cap product

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In
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
the cap product is a method of adjoining a chain of degree ''p'' with a
cochain In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by
Eduard Čech Eduard Čech (; 29 June 1893 – 15 March 1960) was a Czech mathematician. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification (in topo ...
in 1936, and independently by
Hassler Whitney Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration t ...
in 1938.

Definition

Let ''X'' be a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
and ''R'' a coefficient ring. The cap product is a
bilinear map In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example. Definition Vector spaces Let V, W ...
on
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
and
cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
:$\frown\;: H_p\left(X;R\right)\times H^q\left(X;R\right) \rightarrow H_\left(X;R\right).$ defined by contracting a
singular chain In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
$\sigma : \Delta\ ^p \rightarrow\ X$ with a singular
cochain In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
$\psi \in C^q\left(X;R\right),$ by the formula : :$\sigma \frown \psi = \psi\left(\sigma, _\right) \sigma, _.$ Here, the notation $\sigma, _$ indicates the restriction of the simplicial map $\sigma$ to its face spanned by the vectors of the base, see
Simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
.

Interpretation

In analogy with the interpretation of the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
in terms of the
Künneth formula Künneth is a surname. Notable people with the surname include: * Hermann Künneth (1892–1975), German mathematician * Walter Künneth (1901–1997), German Protestant theologian {{DEFAULTSORT:Kunneth German-language surnames ...
, 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 $C_\bullet\left(X\right)$ (and $C^\bullet\left(X\right)$) is the complex of its cellular chains (or cochains, respectively). Consider then the composition $C_\bullet(X) \otimes C^\bullet(X) \overset C_\bullet(X) \otimes C_\bullet(X) \otimes C^\bullet(X) \overset C_\bullet(X)$ where we are taking tensor products of chain complexes, $\Delta \colon X \to X \times X$ is the diagonal map which induces the map $\Delta_* \colon C_\bullet(X)\to C_\bullet(X \times X)\cong C_\bullet(X)\otimes C_\bullet(X)$ on the chain complex, and $\varepsilon \colon C_p\left(X\right) \otimes C^q\left(X\right) \to \mathbb$ is the
evaluation map In general topology and related areas of mathematics, the initial topology (or induced topology or weak topology or limit topology or projective topology) on a set X, with respect to a family of functions on X, is the coarsest topology on ''X'' t ...
(always 0 except for $p=q$). This composition then passes to the quotient to define the cap product $\frown \colon H_\bullet\left(X\right) \times H^\bullet\left(X\right) \to H_\bullet\left(X\right)$, and looking carefully at the above composition shows that it indeed takes the form of maps $\frown \colon H_p\left(X\right) \times H^q\left(X\right) \to H_\left(X\right)$, which is always zero for $p < q$.

Relation with Poincaré duality

For a closed orientable n-manifold M, we can define its fundamental class

The slant product

If in the above discussion one replaces $X\times X$ by $X\times Y$, the construction can be (partially) replicated starting from the mappings $C_\bullet(X\times Y) \otimes C^\bullet(Y)\cong C_\bullet(X) \otimes C_\bullet(Y) \otimes C^\bullet(Y) \overset C_\bullet(X)$ and $C^\bullet(X\times Y) \otimes C_\bullet(Y)\cong C^\bullet(X) \otimes C^\bullet(Y) \otimes C_\bullet(Y) \overset C^\bullet(X)$ to get, respectively, slant products $/$: $H_p(X\times Y;R) \otimes H^q(Y;R) \rightarrow H_(X;R)$ and $H^p(X\times Y;R) \otimes H_q(Y;R) \rightarrow H^(X;R).$ In case ''X = Y'', the first one is related to the cap product by the diagonal map: $\Delta_*\left(a\right)/\phi = a\frown \phi$. 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 : :$\partial\left(\sigma \frown \psi\right) = \left(-1\right)^q\left(\partial \sigma \frown \psi - \sigma \frown \delta \psi\right).$ Given a map ''f'' the induced maps satisfy : :$f_*\left( \sigma \right) \frown \psi = f_*\left(\sigma \frown f^* \left(\psi\right)\right).$ The cap and
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
are related by : :$\psi\left(\sigma \frown \varphi\right) = \left(\varphi \smile \psi\right)\left(\sigma\right)$ where :$\sigma : \Delta ^ \rightarrow X$, $\psi \in C^q\left(X;R\right)$ and $\varphi \in C^p\left(X;R\right).$ An interesting consequence of the last equation is that it makes $H_\left(X;R\right)$ into a right $H^\left(X;R\right)-$
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
.

*
Cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutat ...
*
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
*
Singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
* Homology theory

References

* Hatcher, A.,
Algebraic Topology
''
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
(2002) . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. * {{DEFAULTSORT:Cap Product Homology theory Algebraic topology Binary operations