Nerve theorem

The nerve theorem is a tenet of topology . He gives a "combinatorial model" of the homotopy type of topological spaces through good overlaps of assigned simplicial complexes.

The isomorphism between Čech homology and singular homology of manifolds follows directly from the nerve theorem .

Nerve of an overlap

To a coverage of a topological space by open sets defining their nerve as the simplicial complex , the corners of which correspond to the open amount of the overlap and in which the corners if and only a span simplex, when the average of the respective open amounts is not empty: . ${\displaystyle X=\bigcup _{i\in I}U_{i}}$ ${\displaystyle X}$ ${\displaystyle U_{i}}$ ${\displaystyle N({\mathcal {U}})}$ ${\displaystyle e_{i}(i\in I)}$ ${\displaystyle e_{i_{0}},\ldots ,e_{i_{k}}}$ ${\displaystyle k}$ ${\displaystyle U_{i_{0}}\cap \ldots \cap U_{i_{k}}\not =\emptyset }$

Example: If the geometric realization of a simplicial complex with corners and the overlap of the corners by the open stars is, then is . ${\displaystyle \vert X\vert }$ ${\displaystyle X}$ ${\displaystyle e_{i}(i\in I)}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \vert X\vert }$ ${\displaystyle U_{i}}$ ${\displaystyle e_{i}}$ ${\displaystyle N({\mathcal {U}})\cong X}$

Nerve theorem

For good coverage of paracompact spaces , the geometric realization of homotopy-equivalent is to . ${\displaystyle {\mathcal {U}}}$ ${\displaystyle X}$ ${\displaystyle N({\mathcal {U}})}$ ${\displaystyle X}$

literature

• Chapter 4G in Allen Hatcher: Algebraic topology ( online )
• Karol Borsuk : On the imbedding of systems of compacta in simplicial complexes , Fund. Math. 35, (1948) 217-234
• Jean Leray : L'anneau spectral et l'anneau filtré d'homologie d'un espace localement compact et d'une application continue , J. Math. Pures Appl. (9) 29: 1-139 (1950)
• André Weil : Sur les théorèmes de de Rham , Comment. Math. Helv. 26 (1952), 119-145