Poincare duality

While looking at the lists of Betti numbers of manifolds one notices certain patterns.

To begin with, we have for all connected orientable $n$-manifolds: $$b_0=b_n=1.$$ Further we have

• the sphere: $1,0,1$,
• the torus: $1,2,1$.

So, we immediately have this simple symmetry:

• surfaces: $1,s,1.$

Now for the intermediate dimensions we have to go higher. For the $n$-torus: $$1,1,...,1,1,0,...$$ For the product of the $n$-sphere and the $m$-sphere, $n<m$, we have: $$\begin{array}{ccc} k&0&1&...&n-1&n&n+1&...&m-1&m&m+1&...&n+m-1&n+m \\ b_k&1&0&...&0&1&0&...&0&1&0&...&0&1 \end{array}$$ The symmetry with respect to the center of the string from $0$ to the dimension of the manifold remains. In fact, when $n=m$ there will be a single $2$ in the middle.

The apparent pattern is summarized as follows: $$b_k=b_{N-k}.$$

As it turns out this equation isn't about the symmetry of the homology groups but rather about the one between homology on the right and cohomology on the left, or vice versa: $H^k(M)$ and $H_{n-k}(M)$.

Theorem (Poincare Duality). Suppose $M$ is a compact, oriented, $N$-dimensional manifold, then there is an isomorphism $$D:H^k(M)\to H_{N-k}(M),$$ for all $k$. The isomorphism is given by the cap product with a generator $O_M$ of $H_N(M)$, i.e., the fundamental class of $M$: $$D(a)=a \frown O_M.$$

Further if $(S,\partial S)$ is a compact oriented $n$-manifold with boundary, then the Poincare-Lefschetz duality isomorphism $$D:H^{n-k}(S,\partial S)\longrightarrow H_k(S)$$ is still given by $D(a)=a\frown O_S$.

Poincare duality allows one to "reverse" a homology operator.

Suppose $f:(S_1,\partial S_1)\longrightarrow (S_2,\partial S_2),$ where $(S_i,\partial S_i),\ i=1,2,$ are $n$-manifolds, is a map. We could define the backward homology operator $f_{!}$ as follows. If $$D_i:H^{n-k}(S_i,\partial S_i)\longrightarrow H_k(S_i),i=1,2, $$ denote the duality isomorphisms, we let $$f_{!}=D_1f^{*}D_2^{-1}, $$ so that $f_{!}$ is the composition of the following maps: $$H_k(S_2)\stackrel{D_2^{-1}}{\longrightarrow }H^{n-k}(S_2,\partial S_2)\stackrel{f^{*}}{\longrightarrow }H^{n-k}(S_1,\partial S_1)\stackrel{D_1}{\longrightarrow }H_k(S_1). $$

Similarly we define $f_{!}$ for $f:(X,X\backslash N)\rightarrow (S,\partial S)$, where $X$ is an arbitrary topological space: the backward homology operator of $f$ is the homomorphism $f_{!}:H(S)\longrightarrow H(X)$ given by $$ f_{!}=(f^{*}D_2^{-1})\frown \mu , $$ where $D_2:H^{*}(S,\partial S)\rightarrow H(S)$ is the Poincare-Lefschetz duality isomorphism.

Theorem. If $f_{*}(\mu )=O_S$ then $f_{*}f_{!}=Id.$

Alternatively, we can start with "Poincare duality pairing" $$(,):\Omega ^{j}(M)\otimes \Omega ^{n-j}(M) \rightarrow {\bf R}$$ defined by $$(\omega ,\eta )=\int _M \omega \wedge \eta.$$ Since this pairing is bilinear and non-degenerate, it induces an isomorphism: $$\phi :\Omega ^j\rightarrow (\Omega^{n-j})^*,$$ where the star indicates the dual space. Meanwhile the inner product defines another isomorphism $$\psi:\Omega ^{n-j}(M)\rightarrow (\Omega ^{n-j})^*.$$ The composition of these two produces the Hodge duality operator $$\star :\Omega^j(M)\rightarrow \Omega^{n-j}(M)$$ defined by $$< \star \omega,\eta>=(\omega ,\eta ).$$