Cap product

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 $${\displaystyle C_{\bullet }(X)\otimes C^{\bullet }(X)\stackrel{{\mathrm{\Delta }}_{\ast }\otimes \mathrm{I}\mathrm{d}}{⟶}{C}_{\bullet }(X)\otimes C_{\bullet }(X)\otimes C^{\bullet }(X)\stackrel{\mathrm{I}\mathrm{d}\otimes \epsilon }{⟶}{C}_{\bullet }(X)}$$ 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 $${\displaystyle {\mathrm{\Delta }}_{\ast }:C_{\bullet }(X)\to C_{\bullet }(X\times X)\cong C_{\bullet }(X)\otimes C_{\bullet }(X)}$$ on the chain complex, and ${\displaystyle \epsilon :C_p(X)\otimes C^q(X)\to \mathbb{Z}}$ 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 ${\displaystyle p<q}$.

Fundamental class

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

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

An element ${\displaystyle [M]}$ of ${\displaystyle H_n(M;R)}$ is called the fundamental class for ${\displaystyle M}$ if ${\displaystyle j_{\ast }([M])}$ 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 ${\displaystyle H_n(M;R)\stackrel{j_{\ast }}{\to }{H}_n(M,M-x;R)}$ 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.

Relation with Poincaré duality

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.

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 $${\displaystyle C_{\bullet }(X\times Y)\otimes C^{\bullet }(Y)\cong C_{\bullet }(X)\otimes C_{\bullet }(Y)\otimes C^{\bullet }(Y)\stackrel{\mathrm{I}\mathrm{d}\otimes \epsilon }{⟶}{C}_{\bullet }(X)}$$ 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 ${\displaystyle X=Y}$, the first one is related to the cap product by the diagonal map: ${\displaystyle {\mathrm{\Delta }}_{\ast }(a)/\varphi =a⌢\varphi }$.

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 :

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

Given a map f the induced maps satisfy :

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

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 ${\displaystyle \phi \in C^p(X;R).}$

If ${\displaystyle \sigma }$ is allowed to be of higher degree than ${\displaystyle p+q}$, the last identity takes a more general form

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

which makes ${\displaystyle H_{\ast }(X;R)}$ into a right ${\displaystyle H^{\ast }(X;R)}$-module.