Def. Let $A \subset X$ be a subspace. A retraction is a continuous map $r: X \to A$ s.t. $r|A = id_A$. (I think it may not need to be continuous depending on definition).

Can be used to define a Retraction deformation which is a Homotopy between the identity map of a space $X$ and a retraction to a subspace $A \subset X$.

Lemma. If $r: X \to A$ is a retraction, then $r_*: \pi_1(X,a_0) \to \pi_1(A,a_0)$ is surjective and $i_*: \pi(A,a_0) \to \pi_1(X,a_0)$ (inducded by $i: A \to X$ the Inclusion map) is injective. ($a_0 \in A \subset X$). Proved using Functorial properties of induced map (denoted by $*$). See Fundamental group

Proposition. There is no retraction $r: B^2 \to S_1$ ($= \partial B^2$) ($B^2 = \{x \in \mathbb{R}^2 : ||x|| \leq 1\}$, Unit ball). But there is one from $B - \{0\}$.

Brouwer fixed-point theorem