In algebraic topology, the pushforward of a continuous function f : X → Y between two topological spaces is a homomorphism f ∗ : H n ( X ) → H n ( Y ) between the homology groups for n ≥ 0 .
Homology is a functor which converts a topological space X into a sequence of homology groups H n ( X ) . (Often, the collection of all such groups is referred to using the notation H ∗ ( X ) ; 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 C n ( X ) and C n ( Y ) defined by composing each singular n-simplex σ X : Δ n → X with f to obtain a singular n-simplex of Y , f # ( σ X ) = f σ X : Δ n → Y . Then we extend f # linearly via f # ( ∑ t n t σ t ) = ∑ t n t f # ( σ t ) .
The maps f # : C n ( X ) → C n ( Y ) satisfy f # ∂ = ∂ f # where ∂ is the boundary operator between chain groups, so ∂ f # defines a chain map.
We have that f # takes cycles to cycles, since ∂ α = 0 implies ∂ f # ( α ) = f # ( ∂ α ) = 0 . Also f # takes boundaries to boundaries since f # ( ∂ β ) = ∂ f # ( β ) .
Hence f # induces a homomorphism between the homology groups f ∗ : H n ( X ) → H n ( Y ) for n ≥ 0 .
Properties and homotopy invariance
Two basic properties of the push-forward are:
- ( f ∘ g ) ∗ = f ∗ ∘ g ∗ for the composition of maps X → f Y → g Z .
- ( i d X ) ∗ = i d where i d X : X → X refers to identity function of X and i d : H n ( X ) → H n ( X ) refers to the identity isomorphism of homology groups.
A main result about the push-forward is the homotopy invariance: if two maps f , g : X → Y are homotopic, then they induce the same homomorphism f ∗ = g ∗ : H n ( X ) → H n ( Y ) .
This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:
The maps f ∗ : H n ( X ) → H n ( Y ) induced by a homotopy equivalence f : X → Y are isomorphisms for all n .
