Induced homomorphism

In mathematics, especially in algebraic topology, an induced homomorphism is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space X to a topological space Y induces a group homomorphism from the fundamental group of X to the fundamental group of Y. More generally, in , any functor by definition provides an induced morphism in the target for each morphism in the source category. For example, fundamental groups, higher homotopy groups, singular homology, and De Rham cohomology are algebraic structures that are functorial, meaning that their definition provides a functor from (e.g.) the to (e.g.) the or . This means that each space is associated with an algebraic structure, while each continuous map between spaces is associated with a structure-preserving map between structures, called an induced homomorphism. A homomorphism induced from a map is often denoted . Induced homomorphisms often inherit properties of the maps they come from; for example, two maps that are inverse to each other up to homotopy induce homomorphisms that are inverse to each other. A common use of induced homomorphisms is the following: by showing that a homomorphism with certain properties cannot exist, one concludes that there cannot exist a continuous map with properties that would induce it. Thanks to this, relations between spaces and continuous maps, often very intricate, can be inferred from relations between the homomorphisms they induce. The latter may be simpler to analyze, since they involve algebraic structures which can be often easily described, compared, and calculated in. Fundamental group#Functoriality Let X and Y be topological spaces with points x0 in X and y0 in Y. Let h : X→Y be a continuous map such that h(x0) = y0. Then we can define a map from the fundamental group pi1(X, x0) to the fundamental group pi1(Y, y0) as follows: any element of pi1(X, x0), represented by a loop f in X based at x0, is mapped to the loop in pi1(Y, y0) obtained by composing with h: Here [f] denotes the equivalence class of f under homotopy, as in the definition of the fundamental group.