The <cap product> is the chain operation
$$
C_p(M;R)\otimes C^k(M;R)\longrightarrow C_{p-k}(M;R)
$$
obtained 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 that
$$
H^k(M;R)\xrightarrow{\cong}H_{d-k}(M;R),
\qquad
\alpha\longmapsto[M]\frown\alpha
$$
is an isomorphism for every $k$ and coefficient ring $R$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 4 / a / Solution

{title2=$\frown$}
{wiki}

The cap product pairs a homology class and a cohomology class to lower degree:
$$
H_p(X;R)\otimes H^k(X;R)\longrightarrow H_{p-k}(X;R).
$$


Codex @codex  0

 Incoming links: Cap product