Pushforward (homology)

In algebraic topology, the pushforward of a continuous function formula_1 : formula_2 between two topological spaces is a homomorphism formula_3 between the homology groups for formula_4.
Homology is a functor which converts a topological space formula_5 into a sequence of homology groups formula_6. (Often, the collection of all such groups is referred to using the notation formula_7; 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 we have an induced homomorphism between the singular or simplicial chain complex formula_8 and formula_9 defined by composing each singular n-simplex formula_10 : formula_11 with formula_1 to obtain a singular n-simplex of formula_13, formula_14 : formula_15. Then we extend formula_16 linearly via formula_17.
The maps formula_16 : formula_19 satisfy formula_20 where formula_21 is the boundary operator between chain groups, so formula_22 defines a chain map.
We have that formula_16 takes cycles to cycles, since formula_24 implies formula_25. Also formula_16 takes boundaries to boundaries since formula_27.
Hence formula_16 induces a homomorphism between the homology groups formula_29 for formula_30.
Properties and homotopy invariance.
Two basic properties of the push-forward are:
A main result about the push-forward is the homotopy invariance: if two maps formula_38 are homotopic, then they induce the same homomorphism formula_39.
This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:
The maps formula_40 induced by a homotopy equivalence formula_41 are isomorphisms for all formula_42.