Cap product explained

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

\frown : H _{p(X;R) x} H ^{q(X;R)} → H _{p-q} (X;R). \sigma : \Delta ^{p} → X with a singular cochain \psi \in C ^{q(X;R),} by the formula:\sigma \frown \psi = \psi(\sigma| _{[v _{0,} \ldots, v _{q]}} ) \sigma| _{[v _{q,} \ldots, v _{p]}} .

Here, the notation

\sigma| _{[v _{0,} \ldots, v _{q]}} indicates the restriction of the simplicial map \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

X is a CW-complex and C _{\bullet(X)} (and C ^{\bullet(X)} ) is the complex of its cellular chains (or cochains, respectively). Consider then the composition $$C\_\backslash bullet(X)\; \backslash otimes\; C^\backslash bullet(X)\; \backslash overset\; C\_\backslash bullet(X)\; \backslash otimes\; C\_\backslash bullet(X)\; \backslash otimes\; C^\backslash bullet(X)\; \backslash overset\; C\_\backslash bullet(X)$$where we are taking tensor products of chain complexes, \Delta \colon X \to X x X is the diagonal map which induces the map$$\backslash Delta\_*\; \backslash colon\; C\_\backslash bullet(X)\backslash to\; C\_\backslash bullet(X\; \backslash times\; X)\backslash cong\; C\_\backslash bullet(X)\backslash otimes\; C\_\backslash bullet(X)$$on the chain complex, and \varepsilon \colon C _{p(X)} ⊗ C ^{q(X)} \to Z is the evaluation map (always 0 except for p = q ).

This composition then passes to the quotient to define the cap product

\frown \colon H _{\bullet(X)} x H ^{\bullet(X)} \to H _{\bullet(X)} , and looking carefully at the above composition shows that it indeed takes the form of maps \frown \colon H _{p(X)} x H ^{q(X)} \to H _{p-q} (X) , which is always zero for p < q .

Fundamental class

For any point

x in M , we have the long-exact sequence in homology (with coefficients in R ) of the pair (M, M -) (See Relative homology) … \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 … .

An element

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

Relation with Poincaré duality

For a closed

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

The slant product

If in the above discussion one replaces

X x X by X x Y , the construction can be (partially) replicated starting from the mappings$$C\_\backslash bullet(X\backslash times\; Y)\; \backslash otimes\; C^\backslash bullet(Y)\backslash cong\; C\_\backslash bullet(X)\; \backslash otimes\; C\_\backslash bullet(Y)\; \backslash otimes\; C^\backslash bullet(Y)\; \backslash overset\; C\_\backslash bullet(X)$$and$$C^\backslash bullet(X\backslash times\; Y)\; \backslash otimes\; C\_\backslash bullet(Y)\backslash cong\; C^\backslash bullet(X)\; \backslash otimes\; C^\backslash bullet(Y)\; \backslash otimes\; C\_\backslash bullet(Y)\; \backslash overset\; C^\backslash bullet(X)$$

to get, respectively, slant products

/ :$$H\_p(X\backslash times\; Y;R)\; \backslash otimes\; H^q(Y;R)\; \backslash rightarrow\; H\_(X;R)$$ and$$H^p(X\backslash times\; Y;R)\; \backslash otimes\; H\_q(Y;R)\; \backslash rightarrow\; H^(X;R).$$

In case X = Y, the first one is related to the cap product by the diagonal map:

\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 :

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

Given a map f the induced maps satisfy :

f _{*(} \sigma ) \frown \psi = f _{*(\sigma} \frown f ^{*} (\psi)).

The cap and cup product are related by :

\psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)

where

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

If

\sigma is allowed to be of higher degree than p + q , the last identity takes a more general form(\sigma \frown \varphi) \frown \psi = \sigma \frown (\varphi \smile \psi)

which makes

H _{\ast} (X;R) into a right H ^{\ast} (X;R) -module.

See also

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

References

• Hatcher, A., Algebraic Topology, Cambridge University Press (2002) . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
• Book: May, J. Peter . J. Peter May. A Concise Course in Algebraic Topology . 1999 . . https://ghostarchive.org/archive/20221009/http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf . 2022-10-09 . live . 2008-09-27 . vanc . Section 2.7 provides a category-theoretic presentation of the theorem as a colimit in the category of groupoids.