Cap product
From Wikipedia, the free encyclopedia
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. 
is the bilinear map given by :
![\sigma \frown \psi = \psi(\sigma|_{[v_0, \ldots, v_q]}) \sigma|_{[v_q, \ldots, v_p]}](http://upload.wikimedia.org/math/5/c/0/5c0409a76092159c736fdd4eb90aff37.png)
where
and 
The cap product induces a product on the respective homology and cohomology classes, e.g. :
[edit] Interpretation
In analogy with the interpretation of the cup product in terms of the Kunneth formula, we can explain the existence of the cap product by considering the composition
in terms of the chain and cochain complexes of X, where we are taking tensor products of chain complexes, 
is the diagonal map which induces the map Δ * on the chain complex (more precisely, the map Δ is not cellular, but as detailed in that article, any continuous map of CW complexes is homotopic to a cellular map, so we are in effect considering an associated cellular map to Δ, the choice of homotopic map does not end up mattering when we pass to the quotient), and 
is the evaluation map (always 0 except for p = q).
This composition then passes to the quotient to define the cap product 
, and looking carefully at the above composition shows that it indeed takes the form of maps 
, which is always zero for p < q.
[edit] Equations
The boundary of a cap product is given by :
Given a map f the induced maps satisfy :
The cap and cup product are related by :
where
, 
and 
[edit] See also
[edit] References
- Hatcher, A., Algebraic Topology, Cambridge University Press (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
