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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 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 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 poin ...
and ''R'' a coefficient ring. The cap product is a bilinear map on singular homology 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 view ...
:\frown\;: H_p(X;R)\times H^q(X;R) \rightarrow H_(X;R). defined by contracting a singular chain \sigma : \Delta\ ^p \rightarrow\ X with a singular cochain \psi \in C^q(X;R), by the formula : : \sigma \frown \psi = \psi(\sigma, _) \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.


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 commutati ...
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_\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 In category theory, a branch of mathematics, for any object a in any category \mathcal where the product a\times a exists, there exists the diagonal morphism :\delta_a : a \rightarrow a \times a satisfying :\pi_k \circ \delta_a = \operatorna ...
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(X) \otimes C^q(X) \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'' th ...
(always 0 except for p=q). This composition then passes to the quotient to define the cap product \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 \frown \colon H_p(X) \times H^q(X) \to H_(X), which is always zero for p < q.


Relation with Poincaré duality

For a closed orientable n-manifold M, we can define its fundamental class /math> as a generator of H_n(M), and then the cap product map H^k(M)\to H_(M), \alpha\to cap \alpha gives Poincaré duality. This also holds for (co)homology with coefficient in some other ring R.


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_*(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 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 commutati ...
are related by : : \psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma) where :\sigma : \Delta ^ \rightarrow X, \psi \in C^q(X;R) and \varphi \in C^p(X;R). An interesting consequence of the last equation is that it makes H_(X;R) into a right H^(X;R)-
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 * Modul ...
.


See also

*
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 commutati ...
* Poincaré duality * Singular homology * 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 King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
(2002) . Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc. * {{DEFAULTSORT:Cap Product Homology theory Algebraic topology Binary operations