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.
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
defined by composing each singular n-simplex
σ
to obtain a singular n-simplex of
σ
σ
, and extending this linearly via
σ
σ
satisfy
is the boundary operator between chain groups, so
defines a chain map.
takes cycles to cycles, since
takes boundaries to boundaries since
induces a homomorphism between the homology groups
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