Pushforward (homology)


In algebraic topology, the pushforward of a continuous function : between two topological spaces is a homomorphism between the homology groups for.
Homology is a functor which converts a topological space into a sequence of homology groups. 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 :
First, the map induces a homomorphism between the singular or simplicial chain complex and defined by composing each singular n-simplex with to obtain a singular n-simplex of,, and extending this linearly via.
The maps satisfy where is the boundary operator between chain groups, so defines a chain map.
Therefore, takes cycles to cycles, since implies. Also takes boundaries to boundaries since.
Hence induces a homomorphism between the homology groups for.

Properties and homotopy invariance

Two basic properties of the push-forward are:
  1. for the composition of maps.
  2. where : refers to identity function of and refers to the identity isomorphism of homology groups.
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 that the homology groups of homotopy equivalent spaces are isomorphic: The maps induced by a homotopy equivalence are isomorphisms for all.