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.

Definition[]

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

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

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

${\displaystyle \sigma \frown \psi =\psi (\sigma |_{[v_{0},\ldots ,v_{q}]})\sigma |_{[v_{q},\ldots ,v_{p}]}.}$

Here, the notation ${\displaystyle \sigma |_{[v_{0},\ldots ,v_{q}]}}$ indicates the restriction of the simplicial map ${\displaystyle \sigma }$ to its face spanned by the vectors of the base, see Simplex.

Interpretation[]

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

where we are taking tensor products of chain complexes, ${\displaystyle \Delta \colon X\to X\times X}$ is the diagonal map which induces the map on the chain complex, and ${\displaystyle \varepsilon \colon C_{p}(X)\otimes C^{q}(X)\to \mathbb {Z} }$ is the evaluation map (always 0 except for ${\displaystyle p=q}$).

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

The slant product[]

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

and

to get, respectively, slant products ${\displaystyle /}$:

and

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

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

Equations[]

The boundary of a cap product is given by :

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

Given a map f the induced maps satisfy :

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

The cap and cup product are related by :

${\displaystyle \psi (\sigma \frown \varphi )=(\varphi \smile \psi )(\sigma )}$

where

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

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.

See also[]

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

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.
• slant product in nLab