Cap product

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.

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

$${\displaystyle C_{\bullet }(X)\otimes C^{\bullet }(X){\overset {\Delta _{*}\otimes \mathrm {Id} }{\longrightarrow }}C_{\bullet }(X)\otimes C_{\bullet }(X)\otimes C^{\bullet }(X){\overset {\mathrm {Id} \otimes \varepsilon }{\longrightarrow }}C_{\bullet }(X)}$$

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

$${\displaystyle \Delta _{*}\colon C_{\bullet }(X)\to C_{\bullet }(X\times X)\cong C_{\bullet }(X)\otimes C_{\bullet }(X)}$$

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 ${\displaystyle p<q}$.

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

$${\displaystyle H^{k}(M;R)\to H_{n-k}(M;R),\alpha \mapsto [M]\frown \alpha }$$

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

$${\displaystyle C_{\bullet }(X\times Y)\otimes C^{\bullet }(Y)\cong C_{\bullet }(X)\otimes C_{\bullet }(Y)\otimes C^{\bullet }(Y){\overset {\mathrm {Id} \otimes \varepsilon }{\longrightarrow }}C_{\bullet }(X)}$$

and

$${\displaystyle C^{\bullet }(X\times Y)\otimes C_{\bullet }(Y)\cong C^{\bullet }(X)\otimes C^{\bullet }(Y)\otimes C_{\bullet }(Y){\overset {\mathrm {Id} \otimes \varepsilon }{\longrightarrow }}C^{\bullet }(X)}$$

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

$${\displaystyle H_{p}(X\times Y;R)\otimes H^{q}(Y;R)\rightarrow H_{p-q}(X;R)}$$

and

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

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.

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.

• 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 JP (1999). A Concise Course in Algebraic Topology (PDF). University of Chicago Press. Archived (PDF) from the original on 2022-10-09. 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 at the nLab
• Poincaré duality at the nLab