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Poincaré complex

APoincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifold algebraically.

Definition[edit]

Let ${\displaystyle C=\{C_{i}\}}$ be a chain complexofabelian groups, and assume that the homology groups of ${\displaystyle C}$ are finitely generated. Assume that there exists a map ${\displaystyle \Delta \colon C\to C\otimes C}$, called a chain-diagonal, with the property that ${\displaystyle (\varepsilon \otimes 1)\Delta =(1\otimes \varepsilon )\Delta }$. Here the map ${\displaystyle \varepsilon \colon C_{0}\to \mathbb {Z} }$ denotes the ring homomorphism known as the augmentation map, which is defined as follows: if ${\displaystyle n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k}\in C_{0}}$, then ${\displaystyle \varepsilon (n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k})=n_{1}+\cdots +n_{k}\in \mathbb {Z} }$.^[2]

Using the diagonal as defined above, we are able to form pairings, namely:

${\displaystyle \rho \colon H^{k}($C)\otimes H_{n}(C)\to H_{n-k}(C),\ {\text{where}}\ \ \rho (x\otimes y)=x\frown y} ,

where ${\displaystyle \scriptstyle \frown }$ denotes the cap product.^[3]

A chain complex C is called geometric if a chain-homotopy exists between ${\displaystyle \Delta }$ and ${\displaystyle \tau \Delta }$, where ${\displaystyle \tau \colon C\otimes C\to C\otimes C}$ is the transposition/flip given by ${\displaystyle \tau (a\otimes b)=b\otimes a}$.

A geometric chain complex is called an algebraic Poincaré complex, of dimension n, if there exists an infinite-ordered element of the n-dimensional homology group, say ${\displaystyle \mu \in H_{n}($C)} , such that the maps given by

${\displaystyle (\frown \mu )\colon H^{k}($C)\to H_{n-k}(C)}

are group isomorphisms for all ${\displaystyle 0\leq k\leq n}$. These isomorphisms are the isomorphisms of Poincaré duality.^[4][5]

Example[edit]

• The singular chain complex of an orientable, closed n-dimensional manifold ${\displaystyle M}$ is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class ${\displaystyle [$M]\in H_{n}(M;\mathbb {Z} )} .^[1]

See also[edit]

• Poincaré space

References[edit]

External links[edit]

• Classifying Poincaré complexes via fundamental triples on the Manifold Atlas