Bijection between homotopy classes and basepoint-preserving homotopy classes

Question

$[X,Y]$ is the homotopy classes of maps from $X$ to $Y$ and $[X,Y]_0$ is the based homotopy classes of based maps. If $Y$ is path-connected and $\pi_1(Y)$ is abelian, then is the inclusion $$[X,Y]_0 \hookrightarrow [X,Y]$$ a bijection?

Answer 1

This is false if you allow your spaces to be too wild. For example, let $H$ be the Hawaiian Earring space and let $Y=C(H)$ be the cone on $H$. Then $Y$ has trivial (hence abelian) fundamental group. $H$ has a special point $h_0$ where all the circles converge. We let $h_0$ be the basepoint of $H$ and also of $C(H)$ where we consider $H$ naturally included in $C(H)$. Consider the two maps $f_1,f_2\colon H\to C(H)$ where $f_1$ is the inclusion $H\subset C(H)$ and $f_2$ is the constant map $h\mapsto h_0$. These are freely homotopic since they can both be pulled up to the cone point, but I claim they are not homotopic rel basepoint. This follows from work of Cannon and Conner, who showed that the space you get by gluing two copies of $C(H)$ together along $h_0$ is no longer contractible. If you could contract $C(H)$ to $h_0$ while preserving the basepoint, then that would show the Cannon-Conner example is contractible. (See Cannon and Conner's "On the fundamental groups of one-dimensional spaces" Figure 2.)