Poincaré duality

The Poincaré duality , named after Henri Poincaré , is a fundamental connection in algebraic topology between the homology and the cohomology of orientable manifolds .

statement

Let an n - dimensional closed orientable manifold and a natural number be, then the k th singular cohomology group is isomorphic to the ( n  -  k ) th singular homology group . The isomorphism is realized by the cap product with the fundamental class . ${\displaystyle M}$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle H^{k}(M)}$ ${\displaystyle H_{n-k}(M)}$ ${\displaystyle \left[M\right]\in H_{n}(M)}$

This applies in particular to the Betti numbers . ${\displaystyle b_{k}=b_{n-k}}$

history

The identity was first asserted by Poincaré in 1893. In 1895 he gave a proof in Analysis Situs , initially defining Betti numbers using chains of submanifolds (instead of using chains of simplices as in his later work ) and using intersection numbers of submanifolds for the proof. In the Addenda to Analysis Situs he defined homology as the simplicial homology of triangulated manifolds (but did not discuss their independence from triangulation) and then gave the proof of the duality theorem on dual triangulations that is customary today. ${\displaystyle b_{k}=b_{n-k}}$

Smooth manifolds

If the manifold is also smooth , then there is also the De-Rham cohomology in addition to the singular cohomology . According to de Rham's theorem , the corresponding singular cohomology and de Rham cohomology groups are isomorphic. The space of the k differential forms is denoted by. The Hodge star operator ${\displaystyle {\mathcal {A}}^{k}(M)}$

${\displaystyle {\begin{aligned}\star \colon {\mathcal {A}}^{k}(M)\to {\mathcal {A}}^{n-k}(M)\end{aligned}}}$

induces for each an isomorphism between the De Rham cohomology groups. The following diagram commutes : ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle H_{\operatorname {dR} }^{k}(M)\to H_{\operatorname {dR} }^{n-k}(M)}$

${\displaystyle {\begin{array}{rcl}{\mathcal {A}}^{k}(M)&\longrightarrow &H_{dR}^{k}(M)\\\star {\big \downarrow }\cong &&\cong {\big \downarrow }\scriptstyle \operatorname {Poinc.} \\{\mathcal {A}}^{n-k}(M)&\longrightarrow &H_{dR}^{n-k}(M)\end{array}}}$

literature

• Herbert Seifert , William Threlfall : Textbook of Topology , Teubner 1934. Scan of the English translation (PDF; 7.4 MB)
• Schubert, Horst: Topology. An introduction. Mathematical guidelines BG Teubner Verlagsgesellschaft, Stuttgart 1964
• Munkres, James R .: Elements of algebraic topology. Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

Web links

• Igusa: Poincaré Duality (PDF; 234 kB)
• Hatcher: The Duality Theorem (PDF; 140 kB)
• Ranicki: The Poincaré Duality Theorem and its converse (PDF; 276 kB)

Individual evidence

1. ^ Edwin H. Spanier: Algebraic Topology. 1. corrected Springer edition, reprint. Springer, Berlin et al. 1995, ISBN 3-540-90646-0 , 296-297.
2. ^ Henri Poincaré, Analysis Situs , Journal de l'École Polytechnique ser 2, 1 (1895) pages 1–123.
3. ^ Klaus Jänich : Vector analysis . Springer, Berlin March 2005, ISBN 3-540-23741-0 , pp. 129-130.