In algebraic topology, the pushforward of a continuous function f : between two topological spaces is a homomorphism between the homology groups for n \geq 0. Homology is a functor which converts a topological space X into a sequence of homology groups. (Often, the collection of all such groups is referred to using the notation ; this collection has the structure of a graded ring.) In any category, a functor must induce a corresponding morphism. The pushforward is the morphism corresponding to the homology functor.

Definition for singular and simplicial homology

We build the pushforward homomorphism as follows (for singular or simplicial homology): First, the map induces a homomorphism between the singular or simplicial chain complex and defined by composing each singular n-simplex with f to obtain a singular n-simplex of Y,, and extending this linearly via. The maps satisfy where \partial is the boundary operator between chain groups, so defines a chain map. Therefore, f_{#} takes cycles to cycles, since implies. Also f_{#} takes boundaries to boundaries since. Hence f_{#} induces a homomorphism between the homology groups for n\geq0.

Properties and homotopy invariance

Two basic properties of the push-forward are: (This shows the functoriality of the pushforward.) A main result about the push-forward is the homotopy invariance: if two maps are homotopic, then they induce the same homomorphism. This immediately implies (by the above properties) that the homology groups of homotopy equivalent spaces are isomorphic: The maps induced by a homotopy equivalence are isomorphisms for all n.