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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential 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="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism 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> <ul> <li><a href='#as_a_topological_space'>As a topological space</a></li> <li><a href='#as_a_homotopy_type'>As a homotopy type</a></li> </ul> <li><a href='#properties'>Properties</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>The <strong>circle</strong> (the 1-<a class="existingWikiWord" href="/nlab/show/dimensional">dimension</a> <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>) is a fantastic thing with lots and lots of <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">properties and extra structures</a>. It is a:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> (the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a>)</p> </li> <li> <p><span class="newWikiWord">co-H-group<a href="/nlab/new/co-H-group">?</a></span></p> </li> </ul> <p>and it is one of the basic building blocks for lots of areas of mathematics, including:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+spaces">loop spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/knots">knots</a> and <a class="existingWikiWord" href="/nlab/show/links">links</a></p> </li> </ul> <h2 id="definition">Definition</h2> <p>We consider the circle first as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then as the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> represented by that space.</p> <h3 id="as_a_topological_space">As a topological space</h3> <div class="num_defn" id="circle"> <h6 id="definition_2">Definition</h6> <p>The two most common definitions of the circle as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> are:</p> <ol> <li> <p>It is the <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> consisting of those numbers of <a class="existingWikiWord" href="/nlab/show/norm">norm</a>/<a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mo>:</mo><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> U(1) \coloneqq \{z \in \mathbb{C} : {|z|} = 1\} </annotation></semantics></math></div> <p>(of course, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> can be identified with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> and an equivalent formulation of this definition given; also, to emphasise the reason for the notation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> can be replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mo>*</mo></msup><mo>=</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}^* = GL_1(\mathbb{C})</annotation></semantics></math>)</p> </li> <li> <p>It is the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> by the integers:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>≔</mo><mi>ℝ</mi><mo stretchy="false">/</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> S^1 \coloneqq \mathbb{R}/\mathbb{Z} </annotation></semantics></math></div></li> </ol> </div> <p>The standard equivalence of the two definitions is given by the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \to \mathbb{C}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>↦</mo><msup><mi>e</mi> <mrow><mn>2</mn><mi>π</mi><mi>i</mi><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">t \mapsto e^{2 \pi i t}</annotation></semantics></math>.</p> <p>In constructive mathematics, the notions of real numbers and complex numbers split into multiple notions. In general, if the metric space of the real numbers is not <a class="existingWikiWord" href="/nlab/show/sequentially+Cauchy+complete">sequentially Cauchy complete</a>, such as in the <a class="existingWikiWord" href="/nlab/show/modulated+Cantor+real+numbers">modulated Cantor real numbers</a>, then the two definitions cannot be proven to be equivalent, since the <a class="existingWikiWord" href="/nlab/show/exponential+function">exponential function</a> is not well defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>. Only if the real numbers are <a class="existingWikiWord" href="/nlab/show/sequentially+Cauchy+complete">sequentially Cauchy complete</a> are the two definitions the same.</p> <h3 id="as_a_homotopy_type">As a homotopy type</h3> <p>As the bare <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ʃ</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">&#643; S^1 \simeq \mathbf{B}\mathbb{Z}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ʃ</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mo>*</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> &#643; S^1 \simeq * \coprod_{* \coprod * } * \,. </annotation></semantics></math></div> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, this can be formalized as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> generated by a point <code>base</code> and a path from <code>base</code> to itself; see <a class="existingWikiWord" href="/nlab/show/circle+type">circle type</a> for more.</p> <h2 id="properties">Properties</h2> <p>The topological circle is a <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a>, <a class="existingWikiWord" href="/nlab/show/connected+space">connected</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. It is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (indeed, it is the only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional compact, connected smooth manifold). It is <strong>not</strong> <a class="existingWikiWord" href="/nlab/show/simply+connected+space">simply connected</a>.</p> <p>The circle is a model for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/integer">integers</a>. Equivalently, the circle is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+space">Eilenberg-Mac Lane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K({\mathbb{Z}},1)</annotation></semantics></math>. Explicitly, the first <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(S^1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/integer">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>. But the higher <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_n(S^1) \simeq *</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \gt 1</annotation></semantics></math> all vanish (and so is a <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy 1-type</a>). This can be deduced from the result that the loop space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Omega S^1</annotation></semantics></math> of the circle is the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">{\mathbb{Z}}</annotation></semantics></math> of integers and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> is path-connected. Proofs of this in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> are in the <a href="#references">references</a>.</p> <p>The result that <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>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^1\simeq K({\mathbb{Z}},1)</annotation></semantics></math> holds in a general <a class="existingWikiWord" href="/nlab/show/Grothendieck+%28%E2%88%9E%2C1%29-topos">Grothendieck (∞,1)-topos</a>. In fact, more generally, 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 pointed object of a <a class="existingWikiWord" href="/nlab/show/Grothendieck+%28%E2%88%9E%2C1%29-topos">Grothendieck (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">{\mathcal{H}}</annotation></semantics></math>, there is a natural equivalence between the <a class="existingWikiWord" href="/nlab/show/suspension+object">suspension object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma X</annotation></semantics></math> and the classifying space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>ℤ</mi><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> B{\mathbb{Z}}\wedge X</annotation></semantics></math>. In particular, when <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 specifically the 0-truncated two-point space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0</annotation></semantics></math>, we get <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>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^1\simeq K({\mathbb{Z}},1)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/product">product</a> of the circle with itself is the (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>)-<a class="existingWikiWord" href="/nlab/show/torus">torus</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T = S^1 \times S^1 \,. </annotation></semantics></math></div> <p>Generally, 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>-torus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">T^n</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><msup><mo stretchy="false">)</mo> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(S^1)^n</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+circle">unit circle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+action">circle action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/minimal+simplicial+circle">minimal simplicial circle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+type">circle type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudocircle">pseudocircle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conic+section">conic section</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/ellipse">ellipse</a>, <a class="existingWikiWord" href="/nlab/show/parabola">parabola</a>, <a class="existingWikiWord" href="/nlab/show/hyperbola">hyperbola</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circumference+of+a+circle">circumference of a circle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/area+of+a+circle">area of a circle</a></p> </li> </ul> <h2 id="references">References</h2> <p>A formalization of the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ʃ</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">&#643; S^1 \simeq \mathbf{B}\mathbb{Z}</annotation></semantics></math> of the circle as a <a class="existingWikiWord" href="/nlab/show/higher+inductive+type">higher inductive type</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, along with a proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>ʃ</mi><msup><mi>S</mi> <mn>1</mn></msup><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\Omega &#643;S^1\simeq {\mathbb{Z}}</annotation></semantics></math> (and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ʃ</mi><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\pi_1(&#643;S^1) \simeq \mathbb{Z}</annotation></semantics></math>):</p> <ul> <li id="LicataShulman13"> <p><a class="existingWikiWord" href="/nlab/show/Dan+Licata">Dan Licata</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Calculating the Fundamental Group of the Circle in Homotopy Type Theory</em>, LICS ‘13: Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science June 2013 (<a href="http://arxiv.org/abs/1301.3443">arXiv:1301.3443</a>, <a href="https://doi.org/10.1109/LICS.2013.28">doi:10.1109/LICS.2013.28</a>)</p> <blockquote> <p>blog announcement: <a href="https://homotopytypetheory.org/2011/04/29/a-formal-proof-that-pi1s1-is-z/">A formal proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\pi_1(S^1) = \mathbb{Z}</annotation></semantics></math></a></p> </blockquote> </li> <li id="BezemBuchholtzGraysonShulman19"> <p><a class="existingWikiWord" href="/nlab/show/Marc+Bezem">Marc Bezem</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Grayson">Daniel Grayson</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>Construction of the Circle in UniMath</em>, Journal of Pure and Applied Algebra Volume 225, Issue 10, October 2021, 106687 (<a href="https://arxiv.org/abs/1910.01856">arXiv:1910.01856</a>)</p> </li> </ul> <p>The proof is formalized therein using the <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a> <a class="existingWikiWord" href="/nlab/show/proof+assistant">proof assistant</a>. See also</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/HoTT+Book">HoTT Book</a>, section 8.1</p> </li> <li> <p>The HoTT Coq library: <a href="https://github.com/HoTT/Coq-HoTT/blob/V8.18/theories/Spaces/Circle.v">theories/Spaces/Circle.v</a>, <a href="https://github.com/HoTT/Coq-HoTT/blob/V8.18/theories/Homotopy/Pi1S1.v">theories/Homotopy/Pi1S1.v</a></p> </li> <li> <p>The HoTT Agda library: <a href="https://github.com/HoTT/HoTT-Agda/blob/2.0/homotopy/LoopSpaceCircle.agda">homotopy/LoopSpaceCircle.agda</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 31, 2025 at 08:52:28. See the <a href="/nlab/history/circle" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/circle" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3263/#Item_14">Discuss</a><span class="backintime"><a href="/nlab/revision/circle/26" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/circle" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/circle" accesskey="S" class="navlink" id="history" rel="nofollow">History (26 revisions)</a> <a href="/nlab/show/circle/cite" style="color: black">Cite</a> <a href="/nlab/print/circle" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/circle" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>