Pushforward (homology)
In algebraic topology, the pushforward of a continuous function [math]\displaystyle{ f }[/math] : [math]\displaystyle{ X \rightarrow Y }[/math] between two topological spaces is a homomorphism [math]\displaystyle{ f_{*}:H_n\left(X\right) \rightarrow H_n\left(Y\right) }[/math] between the homology groups for [math]\displaystyle{ n \geq 0 }[/math]. Homology is a functor which converts a topological space [math]\displaystyle{ X }[/math] into a sequence of homology groups [math]\displaystyle{ H_{n}\left(X\right) }[/math]. (Often, the collection of all such groups is referred to using the notation [math]\displaystyle{ H_{*}\left(X\right) }[/math]; 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 [math]\displaystyle{ C_n\left(X\right) }[/math] and [math]\displaystyle{ C_n\left(Y\right) }[/math] defined by composing each singular n-simplex [math]\displaystyle{ \sigma_X }[/math] : [math]\displaystyle{ \Delta^n\rightarrow X }[/math] with [math]\displaystyle{ f }[/math] to obtain a singular n-simplex of [math]\displaystyle{ Y }[/math], [math]\displaystyle{ f_{\#}\left(\sigma_X\right) = f\sigma_X }[/math] : [math]\displaystyle{ \Delta^n\rightarrow Y }[/math]. Then we extend [math]\displaystyle{ f_{\#} }[/math] linearly via [math]\displaystyle{ f_{\#}\left(\sum_tn_t\sigma_t\right) = \sum_tn_tf_{\#}\left(\sigma_t\right) }[/math].
The maps [math]\displaystyle{ f_{\#} }[/math] : [math]\displaystyle{ C_n\left(X\right)\rightarrow C_n\left(Y\right) }[/math] satisfy [math]\displaystyle{ f_{\#}\partial = \partial f_{\#} }[/math] where [math]\displaystyle{ \partial }[/math] is the boundary operator between chain groups, so [math]\displaystyle{ \partial f_{\#} }[/math] defines a chain map.
We have that [math]\displaystyle{ f_{\#} }[/math] takes cycles to cycles, since [math]\displaystyle{ \partial \alpha = 0 }[/math] implies [math]\displaystyle{ \partial f_{\#}\left( \alpha \right) = f_{\#}\left(\partial \alpha \right) = 0 }[/math]. Also [math]\displaystyle{ f_{\#} }[/math] takes boundaries to boundaries since [math]\displaystyle{ f_{\#}\left(\partial \beta \right) = \partial f_{\#}\left(\beta \right) }[/math].
Hence [math]\displaystyle{ f_{\#} }[/math] induces a homomorphism between the homology groups [math]\displaystyle{ f_{*} : H_n\left(X\right) \rightarrow H_n\left(Y\right) }[/math] for [math]\displaystyle{ n\geq0 }[/math].
Properties and homotopy invariance
Two basic properties of the push-forward are:
- [math]\displaystyle{ \left( f\circ g\right)_{*} = f_{*}\circ g_{*} }[/math] for the composition of maps [math]\displaystyle{ X\overset{g}{\rightarrow}Y\overset{f}{\rightarrow}Z }[/math].
- [math]\displaystyle{ \left( \text{id}_X \right)_{*} = \text{id} }[/math] where [math]\displaystyle{ \text{id}_X }[/math] : [math]\displaystyle{ X\rightarrow X }[/math] refers to identity function of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ \text{id}\colon H_n\left(X\right) \rightarrow H_n\left(X\right) }[/math] refers to the identity isomorphism of homology groups.
A main result about the push-forward is the homotopy invariance: if two maps [math]\displaystyle{ f,g\colon X\rightarrow Y }[/math] are homotopic, then they induce the same homomorphism [math]\displaystyle{ f_{*} = g_{*}\colon H_n\left(X\right) \rightarrow H_n\left(Y\right) }[/math].
This immediately implies that the homology groups of homotopy equivalent spaces are isomorphic:
The maps [math]\displaystyle{ f_{*}\colon H_n\left(X\right) \rightarrow H_n\left(Y\right) }[/math] induced by a homotopy equivalence [math]\displaystyle{ f\colon X\rightarrow Y }[/math] are isomorphisms for all [math]\displaystyle{ n }[/math].
References
- Allen Hatcher, Algebraic topology. Cambridge University Press, ISBN:0-521-79160-X and ISBN:0-521-79540-0
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