Retraction

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).

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.