Induced homomorphism (algebraic topology)

In mathematics, especially in the area of topology known as algebraic topology, an induced homomorphism is a way of relating the algebraic invariants of topological spaces which are already related by a continuous function. Such homomorphisms exist whenever the algebraic invariants are functorial. For example, they exist for fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology. For the more categorical approach, see induced homomorphism, and for the specific case of fundamental groups, see induced homomorphism (fundamental group).

Definitions

Given some category [math]\displaystyle{ T }[/math] of topological spaces (possibly with some additional structure) such as the category of all topological spaces Top or the category of pointed topological spaces, that is, topological spaces with a distinguished base point, and a functor [math]\displaystyle{ F: T \to A }[/math] from that category into some category [math]\displaystyle{ A }[/math] of algebraic structures such as the category of groups Grp or of abelian groups Ab which then associates such an algebraic structure to every topological space, then for every morphism [math]\displaystyle{ f: X \to Y }[/math] of [math]\displaystyle{ T }[/math] (which is usually a continuous map, possibly preserving some other structure such as the base point) this functor induces an induced morphism [math]\displaystyle{ F(f): F(X) \to F(Y) }[/math] in [math]\displaystyle{ A }[/math] (which is a group homomorphism if [math]\displaystyle{ A }[/math] is a category of groups) between the algebraic structures [math]\displaystyle{ F(X) }[/math] and [math]\displaystyle{ F(Y) }[/math] associated to [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math], respectively.

Examples

A useful example is the induced homomorphism of fundamental groups. Likewise there are induced homomorphisms of higher homotopy groups and homology groups.

Any homology theory comes with induced homomorphisms. For instance, simplicial homology, singular homology, and Borel-Moore homology all have induced homomorphisms. Similarly, any cohomology comes induced homomorphisms. For instance, Čech cohomology, de Rham cohomology, and singular cohomology all have induced homomorphisms. Generalizations such as cobordism also have induced homomorphisms.