Cap product

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.

Let X be a topological space and R a coefficient ring. The cap product is a bilinear map on singular homology and cohomology

${\displaystyle ⌢:H_p(X;R)\times H^q(X;R)\to H_{p-q}(X;R).}$

defined by contracting a singular chain ${\displaystyle \sigma :\mathrm{\Delta }{}^p\to X}$ with a singular cochain ${\displaystyle \psi \in C^q(X;R),}$ by the formula:

${\displaystyle \sigma ⌢\psi =\psi (\sigma {|}_{[v_0,\dots ,v_q]})\sigma {|}_{[v_q,\dots ,v_p]}.}$

Here, the notation ${\displaystyle \sigma {|}_{[v_0,\dots ,v_q]}}$ indicates the restriction of the simplicial map ${\displaystyle \sigma }$ to its face spanned by the vectors of the base, see Simplex.

In analogy with the interpretation of the cup product in terms of the Künneth formula, we can explain the existence of the cap product in the following way. Using CW approximation we may assume that ${\displaystyle X}$ is a CW-complex and ${\displaystyle C_{\bullet }(X)}$ (and ${\displaystyle C^{\bullet }(X)}$) is the complex of its cellular chains (or cochains, respectively). Consider then the composition $${\displaystyle C_{\bullet }(X)\otimes C^{\bullet }(X)\stackrel{{\mathrm{\Delta }}_{*}\otimes \mathrm{I}\mathrm{d}}{⟶}{C}_{\bullet }(X)\otimes C_{\bullet }(X)\otimes C^{\bullet }(X)\stackrel{\mathrm{I}\mathrm{d}\otimes \epsilon }{⟶}{C}_{\bullet }(X)}$$ where we are taking tensor products of chain complexes, ${\displaystyle \mathrm{\Delta }:X\to X\times X}$ is the diagonal map which induces the map $${\displaystyle {\mathrm{\Delta }}_{*}:C_{\bullet }(X)\to C_{\bullet }(X\times X)\cong C_{\bullet }(X)\otimes C_{\bullet }(X)}$$ on the chain complex, and ${\displaystyle \epsilon :C_p(X)\otimes C^q(X)\to \mathbb{\mathbb{Z}}}$ is the evaluation map (always 0 except for ${\displaystyle p=q}$).

This composition then passes to the quotient to define the cap product ${\displaystyle ⌢:H_{\bullet }(X)\times H^{\bullet }(X)\to H_{\bullet }(X)}$, and looking carefully at the above composition shows that it indeed takes the form of maps ${\displaystyle ⌢:H_p(X)\times H^q(X)\to H_{p-q}(X)}$, which is always zero for ${\displaystyle p<q}$.

For any point ${\displaystyle x}$ in ${\displaystyle M}$, we have the long-exact sequence in homology (with coefficients in ${\displaystyle R}$) of the pair (M, M - {x}) (See Relative homology)

${\displaystyle \cdots \to H_n(M-x;R)\stackrel{i_{*}}{\to }{H}_n(M;R)\stackrel{j_{*}}{\to }{H}_n(M,M-x;R)\stackrel{\partial }{\to }{H}_{n-1}(M-x;R)\to \cdots .}$

An element ${\displaystyle [M]}$ of ${\displaystyle H_n(M;R)}$ is called the fundamental class for ${\displaystyle M}$ if ${\displaystyle j_{*}([M])}$ is a generator of ${\displaystyle H_n(M,M-x;R)}$. A fundamental class of ${\displaystyle M}$ exists if ${\displaystyle M}$ is closed and R-orientable. In fact, if ${\displaystyle M}$ is a closed, connected and ${\displaystyle R}$-orientable manifold, the map ${\displaystyle H_n(M;R)\stackrel{j_{*}}{\to }{H}_n(M,M-x;R)}$ is an isomorphism for all ${\displaystyle x}$ in ${\displaystyle R}$ and hence, we can choose any generator of ${\displaystyle H_n(M;R)}$ as the fundamental class.

For a closed ${\displaystyle R}$-orientable n-manifold ${\displaystyle M}$ with fundamental class ${\displaystyle [M]}$ in ${\displaystyle H_n(M;R)}$ (which we can choose to be any generator of ${\displaystyle H_n(M;R)}$), the cap product map $${\displaystyle H^k(M;R)\to H_{n-k}(M;R),\alpha \mapsto [M]⌢\alpha }$$ is an isomorphism for all ${\displaystyle k}$. This result is famously called Poincaré duality.

If in the above discussion one replaces ${\displaystyle X\times X}$ by ${\displaystyle X\times Y}$, the construction can be (partially) replicated starting from the mappings $${\displaystyle C_{\bullet }(X\times Y)\otimes C^{\bullet }(Y)\cong C_{\bullet }(X)\otimes C_{\bullet }(Y)\otimes C^{\bullet }(Y)\stackrel{\mathrm{I}\mathrm{d}\otimes \epsilon }{⟶}{C}_{\bullet }(X)}$$ and $${\displaystyle C^{\bullet }(X\times Y)\otimes C_{\bullet }(Y)\cong C^{\bullet }(X)\otimes C^{\bullet }(Y)\otimes C_{\bullet }(Y)\stackrel{\mathrm{I}\mathrm{d}\otimes \epsilon }{⟶}{C}^{\bullet }(X)}$$

to get, respectively, slant products ${\displaystyle /}$: $${\displaystyle H_p(X\times Y;R)\otimes H^q(Y;R)\to H_{p-q}(X;R)}$$ and $${\displaystyle H^p(X\times Y;R)\otimes H_q(Y;R)\to H^{p-q}(X;R).}$$

In case X = Y, the first one is related to the cap product by the diagonal map: ${\displaystyle {\mathrm{\Delta }}_{*}(a)/\varphi =a⌢\varphi }$.

These ‘products’ are in some ways more like division than multiplication, which is reflected in their notation.

The boundary of a cap product is given by :

${\displaystyle \partial (\sigma ⌢\psi )=(-1{)}^q(\partial \sigma ⌢\psi -\sigma ⌢\delta \psi ).}$

Given a map f the induced maps satisfy :

${\displaystyle f_{*}(\sigma )⌢\psi =f_{*}(\sigma ⌢f^{*}(\psi )).}$

The cap and cup product are related by :

${\displaystyle \psi (\sigma ⌢\phi )=(\phi ⌣\psi )(\sigma )}$

where

${\displaystyle \sigma :{\mathrm{\Delta }}^{p+q}\to X}$, ${\displaystyle \psi \in C^q(X;R)}$ and ${\displaystyle \phi \in C^p(X;R).}$

An interesting consequence of the last equation is that it makes ${\displaystyle H_{\ast }(X;R)}$ into a right ${\displaystyle H^{\ast }(X;R)}$-module.

• Cup product
• Poincaré duality
• Singular homology
• Homology theory

• 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.
• May, J. Peter (1999). A Concise Course in Algebraic Topology. University of Chicago Press. http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf. Retrieved 2008-09-27.  Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.
• slant product in nLab
• Poincaré duality in nLab

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