Pullback (cohomology)

Last updated May 03, 2019

In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism:

Algebraic topology branch of mathematics

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 up to homotopy equivalence.

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

f^{*}:H^{*}(Y;R)\to H^{*}(X;R)

from the cohomology ring of Y with coefficients in R to that of X. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if X, Y are manifolds, R the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms.

In mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant.

De Rham cohomology cohomology with real coefficients computed using differential forms

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties.

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

The homotopy invariance of cohomology states that if two maps f, g: X → Y are homotopic to each other, then they determine the same pullback: f^* = g^*.

In contrast, a pushforward for de Rham cohomology for example is given by integration-along-fibers.

Definition from chain complexes

We first review the definition of the cohomology of the dual of a chain complex. Let R be a commutative ring, C a chain complex of R-modules and G an R-module. Just as one lets $H_{*}(C;G)=H_{*}(C\otimes _{R}G)$ , one lets

H^{*}(C;G)=H^{*}(\operatorname {Hom} _{R}(C,G))

where Hom is the special case of the Hom between a chain complex and a cochain complex, with G viewed as a cochain complex concentrated in degree zero. (To make this rigorous, one needs to choose signs in the way similar to the signs in the tensor product of complexes.) For example, if C is the singular chain complex associated to a topological space X, then this is the definition of the singular cohomology of X with coefficients in G.

Now, let f: C → C' be a map of chain complexes (for example, it may be induced by a continuous map between topological spaces). Then there is

f^{*}:\operatorname {Hom} _{R}(C',G)\to \operatorname {Hom} _{R}(C,G)

which in turn determines

f^{*}:H^{*}(C';G)\to H^{*}(C;G).

If C, C' are singular chain complexes of spaces X, Y, then this is the pullback for singular cohomology theory.

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References

J. P. May (1999), A Concise Course in Algebraic Topology.
S. P. Novikov (1996), Topology I - General Survey.

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