# 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

*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 be a functor from the category of topological spaces to the category of groups.

Suppose is a topological space, is a retraction and is the image of the retraction. Suppose is the inclusion of in . Since is the identity map on , we get by functorality that is the identity on . This forces that is injective, and is surjective. Identifying with its image subgroup , we can view 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.