Retraction

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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

Fill this in later

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.