Poincare Duality

Orientation

Local Orientation

A local orientation of an n-manifold $M$ at $x\in M$ is a choice of generator of ${\mu }_x\in H_n(M{|}_x)=H_n(M,M-\{x\})\cong H_{n-1}(S^{n-1})\cong \mathbb{Z}$

Orientation and orientable

For $M$ n-manifold, an oriented alats is one for which all transition functions are orientation preserving, $M$ is orientable if it admits an oriented atlas, and when n>0, an orientation of $M$ is a maximal oriented atlas. An $R$-orientable is a compatible choice of generator ${\mu }_x\in H_n(M|x;R)\cong R\forall x\in M$.

Theorem

Let $M$ be a closed n-manifold.

1. $M$ is $R$-orientable then $H_n(M;R)\cong R$ and $H_n(M;R)\to H_n(M{|}_x;R)$ is an isomorphism $\forall x\in M$.
2. $M$ is not orientable then $H_n(M;R)\cong R[2]=\{r\in R,2r=0\}$
3. $H_i(M,R)=0$ for $i>n$.

Corollary

$M$ closed, connected n-manifold,

1. $M$ is orientable then $\{\begin{array}{ll}{H}_n(M)=\mathbb{Z},& \\ H_{n-1}(M)& \text{is torsion free}.\end{array}$
2. $M$ is not orientable then $\{\begin{array}{l}{H}_n(M)=0,\\ H_{n-1}(M)\cong \mathbb{Z}/2\mathbb{Z}\oplus \text{ torsion free group}.\end{array}$

Proposition

If $M$ is non-compact then $H_n(M)=0$

Tensor

Tensor product of group

For abelian group $A$, $B$ the tensor product $A\otimes B$ is defined to be abelian group with genertaors $a\otimes b$ for $a\in A$, $b\in B$, and relations $(a+a^{\prime })\otimes b=a\otimes b+a^{\prime }\otimes b$ and $a\otimes (b+b^{\prime })=a\otimes b+a\otimes b^{\prime }$. The zero element of $A\otimes B$ is $0\otimes 0=0\otimes b=a\otimes 0$, and $-(a\otimes b)=-a\otimes b=a\otimes (-b)$. More generally $n(a\otimes b)=na\otimes b=a\otimes nb$ for all $n\in \mathbb{Z}$.

Proposition

1. $A\otimes B\cong B\otimes A$.
2. $\left({⨁}_i{A}_i\right)\otimes B\cong {⨁}_i(A_i\otimes B)$.
3. $(A\otimes B)\otimes C\cong A\otimes (B\otimes C)$.
4. $\mathbb{Z}\otimes A\cong A$.
5. ${\mathbb{Z}}_n\otimes A\cong A/nA$.
6. A pair of homomorphisms $f:A\to A^{\prime }$ and $g:B\to B^{\prime }$ induces a homomorphism $f\otimes g:A\otimes B\to A^{\prime }\otimes B^{\prime }$ via $(f\otimes g)(a\otimes b)=f(a)\otimes g(b)$.
7. A bilinear map $\phi :A\times B\to C$ induces a homomorphism $A\otimes B\to C$ sending $a\otimes b$ to $\phi (a,b)$.

Cap product

For an arbitrary space $X$ and coefficient ring $R$, define an $R$-bilinear cap product $⌢:C_k(X;R)\times C^{\ell }(X;R)\to C_{k-\ell }(X;R)\text{for }k\ge \ell \text{ by setting}$ $\sigma ⌢\phi =\phi (\sigma |[v_0,\dots ,v_{\ell }])\sigma |[v_{\ell },\dots ,v_k]$ for $\sigma :{\Delta }^k\to X$ and $\phi \in C^{\ell }(X;R)$.

Proposition

Cup and cap product are related by the formula $\psi (\alpha ⌢\phi )=(\phi ⌣\psi )(\alpha )$ for $\alpha \in C_{k+\ell }(X;R)$, $\phi \in C^k(X;R)$, and $\psi \in C^{\ell }(X;R)$. This holds since for a singular $(k+\ell )$-simplex $\sigma :{\Delta }^{k+\ell }\to X$ we have $\psi (\sigma ⌢\phi )=\psi \left(\phi (\sigma |[v_0,\dots ,v_k])\sigma |[v_k,\dots ,v_{k+\ell }]\right)$ $$\begin{gathered} =\phi (\sigma |[v_0,\dots ,v_k])\psi (\sigma |[v_k,\dots ,v_{k+\ell }]) \\ =(\phi ⌣\psi )(\sigma ) \end{gathered}$$

$\phi ⌢:C_{k+\ell }(X;R)\to C_k(X;R)$ is equal to the map ${\text{Hom}}_R(C_{\ell }(X;R),R)\to {\text{Hom}}_R(C_{k+\ell }(X;R),R)$ dual to $⌢\phi$. When the maps $h$ are isomorphisms, for example when $R$ is a field or when $R=\mathbb{Z}$ and the homology groups of $X$ are free, then the map $\phi ⌢$ is the dual of $⌢\phi$. Thus in these cases cup and cap product determine each other, at least if one assumes finite generation so that cohomology determines homology as well as vice versa.

The formula says that the map

Poincare Duality

If $M$ is a closed $R$-orientable $n$-manifold with fundamental class $[M]\in H_n(M;R)$, then the map $D:H^k(M;R)\to H_{n-k}(M;R)$ defined by $D(\alpha )=[M]⌢\alpha$ is an isomorphism for all $k$.

Corollary

A closed manifold of odd dimension has Euler characteristic zero.