In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, orientable manifold are related by Poincaré duality, an isomorphism between its homology and cohomology groups. A chain complex is called a Poincaré complex if its homology and cohomology groups have the abstract properties of Poincaré duality. A Poincaré space is a topological space whose singular chain complex is a Poincaré complex. These are used in surgery theory to analyze manifolds algebraically.
Definition Let C = { C i } {\displaystyle C=\{C_{i}\}} be a chain complex of abelian groups, and assume that the homology groups of C {\displaystyle C} are finitely generated. Assume that there exists a map Δ : C → C ⊗ C {\displaystyle \Delta \colon C\to C\otimes C} , called a chain-diagonal, with the property that ( ε ⊗ 1 ) Δ = ( 1 ⊗ ε ) Δ {\displaystyle (\varepsilon \otimes 1)\Delta =(1\otimes \varepsilon )\Delta } . Here the map ε : C 0 → Z {\displaystyle \varepsilon \colon C_{0}\to \mathbb {Z} } denotes the ring homomorphism known as the augmentation map, which is defined as follows: if n 1 σ 1 + ⋯ + n k σ k ∈ C 0 {\displaystyle n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k}\in C_{0}} , then ε ( n 1 σ 1 + ⋯ + n k σ k ) = n 1 + ⋯ + n k ∈ Z {\displaystyle \varepsilon (n_{1}\sigma _{1}+\cdots +n_{k}\sigma _{k})=n_{1}+\cdots +n_{k}\in \mathbb {Z} } . Using the diagonal as defined above, we are able to form pairings, namely:
ρ : H k ( C ) ⊗ H n ( C ) → H n − k ( C ) , where ρ ( x ⊗ y ) = x ⌢ y {\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. A chain complex C is called geometric if a chain homotopy exists between Δ {\displaystyle \Delta } and τ Δ {\displaystyle \tau \Delta } , where τ : C ⊗ C → C ⊗ C {\displaystyle \tau \colon C\otimes C\to C\otimes C} is the transposition/flip given by τ ( a ⊗ b ) = b ⊗ a {\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 μ ∈ H n ( C ) {\displaystyle \mu \in H_{n}(C)} , such that the maps given by
( ⌢ μ ) : H k ( C ) → H n − k ( C ) {\displaystyle (\frown \mu )\colon H^{k}(C)\to H_{n-k}(C)}
are group isomorphisms for all 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . These isomorphisms are the isomorphisms of Poincaré duality.
Example The singular chain complex of an orientable, closed n-dimensional manifold M {\displaystyle M} is an example of a Poincaré complex, where the duality isomorphisms are given by capping with the fundamental class [ M ] ∈ H n ( M ; Z ) {\displaystyle [M]\in H_{n}(M;\mathbb {Z} )} .
References
External links Classifying Poincaré complexes via fundamental triples on the Manifold Atlas
