Okay, I keep reading the definitions over and over, but I don't see the difference between the two ; apparently spaces that deformation retract onto a point are contractible, but the opposite is not necessarily true.
Definition of contractible : A space $X$ is said to be contractible if it has the homotopy-type of a point, i.e. there is a map $f : X \to \{p\}$ and a map $g : \{p\} \to X$ such that $fg \simeq \mathbb 1_p$ and $gf \simeq \mathbb 1_X$. Since $fg = \mathbb 1_p$, it suffices to show that there is some point in $X$ such that the identity map $\mathbb 1_X$ is homotopic to a constant map at some point $g(p) \in X$.
Definition of deformation retract onto a point : there exists a map $f : X \times I \to X$ such that $f(x,0) = \mathbb 1_X$ and $f(x,1) = p$ for some fixed point $p \in X$ and $p$ does not depend on $x$.
I believe an homotopy from the identity map to the constant map is precisely a deformation retract, so I don't see the difference. Or maybe I got the definitions wrong. Does anybody see it? I am quite stuck, since one exercise asks to show that some weird space is contractible but does not deformation retract onto a point.
Best Answer
For a contractible space, the homotopy is allowed to move the point, we don't need to have
$$h(p,t) = p \text{ for all } t \in [0,1].$$
For the (strong(1)) deformation retract, $p$ must be kept fixed by the homotopy.
So (strong) deformation-retractability is a stronger condition.
(1) Different authors use different terminology. Some call simply deformation retract what others call strong deformation retract, and that seems to be the case if "spaces that deformation retract onto a point are contractible, but the opposite is not necessarily true". If a deformation retract need not keep the retract fixed in the homotopy, the two properties are exactly the same.