Induced Homomorphism

In mathematics, an induced homomorphism is a structure-preserving map between a pair of objects that is derived in a canonical way from another map between another pair of objects. A particularly important case arises in algebraic topology, where any continuous function between two pointed topological spaces induces a group homomorphism between the fundamental groups of the two spaces. Likewise, the same continuous map induces a group homomorphism between the respective homotopy groups, the respective homology groups and a homomorphism going in the opposite direction between the corresponding cohomology groups.

A homomorphism is a structure-preserving map between two mathematical objects of the same type: a group homomorphism, for instance, is a map between two groups such that the image of the product of any two group items is the same as the product of their images, while a graph homomorphism is a map from the vertices of one undirected graph to the vertices of another such that any edge of the first graph is mapped to an edge of the second. Families of objects, and maps between them, are generally formalized as objects and morphisms in a category; by convention, the morphisms in categories are depicted as arrows in diagrams. In many of the important categories of mathematics, the morphisms are called homomorphisms. In category theory, a functor is itself a structure-preserving map, between categories: it must map objects to objects, and morphisms to morphisms, in a way that is compatible with the composition of morphisms within the category. If F is a functor from category A to category B, ƒ is a morphism in category A, and the morphisms of category B are called homomorphisms, then F(ƒ) is the homomorphism induced from ƒ by F.

For example, let X and Y be topological spaces with fundamental groups π(X,x₀) and π(Y,y₀) respectively, with specified base points x₀ and y₀. If ƒ is a continuous function from X to Y that maps the base points to each other (that is, ƒ(x₀) = y₀) then any loop based at x₀ may be composed with ƒ to make a loop based at y₀. This map of loops respects homotopy equivalence of loops: one can map any element of π(X,x₀) to π(Y,y₀) by choosing a loop representing the element, using ƒ to map that representative loop to Y, and selecting the homotopy equivalence class of the resulting mapped loop. Thus, ƒ corresponds to a homomorphism of fundamental groups; this homomorphism is called the induced homomorphism of ƒ. The construction of a fundamental group for each topological space, and of an induced homomorphism of fundamental groups for each continuous function, forms a functor from the category of topological spaces to the category of groups. See fundamental group#Functoriality for more on this type of induced homomorphism.