Retraction

This article defines a property of continuous maps between topological spaces

Definition

Symbol-free definition

A retraction of a topological space is an idempotent continuous map from the topological space to itself. In other words, it is a continuous map from the topological space to a subspace, such that the restriction to that subspace is the identity map.

The image of a retraction is termed a retract (this is a subspace property).

Definition with symbols

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Facts

Given any functor from the category of topological spaces to the category of groups, or the category of modules over a ring, the topological space notion of retract gets mapped to the notion of retract in the relevant category.

Let $F$ be a functor from the category of topological spaces to the category of groups.

Suppose $X$ is a topological space, $r$ is a retraction and $Y$ is the image of the retraction. Suppose $i$ is the inclusion of $Y$ in $X$. Since $r\circ i$ is the identity map on $Y$, we get by functorality that $F(r)\circ F(i)$ is the identity on $F(Y)$. This forces that $F(Y)\to F(X)$ is injective, and $F(X)\to F(Y)$ is surjective. Identifying $F(Y)$ with its image subgroup $F(X)$, we can view $F(r)$ as a retraction of groups.

The above can be applied to the fundamental group, any of the higher homotopy groups, or any of the homology groups.