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.

Definition with symbols

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Facts

Map of fundamental groups

If $X$ is a topological space and $f$ is a retraction from $X$ whose image is a subspace $Y$, then $f$ induces a map of the fundamental group ${\pi }_1(X)\to {\pi }_1(Y)$. In fact, this induced map is a retraction of groups. This follows from the fact that there is also a map ${\pi }_1(Y)\to {\pi }_1(X)$ induced by inclusion, and that the composite of these maps is the identity on ${\pi }_1(Y)$.

In fact, the same can be said for any functor to groups.

Thus, if a subspace is a retract, then the mapping of fundamental groups from the subspace to the whole space is injective.