Statement of Poincaré duality. Poincaré's original "proof" generalizing duality of convex polyhedra. Modern formulation of the isomorphism using the cap product. Definition of the cap product C_n(X) \otimes C^k(X) \to C_{n-k}(X). Boundary formula. The cap product induces a cap product on (co)homology. Properties: functoriality, relative cap product, relation to cup product. 

Proposition: Let M a closed R-orientable manifold. If H^k(M;R) = \Hom_R(H_k(M;R),R) then the cup product pairing H^k(M)\otimes H^{n-k}(M) \to R given by \alpha \otimes \beta \to (\alpha \cup \beta)( [M] ) is a perfect pairing. 

This holds in particular if R is a field. The same proof also gives that the cup product pairing is perfect on the torsion free quotient of cohomology with integral coefficients. 

Example: The cohomology ring of \RP^n with \Z/2 coefficients.

Applications: There are no antipode preserving maps from S^n \to S^{n-1}; Borsuk-Ulam Theorem: If f:S^n \to \R^n is continuous then there exists x\in S^n such that f(x)=f(-x). The (generalized) Ham Sandwich theorem. 

Remark: Relation between the cup product pairing and intersection of oriented submanifolds (reference: Dold).

Example: The cohomology ring of a lens space with \Z/p coefficients.

Compactly supported cohomology of a space. Directed colimits can be computed using cofinal subsets of indices.

Examples of compactly generated cohomology: (0) If X is compact, it agrees with usual cohomology; (1) H^0_c(X) is the set of compactly supported functions which are constant on path components; (2)H^*_c(\R^n) is \Z in degree n and 0 otherwise.

Functoriality of compactly supported cohomology: contravariant for proper maps; covariant via excision for open embeddings. If X = \colim_\alpha U_\alpha is a directed colimit of open sets then H^*_c(X) is the directed colimit of H^*_c(U_\alpha).

An R-orientation of an n-manifold M determines a coherent system of orientation classes [M]_k \in H_n(M|K;R). Construction of the Poincaré duality isomorphism for an arbitrary R-oriented manifold. Sketch proof of Poincaré duality.

Other forms of duality: Poincaré-Lefschetz and Alexander duality.

Theorem: Let M be an n-manifold

(i) H_i(M;R)=0 for i>n

(ii) If M is connected and not compact then H_n(M;R)=0 

(iii) If M is compact (closed) then the canonical map from H_n(M;R) to the R-module of sections \Gamma(M_R) sending a class \alpha to the corresponding section of M_R is an isomorphism. So if M is R-orientable and connected, H_n(M;R) is an iso with H_n(M,M\x;R) for all x while if M is not orientable then H_n(M;R) injects in H_n(M,M\x;R) as the 2-torsion.

In the R-orientable case, a class in H_n(M;R) mapping to an R-orientation is called a fundamental class of M (or orientation class) and denoted [M]. If M is triangulable and the n-simplices of M can be oriented so that their sum is a cycle. Then the sum represents the fundamental class by (iii).

The proof of the theorem follows from 

Lemma: Let A be a compact set in a manifold M. Then 

(i) H_i(M,M\A;R)=0 for i>n

(ii) If \alpha in H_n(M,M\A;R) has image 0 in H_n(M,M\x;R) for all x in A then \alpha is 0.

(iii) Given a section s of M_R, there is a unique class \alpha \in H_n(M, M\A;R) with \alpha_x=s(x) for x in A

The statements (ii) and (iii) identify H_n(M,M\A;R) with the germs at A of sections of M_R.

Properties of the cohomology cross product: functoriality, unitality, associativity, graded commutativity. Relation to the homology cross product <a x b, c x d> = <a,c> <b,d> where < , > is the evaluation of a cohomology class on a homology class. 

Example: It follows from UCT and the Kunneth formula for homology that the cross product H^k(S^k;R) \otimes_R H^l(S^l;R) \to H^{k+l}(S^k \times S^l;R) is an isomorphism.

Kunneth formula for cohomology: Let R be a PID and X be a space with H_*(X) of finite type (i.e. finitely generated in each degree). Then there is a natural short exact sequence 

0 --> \oplus_{k+l=n} H^k(X;R)\otimes_R H^l(Y;R) --> H^n(X\times Y;R) --> \oplus_{k+l=n-1} Tor^R(H^k(X;R), H^l(Y;R)) --> 0 

The cup product is defined by a \cup b = \Delta^*(a \times b). Properties of the cup product: functoriality, unitality, associativity, commutativity. Relation with the cross product: (a\times b) \cup (c \times d) = (-1)^|b||c| (a\cup c) \times (b\cup d). In particular axb = \pi_1^*a \cup \pi_2^*b so the cross product is determined by the cup product.

The relative cup product H^k(X,A) \otimes H^l(X,B) \to H^k(X,A\cup B).

Examples: The cohomology ring of S^n \times S^m; of \coprod_\alpha X_\alpha; of a wedge of well pointed spaces; of the surface of genus 2, of a product of \CP^2 with a space X (more generally of the product of two spaces one of which has torsion free homology).