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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> <a class="existingWikiWord" href="/nlab/show/topology">topology</a> (<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>) see also <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a> Basic concepts <ul> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <a href="Top#UniversalConstructions">Universal constructions</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>, </li> <li> fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a> </li> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> </li> </ul> Examples <ul> <li> <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <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> <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a> </li> <li> <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>) </li> <li> <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a> </li> <li> <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> <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> <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a> </li> </ul> Basic statements <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a> </li> <li> <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> </li> <li> <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> </li> <li> <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> </li> <li> <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> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a> </li> <li> <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> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a> </li> </ul> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a> </li> </ul> Analysis Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> </li> </ul> </div></div> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> Introductions <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a> Differentials <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a> </li> </ul> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a> </li> <li> <a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> Models <ul> <li> <a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring) </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>) </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a> </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='#examples'>Examples</a></li> <li><a href='#entries_in_differential_topology'>Entries in differential topology</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> Differential topology is the subject devoted to the study of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebro</a>-<a class="existingWikiWord" href="/nlab/show/topology">topological</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy-theoretic</a> properties of <a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and related <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometric</a> <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> such as <a class="existingWikiWord" href="/nlab/show/stratifolds">stratifolds</a>, <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> and more generally <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>. A key part of differential topology is <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, where the <a class="existingWikiWord" href="/nlab/show/Pontryagin+theorem">Pontryagin-</a><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+theorem">Thom theorem</a> relates the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> to <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> (<a class="existingWikiWord" href="/nlab/show/submanifold">sub</a>-)<a class="existingWikiWord" href="/nlab/show/smooth+manifold">manifolds</a> (for instance <a class="existingWikiWord" href="/nlab/show/cohomotopy">cohomotopy</a> to <a class="existingWikiWord" href="/nlab/show/normal+framing">normally framed</a> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>). Differential topology is also concerned with the problem of finding out which topological (or PL) manifolds allow a <a class="existingWikiWord" href="/nlab/show/smooth+structure">differentiable structure</a> and the degree of nonuniqueness of that structure if they do (e.g. <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structures">exotic smooth structures</a>). It is also concerned with concrete constructions of <a class="existingWikiWord" href="/nlab/show/cohomology">(co)</a><a class="existingWikiWord" href="/nlab/show/homology">homology</a> classes (e.g. <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>es) for differentiable manifolds and of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential refinements of cohomology theories</a>. More recently, the <a class="existingWikiWord" href="/nlab/show/smooth+Oka+principle">smooth Oka principle</a> reveals a deep structure in differential topology which is visible in the full generality of <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> (<a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a>). <h2 id="examples">Examples</h2> Many considerations, and classification problems, depend crucially on <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, and the case of high-dimensional manifolds (the notion of ‘high’ depends on the problem) is often very different from the situation in each of the low dimensions; thus there are specialists’ subjects like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-(dimensional) topology and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math>-topology. There are restrictions on an underlying topology which is allowed for some sorts of additional structures on a differentiable manifold. For example, only some even-dimensional differentiable manifolds allow for <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic structure</a> and only some odd-dimensional one allow for a <a class="existingWikiWord" href="/nlab/show/contact+manifold">contact structure</a>; in these cases moreover special constructions of topological invariants like <a class="existingWikiWord" href="/nlab/show/Floer+homology">Floer homology</a> and <a class="existingWikiWord" href="/nlab/show/symplectic+field+theory">symplectic field theory</a> exist. This yields the relatively young subjects of symplectic and contact topologies, with the first significant results coming from Gromov. Any (Hausdorff paracompact finite-dimensional) differentiable manifold allows for <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">riemannian structure</a> however; therefore there is no special subject of ‘riemannian topology’. <h2 id="entries_in_differential_topology">Entries in differential topology</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hopf+degree+theorem">Hopf degree theorem</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+Hopf+degree+theorem">equivariant Hopf degree theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9%E2%80%93Hopf+theorem">Poincaré–Hopf theorem</a> </li> <li> <a href="cohomotopy#RelationToCobordismGroup">Pontrjagin-Thom theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Sard%27s+theorem">Sard's theorem</a>, <a class="existingWikiWord" href="/nlab/show/transversality">transversality</a>, <a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeb+sphere+theorem">Reeb sphere theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> … </li> </ul> <h2 id="related_entries">Related entries</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/low+dimensional+topology">low dimensional topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+topology">synthetic differential topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/equivariant+differential+topology">equivariant differential topology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/surgery">surgery</a> </li> </ul> <h2 id="references">References</h2> Though some of the basic results, methods and conjectures of differential topology go back to <a class="existingWikiWord" href="/nlab/show/Henri+Poincar%C3%A9">Poincaré</a>, <a class="existingWikiWord" href="/nlab/show/Hassler+Whitney">Whitney</a>, <a class="existingWikiWord" href="/nlab/show/Anthony+P.+Morse">Morse</a> and <a class="existingWikiWord" href="/nlab/show/Lev+Pontrjagin">Pontrjagin</a>, it became an independent field only in the late 1950s and early 1960s with the seminal works of <a class="existingWikiWord" href="/nlab/show/Stephen+Smale">Smale</a>, <a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Thom">Thom</a>, <a class="existingWikiWord" href="/nlab/show/John+Milnor">Milnor</a> and <a class="existingWikiWord" href="/nlab/show/Morris+Hirsch">Hirsch</a>. Soon after the initial effort on foundations, mainly in the American school, a strong activity started in Soviet Union (<a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <a class="existingWikiWord" href="/nlab/show/Alexandr+S.+Mishchenko">A. S. Mishchenko</a>, <a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">S. Novikov</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Abramovich+Rokhlin">V. A. Rokhlin</a>, <a class="existingWikiWord" href="/nlab/show/Mikhail+Gromov">M. Gromov</a>,…). Introductions and monographs: <ul> <li id="Milnor64"> <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>: Differential topology, chapter 6 in T. L. Saaty (ed.), Lectures On Modern Mathematic II (1964) [<a class="existingWikiWord" href="/nlab/files/Milnor_DifferentialTopology.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, Lectures on the h-cobordism theorem (1965) [<a href="https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/James+R.+Munkres">James R. Munkres</a>: Elementary Differential Topology, Annals of Mathematics Studies 54, Princeton University Press (1966) [<a href="http://dx.doi.org/10.1515/9781400882656">doi:10.1515/9781400882656</a>] </li> <li> Andrew H. Wallace: Differential topology: first steps, Benjamin (1968) [<a href="https://archive.org/details/differentialtopo0000wall/page/n5/mode/2up">ark:/13960/t5s830222</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, Alan Pollack: Differential topology, Prentice-Hall (1974) [<a href="https://doi.org/10.1090/chel/370">doi:10.1090/chel/370</a>, <a href="https://math.ucr.edu/~res/math260s10/old/difftopGP.pdf">pdf</a>, <a href="https://www.cimat.mx/~gil/docencia/2020/topologia_diferencial/%5BGuillemin,Pollack%5DDifferential_Topology(1974).pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Morris+Hirsch">Morris Hirsch</a>, Differential topology, Springer Graduate Texts in Mathematics 33 (1976) [<a href="https://link.springer.com/book/10.1007/978-1-4684-9449-5">doi:10.1007/978-1-4684-9449-5</a>, <a href="http://books.google.com/books/about/?id=iSvnvOodWl8C">gBooks</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+J%C3%A4nich">Klaus Jänich</a>, Introduction to differentiable topology (1982) [<a href="https://www.cambridge.org/ae/universitypress/subjects/mathematics/differential-and-integral-equations-dynamical-systems-and-co/introduction-differential-topology?format=PB&isbn=9780521284707">ISBN:9780521284707</a>] <blockquote> (translated from the German 1973 edition) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, <a class="existingWikiWord" href="/nlab/show/Differential+Forms+in+Algebraic+Topology">Differential Forms in Algebraic Topology</a>, Graduate Texts in Math. 82, Springer (1982) [<a href="https://link.springer.com/book/10.1007/978-1-4757-3951-0">doi:10.1007/978-1-4757-3951-0</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>: Topology from the differential viewpoint, Princeton University Press (1997) [<a href="https://press.princeton.edu/books/paperback/9780691048338/topology-from-the-differentiable-viewpoint">ISBN:9780691048338</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/milnortop.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Mladen+Bestvina">Mladen Bestvina</a> (notes by <a class="existingWikiWord" href="/nlab/show/Adam+Keenan">Adam Keenan</a>): Differentiable Topology and Geometry (2002) [<a href="http://www.math.utah.edu/~keenan/manifoldsnotes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BestvinaKeenanDifferentialTopology.pdf" title="pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/C.+T.+C.+Wall">C. T. C. Wall</a>: Differential topology, Cambridge Studies in Advanced Mathematics 154, Cambridge University Press (2016) [<a href="https://doi.org/10.1017/CBO9781316597835">doi:10.1017/CBO9781316597835</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/surgery/wall.pdf">pdf</a>] </li> <li> <a class="existingWikiWord" href="/nlab/show/Joel+W.+Robbin">Joel W. Robbin</a>, <a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar Salamon</a>, Introduction to differential topology, webdraft (2018) 294 pp [<a href="https://people.math.ethz.ch/~salamon/PREPRINTS/difftop.pdf">pdf</a>] </li> <li id="Benedetti19"> <a class="existingWikiWord" href="/nlab/show/Riccardo+Benedetti">Riccardo Benedetti</a>, Lectures on Differential Topology, Graduate Studies in Mathematics 218, AMS 2021 (<a href="https://arxiv.org/abs/1907.10297">arXiv:1907.10297</a>, <a href="https://bookstore.ams.org/cdn-1631100821985/gsm-218/">ISBN: 978-1-4704-6674-9</a>) </li> </ul> Survey with connections to <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (<a href="https://link.springer.com/book/10.1007/978-3-662-10579-5">doi:10.1007/978-3-662-10579-5</a>, <a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf">pdf</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Jean+Dieudonn%C3%A9">Jean Dieudonné</a>, A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics (2009) [<a href="https://www.springer.com/de/book/9780817649067">ISBN:978-0-8176-4907-4</a>] </li> </ul> See also: <ul> <li>Wikipedia: <a href="https://en.wikipedia.org/wiki/Differential_topology">Differential topology</a></li> </ul> Generalization to <a class="existingWikiWord" href="/nlab/show/equivariant+differential+topology">equivariant differential topology</a>: <ul> <li id="Wasserman69"><a class="existingWikiWord" href="/nlab/show/Arthur+Wasserman">Arthur Wasserman</a>, Equivariant differential topology, Topology 8 (1969) 127-150 [<a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf">pdf</a>]</li> </ul> </body></html> </div> <div class="revisedby"> Last revised on December 10, 2024 at 11:05:53. See the <a href="/nlab/history/differential+topology" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/differential+topology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3272/#Item_10">Discuss</a><a href="/nlab/revision/differential+topology/25" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/diff/differential+topology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/differential+topology" accesskey="S" class="navlink" id="history" rel="nofollow">History (25 revisions)</a> <a href="/nlab/show/differential+topology/cite" style="color: black">Cite</a> <a href="/nlab/print/differential+topology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/differential+topology" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>