I am currently reading through Hatcher's Algebraic Topology book. I am having some trouble understanding the difference between a deformation retraction and just a retraction. Hatcher defines them as follows:
A deformation retraction of a space $X$ onto a subspace $A$ is a family of maps $f_t:X \to X$, $t \in I$, such that $f_0=\mathbb{1}$ (the identity map), $f_1(X)=A$, and $f_t|A=\mathbb{1}$ for all $t$.
A retraction of $X$ onto $A$ is a map $r:X \to X$ such that $r(X)=A$ and $r|A=\mathbb{1}$.
Is the notion of time the important characteristic that sets the two ideas apart? (It seems that the definition of deformation retraction utilizes time in its definition, whereas retraction seems to not.)
Any insight is appreciated. Also, if anyone have additional suggested reading material to help with concepts in Algebraic topology, that would be much appreciated.
Best Answer
The difference between a retraction and a deformation retraction does have to do with the "notion of time" as you suggest.
Here's a strong difference between the two:
1) For any $x_0 \in X$, $\{x_0\} \subset X$ has a retract. Choose $r : X \to \{x_0\}$ to be the unique map to the one-point set. Then, certainly, $r(x_0) = x_0$.
2) However, $\{x_0\} \subset X$ only has a deformation retraction if $X$ is contractible. To see, why, notice there has to be a family of maps $f_t : X \to X$ such that $f_0(x) = x$, $f_1(x) = x_0$, and $f_{t}(x_0) = x_0$ for every $t$. This gives a homotopy from $id_X$ to the constant map at $x_0$, which makes $X$ contractible.
In fact, showing a deformation retract from $X$ onto a subspace $A$ always exhibits that $A$ and $X$ are homotopy equivalent, whereas $A$ being a retract of $X$ is weaker. (But often, still useful! Two spaces being homotopy equivalent is very strong indeed!)