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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="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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#existence_and_examples'>Existence and Examples</a></li> <ul> <li><a href='#no_exotic_smooth_structure_in_dimensions_'>No exotic smooth structure in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\leq 3</annotation></semantics></math></a></li> <li><a href='#no_exotic_euclidean_space_away_from_dimension_4'>No exotic Euclidean space away from dimension 4</a></li> <li><a href='#ExoticEuclideal4Space'>Exotic Euclidean 4-space</a></li> <li><a href='#exotic_'>Exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \times S^2</annotation></semantics></math></a></li> <li><a href='#Exotic4Spheres'>Exotic 4-spheres?</a></li> <li><a href='#Exotic7Sphere'>Exotic 7-sphere</a></li> <li><a href='#Exotic8Sphere'>Exotic 8-sphere</a></li> <li><a href='#ExoticNSpheresForNGreaterThanFour'>Exotic <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>-spheres for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n \geq 5</annotation></semantics></math></a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_the_mathematical_theory'>For the mathematical theory</a></li> <li><a href='#ReferencesForApplicationsToPhysics'>For applications to physics</a></li> <ul> <li><a href='#general_relativity'>General relativity</a></li> <li><a href='#generation_of_source_terms_fields'>Generation of source terms (fields)</a></li> <li><a href='#quantum_field_theory'>Quantum (field) theory</a></li> <li><a href='#ReferencesInStringTheory'>String theory</a></li> <li><a href='#cosmology'>Cosmology</a></li> <li><a href='#quantum_gravity'>Quantum gravity</a></li> </ul> </ul> </ul> </div> <h2 id="idea">Idea</h2> An exotic smooth structure is, roughly speaking, a <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> on a <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which makes the resulting <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> be <a class="existingWikiWord" href="/nlab/show/diffeomorphism">non-diffeomorphic</a> to the smooth manifold given by some evident ‘standard’ smooth structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Mostly the term is used for smooth structures on <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and on the <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. The standard smooth structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is exhibited by the identity <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>, and the standard smooth structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> is that given by the atlas of the two <a class="existingWikiWord" href="/nlab/show/hemispheres">hemispheres</a> as given by <a class="existingWikiWord" href="/nlab/show/stereographic+projection">stereographic projection</a>. For special values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> there may exist smooth structure not equivalent to these. They are the exotic smooth structures. A classification of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a>, <a class="existingWikiWord" href="/nlab/show/piecewise-linear+manifold">PL</a> and <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological</a> structures on manifolds in dimension 5 and higher, in terms of various groups from <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> (many not known) was established by <a href="#KirbSieb">Kirby and Siebenmann (1977)</a> using <a class="existingWikiWord" href="/nlab/show/obstruction+theory">obstruction theory</a>. <h2 id="existence_and_examples">Existence and Examples</h2> <h3 id="no_exotic_smooth_structure_in_dimensions_">No exotic smooth structure in dimensions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">\leq 3</annotation></semantics></math></h3> <a href="#Rad">Rado (1925)</a> proved that in dimension 2 there are no exotic differentiable structures (or the uniqueness of the standard structure). The classification of 1-dimensional manifolds and the uniqueness of the smooth structure can be found in the Appendix of <a href="#Milnor1965b">Milnor (1965b)</a>. <a href="#Moise">Moise (1952)</a> proved that in dimension 3 there are no exotic differentiable structures, or to put in another way, 3-dimensional differentiable manifolds which are <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a>. In this way the 3-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math> inherits a unique differentiable structure, no matter which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> it is considered to be embedded in. <h3 id="no_exotic_euclidean_space_away_from_dimension_4">No exotic Euclidean space away from dimension 4</h3> There exists a unique smooth structure on the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≠</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">n\neq 4</annotation></semantics></math> (<a href="#Stallings62">Stallings 1962</a>). <h3 id="ExoticEuclideal4Space">Exotic Euclidean 4-space</h3> There exists uncountably many exotic smooth structures on the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> of dimension 4 (<a href="#Gompf">Gompf 1985</a>, <a href="#FreedTay">Freedman/Taylor 1986</a>, <a href="#Taubes">Taubes 1987</a>). See also at <a class="existingWikiWord" href="/nlab/show/exotic+R%5E4">exotic R^4</a>. There is a unique maximal exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> into which all other ‘versions’ of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> smoothly embed as open subsets (Freedman/Taylor 1986, <a href="#DeMFreed">DeMichelis/Freedman 1992</a>). There are two classes of exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>‘s: large and small. A large exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> cannot be embedded in the 4-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math> (Gompf 1985, Taubes 1987) whereas a small exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> admits such an embedding (DeMichelis/Freedman 1992): <ul> <li> A large exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> is constructed by using the failure to smoothly split a smooth 4-manifold (the <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a> for instance) as a <a class="existingWikiWord" href="/nlab/show/connected+sum">connected sum</a> of some factors (where a topological splitting exits). </li> <li> The small exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> (or ribbon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>) is constructed by using the failure of the smooth <a class="existingWikiWord" href="/nlab/show/h-cobordism+theorem">h-cobordism theorem</a> in dimension 4 (<a href="#Donaldson1">Donaldson 1987</a>, <a href="#Donaldson2">1990</a>). <a href="#BizGompf">Bizaca and Gompf (1996)</a> are able to present an infinite handle body of a small exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> which serve as a coordinate representation. </li> </ul> <h3 id="exotic_">Exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \times S^2</annotation></semantics></math></h3> There exists an infinite family of mutually non-diffeomorphic irreducible smooth structures on the topological 4-manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \times S^2</annotation></semantics></math> (<a href="#AkhmedovPark10">Akhmedov-Park 10</a>). <h3 id="Exotic4Spheres">Exotic 4-spheres?</h3> It is open whether the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> admits an exotic smooth structure. See (<a href="#FreedmanGompfMorrisonWalker09">Freedman-Gompf-Morrison-Walker 09</a> for review). <h3 id="Exotic7Sphere">Exotic 7-sphere</h3> <a href="#Milnor1956">Milnor (1956)</a> gave the first examples of exotic smooth structures on the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>, finding at least seven. The <a class="existingWikiWord" href="/nlab/show/exotic+7-spheres">exotic 7-spheres</a> constructed in <a href="#Milnor1956">Milnor 1956</a> are all examples of <a class="existingWikiWord" href="/nlab/show/fibre+bundles">fibre bundles</a> over the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">S^4</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/fibre">fibre</a> the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math>, with <a class="existingWikiWord" href="/nlab/show/structure+group">structure group</a> the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a> (see also at <a class="existingWikiWord" href="/nlab/show/8-manifold">8-manifold</a> the section <a href="8-manifold#ExoticBoundary7Spheres">With exotic boundary 7-spheres</a>): By the classification of bundles on spheres via the <a class="existingWikiWord" href="/nlab/show/clutching+construction">clutching construction</a>, these correspond to <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup><mo>→</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^3 \to SO(4)</annotation></semantics></math>, i.e. elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_3(SO(4))</annotation></semantics></math>. From the table at <a href="orthogonal+group#HomotopyGroups">orthogonal group – Homotopy groups</a>, this latter group is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}\oplus\mathbb{Z}</annotation></semantics></math>. Thus any such bundle can be described, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, by a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,m)</annotation></semantics></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mi>m</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+m=1</annotation></semantics></math>, then one can show there is a <a class="existingWikiWord" href="/nlab/show/Morse+function">Morse function</a> with exactly two <a class="existingWikiWord" href="/nlab/show/critical+points">critical points</a> on the total space of the bundle, and hence this 7-manifold is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to a sphere. The <a class="existingWikiWord" href="/nlab/show/fractional+first+Pontryagin+class">fractional first Pontryagin class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><mn>2</mn></mfrac><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\frac{p_1}{2} \in H^4(S^4) \simeq \mathbb{Z}</annotation></semantics></math> of the bundle is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">n-m</annotation></semantics></math>. Milnor constructs, using <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> theory and <a class="existingWikiWord" href="/nlab/show/Hirzebruch%27s+signature+theorem">Hirzebruch's signature theorem</a> for 8-manifolds, a mod-7 diffeomorphism invariant of the manifold, so that it is standard 7-sphere precisely when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mfrac><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><mn>2</mn></mfrac> <mn>2</mn></msup><mo>−</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mi>mod</mi><mn>7</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{p_1}{2}^2 -1 = 0 (mod 7)</annotation></semantics></math>. By using the <a class="existingWikiWord" href="/nlab/show/connected+sum">connected sum</a> operation, the set of smooth, non-diffeomorphic structures on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere has the structure of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>. For the <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a>, it is the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">/</mo><mn>28</mn></mrow><annotation encoding="application/x-tex">Z/{28}</annotation></semantics></math> and Brieskorn (1966) found the generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><mrow><mi>Σ</mi><mo>#</mo><mi>⋯</mi><mo>#</mo><mi>Σ</mi></mrow><mo>⏟</mo></munder> <mn>28</mn></msub></mrow><annotation encoding="application/x-tex">\underbrace{\Sigma\#\cdots\#\Sigma}_28</annotation></semantics></math> is the standard sphere. For review see for instance (<a href="#Kreck10">Kreck 10, chapter 19</a>, <a href="#McEnroe15">McEnroe 15</a>). For more see at <a class="existingWikiWord" href="/nlab/show/exotic+7-sphere">exotic 7-sphere</a>. From the point of view of <a class="existingWikiWord" href="/nlab/show/M-theory+on+8-manifolds">M-theory on 8-manifolds</a>, these <a class="existingWikiWord" href="/nlab/show/8-manifolds">8-manifolds</a> <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> with (exotic) <a class="existingWikiWord" href="/nlab/show/7-sphere">7-sphere</a> <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> correspond to <a class="existingWikiWord" href="/nlab/show/near+horizon+limits">near horizon limits</a> of <a class="existingWikiWord" href="/nlab/show/black+brane">black</a> <a class="existingWikiWord" href="/nlab/show/M2+brane">M2 brane</a> spacetimes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2,1} \times X</annotation></semantics></math>, where the <a class="existingWikiWord" href="/nlab/show/M2-branes">M2-branes</a> themselves would be sitting at the center of the <a class="existingWikiWord" href="/nlab/show/7-spheres">7-spheres</a> (if that were included in the spacetime, see also <a class="existingWikiWord" href="/nlab/show/Dirac+charge+quantization">Dirac charge quantization</a>). (<a href="M-theory+on+8-manifolds#MorrisonPlesser99">Morrison-Plesser 99, section 3.2</a>) <h3 id="Exotic8Sphere">Exotic 8-sphere</h3> The abelian group of non-diffeomorphic structures with connected sum on the <a class="existingWikiWord" href="/nlab/show/8-sphere">8-sphere</a> is the cyclic group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">Z/2</annotation></semantics></math>. The unique exotic 8-sphere corresponds to the nontrivial element of the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of the <a class="existingWikiWord" href="/nlab/show/J-homomorphism">J-homomorphism</a> and is the first instance of an exotic sphere that does not bound a <a class="existingWikiWord" href="/nlab/show/parallelizable+manifold">parallelizable manifold</a> (<a href="#Amabel17">Amabel 17, Sec. 11</a>). It admits a metric of positive <a class="existingWikiWord" href="/nlab/show/Ricci+curvature">Ricci curvature</a>. <h3 id="ExoticNSpheresForNGreaterThanFour">Exotic <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>-spheres for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n \geq 5</annotation></semantics></math></h3> Via the celebrated <a class="existingWikiWord" href="/nlab/show/h+cobordism+theorem">h cobordism theorem</a> of Smale (<a href="#Smale">Smale 1962</a>, <a href="#Milnor1965a">Milnor 1965</a>) one gets a relation between the number of smooth structures on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n \geq 5</annotation></semantics></math>) and the number of <a class="existingWikiWord" href="/nlab/show/isotopy">isotopy</a> classes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0 (Diff(S^{n-1}))</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/equator">equator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math>. Then <a href="#KervMil">Kervaire and Milnor (1963)</a> proved that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n \geq 5</annotation></semantics></math> there are only finitely many exotic smooth structures on the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math> (possibly none). By using the <a class="existingWikiWord" href="/nlab/show/connected+sum">connected sum</a> operation, the set of smooth, non-diffeomorphic structures on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-sphere has the structure of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>. The only odd-dimensional spheres with no exotic smooth structure are the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/3-sphere">3-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^3</annotation></semantics></math>, as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">S^5</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>61</mn></msup></mrow><annotation encoding="application/x-tex">S^{61}</annotation></semantics></math> (<a href="#WangXu16">Wang-Xu 16, corollary 1.13</a>) In the range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>5</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>61</mn></mrow><annotation encoding="application/x-tex">5 \leq n \leq 61</annotation></semantics></math> the only <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>-spheres with no exotic smooth structures are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">S^5</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">S^6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>12</mn></msup></mrow><annotation encoding="application/x-tex">S^{12}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>56</mn></msup></mrow><annotation encoding="application/x-tex">S^{56}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>61</mn></msup></mrow><annotation encoding="application/x-tex">S^{61}</annotation></semantics></math> (<a href="#WangXu16">Wang-Xu 16, corollary 1.15</a>). It is conjectured that this exhausts in fact all examples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-spheres without exotic smooth structure for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">n \geq 5</annotation></semantics></math> (<a href="#WangXu16">Wang-Xu 16, conjecture 1.17</a>). <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></li> </ul> <h2 id="references">References</h2> See also <ul> <li>Wikipedia, <a href="http://en.wikipedia.org/wiki/Exotic_sphere">Exotic sphere</a></li> </ul> <h3 id="for_the_mathematical_theory">For the mathematical theory</h3> The first construction of exotic smooth structures was on the 7-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> in <ul> <li id="Milnor1956"><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (2): 399–405 (1956) (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/exotic.pdf">pdf</a>, <a href="https://doi.org/10.1142/9789812836878_0001">doi:10.1142/9789812836878_0001</a>)</li> </ul> (…) <ul> <li id="Smale"> <a class="existingWikiWord" href="/nlab/show/Stephen+Smale">Stephen Smale</a>, On the structure of manifolds, Amer. J. of Math. 84 : 387-399 (1962) </li> <li id="Milnor1965a"> <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a> (1965), Lectures on the h-cobordism theorem (Princeton Univ. Press, Princeton) </li> <li id="KervMil"> <a class="existingWikiWord" href="/nlab/show/Michel+Kervaire">Michel Kervaire</a>, ; <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, (1963) “Groups of homotopy spheres: I”, Ann. Math. 77, pp. 504 - 537. </li> <li id="KirbSieb"> Kirby, R.; Siebenmann, L. (1977) Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Studies (Princeton University Press, Princeton). </li> <li id="Stallings62"> John R. Stallings, The piecewise-linear structure of Euclidean space, Proceedings of the Cambridge Philosophical Society 58: 481–488 (1962) (<a href="http://www.maths.ed.ac.uk/~aar/papers/stallings2.pdf">pdf</a>) </li> <li id="FreedTay"> Freedman, Michael H.; Taylor, Laurence (1986) “A universal smoothing of four-space”, J. Diff. Geom. 24, pp. 69-78 </li> <li id="DeMFreed"> De Michelis, Stefano; Freedman, Michael H. (1992) “Uncountably many exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>‘s in standard 4-space”, J. Diff. Geom. 35, pp. 219-254. </li> <li id="Donaldson1"> <a class="existingWikiWord" href="/nlab/show/Simon+Donaldson">Simon Donaldson</a> (1987) “Irrationality and the h-cobordism conjecture”, J. Diff. Geom. 26, pp. 141-168. </li> <li id="Donaldson2"> <a class="existingWikiWord" href="/nlab/show/Simon+Donaldson">Simon Donaldson</a>, (1990) “Polynomial invariants for smooth four manifolds”, Topology 29, pp. 257-315. </li> <li id="Gompf"> Gompf, Robert (1985) “An infinite set of exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>‘s”, J. Diff. Geom. 21, pp. 283-300. </li> <li id="Taubes"> Taubes, Clifford H. (1987) “Gauge theory on asymptotically periodic 4-manifolds”, J. Diff. Geom. 25, pp. 363-430 </li> <li id="BizGompf"> Bizaca, Z.; Gompf, Robert (1996) “Elliptic surfaces and some simple exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>‘s”, J. Diff. Geom. 43, pp. 458-504. </li> <li id="Rad"> Rado, T. (1925) “Über den Begriff der Riemannschen Fläche” , Acta Litt. Scient. Univ. Szegd 2, pp. 101-121 </li> <li id="Milnor1965b"> <a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, (1965b) Topology from the Differentiable Viewpoint (University Press of Virginia) </li> <li id="WangXu16"> Guozhen Wang, Zhouli Xu, The triviality of the 61-stem in the stable homotopy groups of spheres (<a href="https://arxiv.org
/abs/1601.02184">arXiv:1601.02184</a>) </li> <li> Llohann D. Sperança, Pulling back the Gromoll-Meyer construction and models of exotic spheres, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 (<a href="https://arxiv.org/abs/1010.6039">arXiv:1010.6039</a>) </li> <li> Llohann D. Sperança, Explicit Constructions over the Exotic 8-sphere (<a href="https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SperancaExoticSpheres.pdf" title="pdf">pdf</a>) </li> <li id="Amabel17"> , Exotic spheres (<a href="https://math.mit.edu/~araminta/Exotic.pdf">pdf</a>) </li> <li id="DRL10"> C. Duran, A. Rigas, Llohann D. Sperança, Bootstrapping Ad-equivariant maps, diffeomorphisms and involutions, Matematica Contemporanea, 35:27–39, 2010 (<a href="http://www.ime.unicamp.br/~rigas/bootstrapping">pdf</a>) </li> </ul> On the open issue of exotic <a class="existingWikiWord" href="/nlab/show/4-spheres">4-spheres</a>: <ul> <li id="FreedmanGompfMorrisonWalker09"><a class="existingWikiWord" href="/nlab/show/Michael+Freedman">Michael Freedman</a>, <a class="existingWikiWord" href="/nlab/show/Robert+Gompf">Robert Gompf</a>, <a class="existingWikiWord" href="/nlab/show/Scott+Morrison">Scott Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Kevin+Walker">Kevin Walker</a>, Man and machine thinking about the smooth 4-dimensional Poincaré conjecture, Quantum Topology, Volume 1, Issue 2 (2010), pp. 171-208 (<a href="http://arxiv.org/abs/0906.5177">arXiv:0906.5177</a>)</li> </ul> On exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \times S^2</annotation></semantics></math> <ul> <li id="AkhmedovPark10">Anar Akhmedov, B. Doug Park, Exotic smooth structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \times S^2</annotation></semantics></math> (<a href="https://arxiv.org/abs/1005.3346">arXiv:1005.3346</a>)</li> </ul> Review: <ul> <li id="Kreck10"> <a class="existingWikiWord" href="/nlab/show/Matthias+Kreck">Matthias Kreck</a>, chapter 19 “Exotic 7-spheres” of Differential Algebraic Topology – From Stratifolds to Exotic Spheres, AMS 2010 </li> <li> <a class="existingWikiWord" href="/nlab/show/Rustam+Sadykov">Rustam Sadykov</a>, Sections 7,8 of: Elements of Surgery Theory, 2013 (<a href="https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf">pdf</a>) </li> <li id="McEnroe15"> Rachel McEnroe, Milnor’ construction of exotic 7-spheres, 2015 (<a href="http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf">pdf</a>) </li> </ul> See also <ul> <li>Wikipedia (<a href="http://en.wikipedia.org/wiki/Exotic_sphere#References">spheres</a>, <a href="http://en.wikipedia.org/wiki/Exotic_R4#References"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">R^4</annotation></semantics></math></a>)</li> </ul> <h3 id="ReferencesForApplicationsToPhysics">For applications to physics</h3> The relevance of exotic smooth structure to <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is tantalizing but remains by and large unclear. Some of the following references probably ought to be handled with care. <h4 id="general_relativity">General relativity</h4> The argument that <a class="existingWikiWord" href="/nlab/show/exotic+spheres">exotic spheres</a> are to be regarded as <a class="existingWikiWord" href="/nlab/show/gravitational+instantons">gravitational instantons</a>: <ul> <li id="Witten85"> <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, p. 12 of: Global gravitational anomalies, Comm. Math. Phys. Volume 100, Number 2 (1985), 197–229. (<a href="http://projecteuclid.org/euclid.cmp/1103943444">EUCLID</a>) </li> <li> Randy A. Baadhio, On the global gravitational instanton and soliton that are homotopy spheres, Journal of Mathematical Physics 32, 2869 (1991) (<a href="https://doi.org/10.1063/1.529078">doi:10.1063/1.529078</a>) </li> </ul> Further discussion of exotic <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>-manifolds from the <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> point of view is in <ul> <li> <a class="existingWikiWord" href="/nlab/show/Carl+Brans">Carl Brans</a>, Duane Randall, Exotic differentiable structures and general relativity Gen. Rel. Grav., 25 (1993) 205–220 </li> <li> <a class="existingWikiWord" href="/nlab/show/Carl+Brans">Carl Brans</a>, Exotic smoothness and physics J. Math. Phys. 35, (1994), 5494–5506. </li> </ul> The following paper contained a first proof to localize exotic smoothness in an exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>: <ul> <li><a class="existingWikiWord" href="/nlab/show/Carl+Brans">Carl Brans</a>, Localized exotic smoothness Class. Quant. Grav., 11, (1994), 1785–1792.</li> </ul> A more philosophical discussion can be found in: <ul> <li><a class="existingWikiWord" href="/nlab/show/Carl+Brans">Carl Brans</a>, Absolute spacetime: the twentieth century ether Gen. Rel. Grav. 31, (1999), 597–609</li> </ul> <h4 id="generation_of_source_terms_fields">Generation of source terms (fields)</h4> Brans conjectured in the papers above, that exotic smoothness should be a source of an additional <a class="existingWikiWord" href="/nlab/show/gravity">gravitational field</a> (Brans conjecture). This conjecture was confirmed for compact <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>-manifolds (using implicitly a mapping of basic classes): Using the invariant of L. Taylor <a href="http://de.arxiv.org/abs/math/9807143">arXiv</a>, Sladkowski confirmed the conjecture for the exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> in: <ul> <li>Jan Sładkowski Gravity on exotic R4 with few symmetries Int.J. Mod. Phys. D, 10, (2001) 311–313</li> </ul> <h4 id="quantum_field_theory">Quantum (field) theory</h4> The first real connection between exotic smoothness and <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is Witten’s TQFT: <ul> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Topological quantum field theory Comm. Math. Phys., 117, (1988), 353–386.</li> </ul> and the whole work of Seiberg and Witten leading to the celebrated invariants. The relation to particle physics by using the algebra of smooth functions can be found in <ul> <li> Jan Sładkowski, Exotic smoothness, noncommutative geometry and particle physics Int. J. Theor. Phys., 35, (1996), 2075–2083 </li> <li> Jan Sładkowski, Exotic smoothness and particle physics Acta Phys. Polon., B 27, (1996), 1649–1652 </li> <li> Jan Sładkowski, Exotic smoothness, fundamental interactions and noncommutative geometry <a href="http://arxiv.org/abs/hep-th/9610093">arXiv</a> </li> </ul> The relation between TQFT and differential-topological invariants of smooth manifolds was clarified in: <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hendryk+Pfeiffer">Hendryk Pfeiffer</a>: Quantum general relativity and the classification of smooth manifolds <a href="http://arxiv.org/abs/gr-qc/0404088">arXiv</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hendryk+Pfeiffer">Hendryk Pfeiffer</a>: Diffeomorphisms from finite triangulations and absence of ‘local’ degrees of freedom Phys.Lett. B, 591, (2004), 197-201 </li> </ul> On the impact of exotic smooth structures on the spectrum of <a class="existingWikiWord" href="/nlab/show/Dirac+operators">Dirac operators</a>: <ul> <li>Ulrich Chiapi-Ngamako, M. B. Paranjape: Physics on manifolds with exotic differential structures [<a href="https://arxiv.org/abs/2501.13328">arXiv:2501.13328</a>]</li> </ul> <h4 id="ReferencesInStringTheory">String theory</h4> An argument for interpreting exotic smooth spheres as <a class="existingWikiWord" href="/nlab/show/gravitational+instantons">gravitational instantons</a> and to cancel the gravitational <a class="existingWikiWord" href="/nlab/show/anomalies">anomalies</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is in (<a href="#Witten85">Witten 85</a>). The influence of exotic smoothness for <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein models</a> was discussed here: <ul> <li><a class="existingWikiWord" href="/nlab/show/Matthias+Kreck">Matthias Kreck</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a>, A diffeomorphism classification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>7</mn></mrow><annotation encoding="application/x-tex">7</annotation></semantics></math>-dimensional homogeneous Einstein manifolds with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>𝔰𝔲</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>𝔲</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{su}(3) \times \mathfrak{su}(2) \times \mathfrak{u}(1)</annotation></semantics></math>-symmetry Ann. Math. 127, (1988), 373–388.</li> </ul> A discussion of topological effects (also of string theory) in relation to exotic smoothness is in <ul> <li>Ryan Rohm, Topological Defects and Differential Structures Annals Of Physics, 189, (1989), 223–239.</li> </ul> <h4 id="cosmology">Cosmology</h4> An overview can be also found in <ul> <li>Jan Sładkowski, Exotic smoothness and astrophysics Act. Phys. Polon. B, 40, (2009), 3157–3163</li> </ul> <h4 id="quantum_gravity">Quantum gravity</h4> A first calculation of the state sum in <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a> by inclusion of exotic smoothness <ul> <li>Kristin Schleich, Donald Witt, Exotic Spaces in Quantum Gravity I: Euclidean Quantum Gravity in Seven Dimensions Class.Quant.Grav., 16, (1999), 2447–2469</li> </ul> A semi-classical approach to the functional integral is discussed here: <ul> <li>Christofer Duston, Exotic smoothness in 4 dimensions and semiclassical Euclidean quantum gravity <a href="http://arxiv.org/abs/0911.4068">arxiv</a></li> </ul> The inclusion of singularities for asymptotically flat spacetimes is discussed here (with an example of a singularity coming from exotic smoothness): <ul> <li>Kristin Schleich, Donald Witt, Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes, <a href="http://arxiv.org/abs/1006.2890">arxiv</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on January 24, 2025 at 07:02:07. 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