Cap product

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In algebraic topology the cap product is a method of adjoining a chain of degree p with a cochain of degree q, such that q ≤ p, to form a composite chain of degree p - q. It was introduced by Eduard Čech in 1936, and independently by Hassler Whitney in 1938.

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[edit] Definition

Let X be a topological space and R a coefficient ring. \frown is the bilinear map given by :

\sigma \frown \psi = \psi(\sigma|[v_0, ..., v_q]) \sigma|[v_q, ..., v_p]

where

\sigma : \Delta\ ^p \rightarrow\ X and \psi \in C^q(X;R).

The cap product induces a product on the respective Homology and Cohomology classes, e.g. :

\frown\ : H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R).

[edit] Equations

The boundary of a cap product is given by :

\partial(\sigma \frown \psi) = (-1)^q(\partial \sigma \frown \psi - \sigma \frown \delta \psi).

Given a map f the induced maps satisfy :

f_*( \sigma ) \frown \psi = f_*(\sigma \frown f^* (\psi)).

The cap and cup product are related by :

\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)

where

Failed to parse (Cannot write to or create math output directory): \sigma : \Delta ^{p+q} \rightarrow X
,   \psi \in C^q(X;R)and  \varphi \in C^p(X;R).

[edit] References

[edit] See also