.</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Every CW complex is homotopy equivalent to a space that admits a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>.</p> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of a CW complex and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+CW+complex">finite CW complex</a>, then the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> has the homotopy type of a CW complex.</p> </div> <p>(<a href="#Milnor59">Milnor 59</a>)</p> <h3 id="Subcomplexes">Subcomplexes</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a CW complex, the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X' \hookrightarrow X</annotation></semantics></math> of any subcomplex has an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which is a <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">X'</annotation></semantics></math>. In particular such an inclusion is a <em><a href="relative+homology#GoodPair">good pair</a></em> in the sense of <a class="existingWikiWord" href="/nlab/show/relative+homology">relative homology</a>.</p> </div> <p>For instance (<a href="#Hatcher02">Hatcher 2002, prop. A.5</a>).</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> the inclusion of a subcomplex into a CW complex, then the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> is often called a <em><a class="existingWikiWord" href="/nlab/show/CW-pair">CW-pair</a></em>. This appears notably in the <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> for <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a>.</p> </div> <p>e.g. (<a href="#AGP02">AGP 02, def. 5.1.11</a>)</p> <h3 id="CellularApproximationTheorem">Cellular approximation theorem</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/cellular+approximation+theorem">cellular approximation theorem</a></em> states that every <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> between <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a> (with chosen CW presentations) is <a class="existingWikiWord" href="/nlab/show/homotopy">homotopic</a> to a <a class="existingWikiWord" href="/nlab/show/cellular+map">cellular map</a> (a map induced by a morphism of <a class="existingWikiWord" href="/nlab/show/cell+complexes">cell complexes</a>).</p> <p>This is the analogue for <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> of the <a class="existingWikiWord" href="/nlab/show/simplicial+approximation+theorem">simplicial approximation theorem</a> (sometimes also called lemma): that every continuous map between the <a class="existingWikiWord" href="/nlab/show/geometric+realizations">geometric realizations</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+complexes">simplicial complexes</a> is <a class="existingWikiWord" href="/nlab/show/homotopy">homotopic</a> to a map induced by a map of <a class="existingWikiWord" href="/nlab/show/simplicial+complexes">simplicial complexes</a> (after subdivision).</p> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/cellular+approximation+theorem">cellular approximation theorem</a></em>.</p> <h3 id="fibrations">Fibrations</h3> <p>Fibrations between CW-complexes also behave particularly well: <a class="existingWikiWord" href="/nlab/show/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration">a Serre fibration between CW-complexes is a Hurewicz fibration</a>.</p> <h3 id="SingularHomology">Singular homology</h3> <p>We discuss aspects of the <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo></mrow><annotation encoding="application/x-tex">H_n(-) \colon </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of CW-complexes. See also at <em><a class="existingWikiWord" href="/nlab/show/cellular+homology">cellular homology</a> of CW-complexes</em>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a CW-complex and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \cdots \hookrightarrow X </annotation></semantics></math></div> <p>for its <a class="existingWikiWord" href="/nlab/show/filtered+topological+space">filtered topological space</a>-structure with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_{n+1}</annotation></semantics></math> the topological space obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> by gluing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-cells. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>nCells</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">nCells \in Set</annotation></semantics></math> for the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <div class="num_prop" id="SkeletalRelativeSingularHomologyOfCW"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/relative+singular+homology">relative singular homology</a> of the filtering degrees is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>nCells</mi><mo stretchy="false">]</mo></mtd> <mtd><mi>if</mi><mspace width="thickmathspace"></mspace><mi>k</mi><mo>=</mo><mi>n</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H_n(X_k , X_{k-1}) \simeq \left\{ \array{ \mathbb{Z}[nCells] & if\; k = n \\ 0 & otherwise } \right. \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>nCells</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[nCells]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/free+abelian+group">free abelian group</a> on the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-cells.</p> </div> <p>The proof is spelled out at <em><a href="relative+homology#RelativeHomologyOfCWComplexes">Relative singular homology - Of CW complexes</a></em>.</p> <div class="num_prop" id="RelativeHomologyOfFilterStep"> <h6 id="proposition_4">Proposition</h6> <p>With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>,</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k,n \in \mathbb{N}</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>></mo><mi>n</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (k \gt n) \Rightarrow (H_k(X_n) \simeq 0) \,. </annotation></semantics></math></div> <p>In particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a CW-complex of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension+of+a+CW-complex">dimension of a CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">dim X</annotation></semantics></math> (the maximum degree of cells), then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>></mo><mi>dim</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (k \gt dim X) \Rightarrow (H_k(X) \simeq 0). </annotation></semantics></math></div> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo><</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \lt n</annotation></semantics></math> the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_k(X_n) \stackrel{\simeq}{\to} H_k(X) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k = n</annotation></semantics></math> we have an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>image</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> image(H_n(X_n) \to H_n(X)) \simeq H_n(X) \,. </annotation></semantics></math></div></div> <p>This is mostly for instance in (<a href="#Hatcher02">Hatcher 2002, lemma 2.34 b),c)</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> in <a class="existingWikiWord" href="/nlab/show/relative+homology">relative homology</a>, discussed at <em><a href="relative+homology#LongExactSequences">Relative homology – long exact sequences</a></em>, we have an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{k+1}(X_n , X_{n-1}) \to H_k(X_{n-1}) \to H_k(X_n) \to H_k(X_n, X_{n-1}) \,. </annotation></semantics></math></div> <p>Now by prop. <a class="maruku-ref" href="#SkeletalRelativeSingularHomologyOfCW"></a> the leftmost and rightmost homology groups here vanish when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≠</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \neq n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≠</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \neq n-1</annotation></semantics></math> and hence exactness implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>H</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_k(X_{n-1}) \stackrel{\simeq}{\to} H_k(X_n) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≠</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \neq n,n-1</annotation></semantics></math>. This implies the first claims by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.</p> <p>Finally for the last claim use that the above exact sequence gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> H_{n-1+1}(X_n , X_{n-1}) \to H_{n-1}(X_{n-1}) \to H_{n-1}(X_n) \to 0 </annotation></semantics></math></div> <p>and hence that with the above the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H_{n-1}(X_{n-1}) \to H_{n-1}(X)</annotation></semantics></math> is surjective.</p> </div> <h2 id="Examples">Examples</h2> <p> <div class='num_remark'> <h6>Example</h6> <p>Any undirected <a class="existingWikiWord" href="/nlab/show/graph">graph</a> (loops and/or multiple edges allowed) has a geometric realization as a 1-dimensional CW complex.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> of any <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> is a CW-complex (<a href="#Milnor57">Milnor 57</a>).</p> <p>In particular, in the context of the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a> the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a> and <a class="existingWikiWord" href="/nlab/show/nice+topological+spaces">nice topological spaces</a> maps each <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> to a CW-complex.</p> </div> </p> <p> <div class='num_remark'> <h6>Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> have a standard CW-complex structure, with exactly 2-cells in each dimension, obtained <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> by <a class="existingWikiWord" href="/nlab/show/space+attachment">attaching</a> two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/hemispheres">hemispheres</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-sphere regarded as the <a class="existingWikiWord" href="/nlab/show/equator">equator</a> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere.</p> </div> </p> <p>The <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+sphere">infinite-dimensional sphere</a> may be realized as the CW-complex which is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> over the resulting <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a>-inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>↪</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>↪</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>↪</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">S^n \hookrightarrow S^{n + 1} \hookrightarrow S^{n + 2} \hookrightarrow \cdots</annotation></semantics></math>. \end{example}</p> <p> <div class='num_remark'> <h6>Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> or <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> has the structure of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> with a single cell i in each dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding="application/x-tex">2k</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi></mrow><annotation encoding="application/x-tex">4k</annotation></semantics></math>, respectively. See at <em><a class="existingWikiWord" href="/nlab/show/cell+structure+of+K-projective+space">cell structure of K-projective space</a></em>.</p> </div> </p> <p> <div class='num_remark' id='SmoothManifoldsAdmitCWComplexStructure'> <h6>Example</h6> <p><strong>(smooth manifolds)</strong> Every <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> admits a smooth <a class="existingWikiWord" href="/nlab/show/triangulation">triangulation</a> (by the <a class="existingWikiWord" href="/nlab/show/triangulation+theorem">triangulation theorem</a>) and hence a CW-complex structure.</p> <p>In the generality of manifolds with group actions see at <em><a href="G-CW+complex#GManifolds">G-CW complex – G-manifolds</a></em>.</p> <p>Every noncompact <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a> to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-dimensional CW-complex. (<a href="#NapierRamachandran">Napier & Ramachandran</a>).</p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension+of+a+CW-complex">dimension of a CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+map">cellular map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW+approximation">CW approximation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-finite+CW-complex">quasi-finite CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+homology">cellular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-spectrum">CW-spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-CW+complex">G-CW complex</a></p> </li> </ul> <div> <p><strong>examples of <a href="Top#UniversalConstructions">universal constructions of topological spaces</a>:</strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_1"><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAAA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/limits">limits</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_2"><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAAA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/colimits">colimits</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_3"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/point+space">point space</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_4"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_5"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_6"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_7"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_8"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_9"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/disjoint+union+topological+space">disjoint union topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_10"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_11"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_12"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_13"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_14"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_15"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> fiber space <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_16"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_17"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_18"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_19"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_20"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_21"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_22"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_23"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_24"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The introduction of the term is contained in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/J.+H.+C.+Whitehead">J. H. C. Whitehead</a>, <em>Combinatorial homotopy I</em> , Bull. Amer. Math. Soc, 55, (1949), 213–245.</li> </ul> <p>Basic textbook accounts:</p> <ul> <li> <p>Brayton Gray, <em>Homotopy Theory: An Introduction to Algebraic Topology</em>, Academic Press, New York (1975).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/George+Whitehead">George Whitehead</a>, chapter II of <em>Elements of homotopy theory</em>, 1978</p> </li> <li id="May"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/show/A+Concise+Course+in+Algebraic+Topology">A Concise Course in Algebraic Topology</a></em>, U. Chicago Press (1999)</p> </li> <li id="Hatcher02"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Algebraic Topology</em>, Cambridge University Press 2002 (<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401">ISBN:9780521795401</a>, <a href="https://pi.math.cornell.edu/~hatcher/AT/ATpage.html">webpage</a>)</p> </li> <li id="HatcherTopologyOfCellComplexes"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Topology of cell complexes</em> (<a href="https://www.math.cornell.edu/~hatcher/AT/ATapp.pdf">pdf</a>) in <em>Algebraic Topology</em></p> </li> <li id="HatcherK"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Vector bundles & K-theory</em> (<a href="https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html">web</a>)</p> </li> <li id="FP"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, <em>Cellular structures in topology</em>, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (<a href="https://doi.org/10.1017/CBO9780511983948">doi:10.1017/CBO9780511983948</a>, <a href="https://epub.ub.uni-muenchen.de/4493/1/4493.pdf">pdf</a>)</p> </li> <li id="AGP02"> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguilar">Marcelo Aguilar</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Prieto">Carlos Prieto</a>, section 5.1 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> </ul> <p>Original articles include</p> <ul> <li id="Milnor59"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <em>On spaces having the homotopy type of a CW-complex</em>, Trans. Amer. Math. Soc. 90 (2) (1959), 272-280.</p> </li> <li id="Milnor57"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <em>The geometric realization of a semi-simplicial complex</em>, Annals of Mathematics, 2nd Ser., <strong>65</strong>, n. 2. (Mar., 1957), pp. 357-362; doi:<a href="https://doi.org/10.2307/1969967">10.2307/1969967</a>, <a href="https://pdfs.semanticscholar.org/7cbe/0482ce422d3adcc84be80b5ab3f68520a247.pdf">Semantic scholar pdf</a></p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/products">products</a> of CW-complexes:</p> <ul> <li id="BrookeTaylor17"><a class="existingWikiWord" href="/nlab/show/Andrew+Brooke-Taylor">Andrew Brooke-Taylor</a>, <em>Products of CW complexes – the full story</em>, 2017 (<a href="http://www.math.helsinki.fi/logic/arctic/2017/Slides/BrookeTaylor_arctic2017.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Brooke-Taylor_ProductsOfCWComplexes.pdf" title="pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li id="NapierRamachandran">Terrance Napier, Mohan Ramachandran, <em><a href="http://www.unige.ch/math/EnsMath/EM_en/">Elementary Construction of Exhausting Subsolutions of Elliptic Operators</a></em> L’Enseignement Mathématique, t. 50 (2004), p. 367 - 389.</li> </ul> <p>An inconclusive discussion <a href="http://nforum.mathforge.org/discussion/4135/simplicial-homology/?Focus=33785#Comment_33785">here</a> about what parts of the definition of a CW complex should be <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">properties</a> and what parts should be structure.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on February 5, 2024 at 03:12:51. 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