Pushforward (homology)

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:

1. ( f ∘ g ) ∗ = f ∗ ∘ g ∗ for the composition of maps X → f Y → g Z .
2. ( 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 .