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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, given a continuous map ''f'': ''X'' → ''Y'' of

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 ...

s and a ring ''R'', the pullback along ''f'' on cohomology theory is a grade-preserving ''R''-algebra homomorphism: :

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

from the

cohomology ring 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 und ...

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 form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

s. 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 In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via "integration". It is also called the fiber integration. Definition Let \pi: E \to B be a fiber bundle ove ...

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_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_R(C', G) \to \operatorname_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.

References

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