A few days ago Pedro asked the following question:
Let $H:X\times [0,1] \rightarrow Y$ be a homotopy between two pointed maps $f, g: (X, x_0) \rightarrow (Y, y_0)$, such that $H(x_0,t)$ is nullhomotopic. Does there exist a pointed homotopy between $f$ and $g$? [i.e. homotopy between $f, g$ such that for every $t$ we have $H(x_0,t)=y_0$]
Initially I tried proving the claim, but after realizing that I can't really prove anything without knowing something about the neighborhood of $x_0$, I've found a counterexample based on a problem in Hatcher's book (see the answer to the linked question).
A key feature of the counterexample was that it wasn't locally path connected.
On the other hand, not all is lost for the original question - for manifolds, the claim is actually true. It can be proved by taking an outer and inner neighborhoods for $x_0$ and constructing a pointed homotopy by hand using a homotopy of $H(x_0,t)$ to $y_0$.
This led me to the question - Is that claim correct if we assume that $X$ is locally simply connected? Locally contractible? Or more generally, under what topological assumptions on $X$ (or $Y$ if needed) is the claim correct?
A natural sufficient hypothesis for this to be true, in my opinion, is that $(X,x_0)$ be cofibered. Indeed, the hypothesis give us a map $H\colon X\times I\rightarrow Y$ and a homotopy rel end points of $H\vert_{\{x_0\}\times I}$ to a constant map and we want to modify $H$ to a map $X\times I\rightarrow Y$ that agrees with $H$ on $X\times\partial I$ and the constant map at the end of that homotopy rel end points on $\{x_0\}\times I$. Now, I believe simply asking for the existence of such a map is not a tractable question in general, but asking that "modification" to mean achieving such a map via homotopy does make it a tractable question. So we combine the nullhomotopy of $H\vert_{\{x_0\}\times I}$ together with the constant homotopy of $H\vert_{X\times\partial I}$ to obtain a homotopy of $H\vert_{\{x_0\}\times I\cup X\times\partial I}$. Now, we want to extend this to a homotopy of $H$ and the condition guaranteeing this is possible in general is precisely that $(X\times I,\{x_0\}\times I\cup X\times\partial I)$ is cofibered, which happens to be the case if and only if $(X,x_0)$ is cofibered.