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 . (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 edit
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 and defined by composing each singular n-simplex : with to obtain a singular n-simplex of , : . Then we extend linearly via .
The maps : satisfy where is the boundary operator between chain groups, so defines a chain map.
We have that 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 edit
Two basic properties of the push-forward are:
- for the composition of maps .
- 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 .
References edit
- Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0