Method in algebraic topology
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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae05f962b47151860a22b5adfcb3d62b21120631)
defined by contracting a singular chain
with a singular cochain
by the formula:
![{\displaystyle \sigma \frown \psi =\psi (\sigma |_{[v_{0},\ldots ,v_{q}]})\sigma |_{[v_{q},\ldots ,v_{p}]}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6882b875de7b66d6b3cf4bbb8de999fe132c21f)
Here, the notation
indicates the restriction of the simplicial map
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
is a CW-complex and
(and
) 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0daa3235e5cfe011fc2c3f8b478bb60b11153f5)
where we are taking
tensor products of chain complexes,
![{\displaystyle \Delta \colon X\to X\times X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ef0fbe54823e8ddcc82e9d702f61f9bc1aaccd)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0df48cf8a497ff723ed33be48c0974393e0c7050)
on the chain complex, and
![{\displaystyle \varepsilon \colon C_{p}(X)\otimes C^{q}(X)\to \mathbb {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/035ef84bf68a94d3de2a0ea99a2975bf7bd902d8)
is the evaluation map (always 0 except for
![{\displaystyle p=q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cf52da7809003406aee7e680eb5476a05a5c546)
).
This composition then passes to the quotient to define the cap product
, and looking carefully at the above composition shows that it indeed takes the form of maps
, which is always zero for
.
Fundamental class
For any point
in
, we have the long-exact sequence in homology (with coefficients in
) 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 .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19917c042852e1a7359693f9f3720c59fe4ac366)
An element
of
is called the fundamental class for
if
is a generator of
. A fundamental class of
exists if
is closed and R-orientable. In fact, if
is a closed, connected and
-orientable manifold, the map
is an isomorphism for all
in
and hence, we can choose any generator of
as the fundamental class.
Relation with Poincaré duality
For a closed
-orientable n-manifold
with fundamental class
in
(which we can choose to be any generator of
), the cap product map
![{\displaystyle H^{k}(M;R)\to H_{n-k}(M;R),\alpha \mapsto [M]\frown \alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/74b87e2023d778bac8acd8436c9774ce7b06e293)
is an isomorphism for all
![{\displaystyle k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40)
. This result is famously called
Poincaré duality.
The slant product
If in the above discussion one replaces
by
, 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcae373ea0a29f1871f3d6a3e39e5d4f99094e61)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca50526dcd688a32978b90956c7ec6e07f4da143)
to get, respectively, slant products
:
![{\displaystyle H_{p}(X\times Y;R)\otimes H^{q}(Y;R)\rightarrow H_{p-q}(X;R)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f909b34c29aab515cbabf08216ca9521d3e51b74)
and
![{\displaystyle H^{p}(X\times Y;R)\otimes H_{q}(Y;R)\rightarrow H^{p-q}(X;R).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe43eb771a95c9c493d110ce5962b097f7217bf6)
In case X = Y, the first one is related to the cap product by the diagonal map:
.
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 ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0feecf0d951a1d1dbbc77b76f7349ef1526032)
Given a map f the induced maps satisfy :
![{\displaystyle f_{*}(\sigma )\frown \psi =f_{*}(\sigma \frown f^{*}(\psi )).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac43892fb9e49906a54931f36067d82b83e900b0)
The cap and cup product are related by :
![{\displaystyle \psi (\sigma \frown \varphi )=(\varphi \smile \psi )(\sigma )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a2d6db65c89a7323bb15a4084848661ca8cdae9)
where
,
and ![{\displaystyle \varphi \in C^{p}(X;R).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ae261bff88d951ef9f54b912a46e311ba437275)
If
is allowed to be of higher degree than
, the last identity takes a more general form
![{\displaystyle (\sigma \frown \varphi )\frown \psi =\sigma \frown (\varphi \smile \psi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63bf01c5c45e37b1e7e68ed47e34efe868012907)
which makes
into a right
-module.
See also
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.
- 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