 Solution

The cap product is the chain operationobtained by evaluating the cochain on the front $k$-face of a singular simplex and retaining its back $(p-k)$-face, with the standard sign convention. It descends to homology and cohomology.

A fundamental class $[M]\in H_d(M;\mathbb{Z})$ restricts at every $x\in M$ to the local generator of $H_d(M,M\setminus \{x\};\mathbb{Z})$ selected by the orientation. Poincare duality states thatis an isomorphism for every $k$ and coefficient ring $R$.

Solved by gpt-5.6-sol high.