When is a freely contractible space also based contractible?

Question

There are spaces, for example the cone of the Hawaiian earring, which are contractible but that have a basepoint such that no contraction fixes that basepoint. Are there any good sufficient conditions which guarantee that a contractible space $X$ with basepoint $x_0$ must have a contraction fixing that basepoint?

One idea I had is, if $(X,x)$ has the homotopy extension property for any $x\in X$, we could take a contraction $H(x,t)$ from $X$ to $x_0$ which doesn't necessarily fix $x_0$ and then compose with a homotopy $G(y,t)$ from the identity taking $H(x_0,t)$ to $x_0$ for each $t$, giving a contraction that does fix $x_0$. The HEP would give such a homotopy for each particular $t$ but the problem is making a choice that simultaneously works for all $t$.

Answer 1

According to the Whitehead theorem (Hatcher, page 346), if $X \hookrightarrow Y$ is an inclusion between connected CW complexes and a weak homotopy equivalence then $X$ must be a (strong) deformation retract of $Y$.

Then if $X$ is a contractible CW complex, $\{x\}\hookrightarrow X$ is a weak homotopy equivalence, so there is a contraction of $X$ fixing $x$.

Answer 2

It is not true that the cone $CH = H \times I/ H \times \{1\}$ of the Hawaiian earring $H$ is contractible but does not have any contractions that fix a particular basepoint. Yes, there are basepoints $x_0 \in CH$ which do not admit a basepoint preserving contraction, but for example the tip of $CH$ admits such a contraction.

Now let $X$ be contractible and $x_0 \in X$. You ask when the pointed space $(X,x_0)$ is pointed contractible. Here is a well-known theorem.

Theorem 1. if $(X,x_0)$ is well-pointed (which means that the inclusion map $\{x_0\} \hookrightarrow X$ is a cofibration), then $(X,x_0)$ is pointed contractible.

The proof is not completely elementary. See for example Tyrone's answer to Strong deformation retraction. (he proves a more general result: If a subspace inclusion $i : A \hookrightarrow X$ is both a cofibration and a homotopy equivalence, then $A$ is a strong deformation retract of $X$). Also see Proposition 4.4.14 in

Aguilar, Marcelo A., et al. Algebraic topology from a homotopical viewpoint. Vol. 2002. New York: Springer, 2002.

Here is partial converse.

Theorem 2. Let $(X,x_0)$ be pointed contractible. If $\{x_0\}$ is a zero-set in $X$ (i.e. $\{x_0\} = \phi^{-1}(0)$ for some continuous $\phi : X \to \mathbb R$, then $(X,x_0)$ is well-pointed.

For example, if $X$ is metrizable, then the zero-set condition is always satisfied.

Proof. In Showing continuity of partially defined map one can find a theorem of Arne Strøm which applies here by taking $U = X$.

Note that if $\{x_0\}$ is a closed subset of $X$, then a necessary condition for $(X,x_0)$ being well-pointed is that $\{x_0\}$ is a zero-set in $X$. See again the theorem of Arne Strøm.

Let us come back to $CH$. One can show that the only $x_0 \in CH$ for which $(CH,x_0)$ is not well-pointed are those of the form $x_0 = [\mathcal O,t]$ with $t < 1$, where $\mathcal O \in H$ is the point common to all circles. The proof is an easy consequence of Strøm's theorem.