- Cap product
In
algebraic topology the cap product is a method of adjoining a chain of degree "p" with acochain of degree "q", such that "q" ≤ "p", to form a composite chain of degree "p" - "q". It was introduced byEduard Čech in 1936, and independently byHassler Whitney in 1938.Definition
Let "X" be a
topological space and "R" a coefficient ring. frown is thebilinear map given by ::sigma frown psi = psi(sigma| [v_0, ..., v_q] ) sigma| [v_q, ..., v_p]
where
:sigma : Delta ^p ightarrow 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) imes H^q(X;R) ightarrow H_{p-q}(X;R).
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
:sigma : Delta ^{p+q} ightarrow X , psi in C^q(X;R)and varphi in C^p(X;R).
ee also
*
Poincaré duality
*singular homology
*homology theory References
*Hatcher, A., " [http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology] ,"
Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
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