Continuous functions on path-connected subsets

Question

Let $X$ be a topological space, and $PX$ the space of all paths on $X$. Then let $G\subset X$ be a path-connected subset and $p\in G$ a point. Let $\sigma:G\rightarrow PX$ be a continuous function such that

$\sigma(x)(0)=x\ \forall x\in G,$

$\sigma(x)(1)=p\ \forall x\in G.$

Then show that there exists a function $\tau:G\rightarrow PX$ such that

$\tau(x)(0)=x\ \forall x\in G$

$\tau(x)(1)=p\ \forall x\in G$

$\tau(p)(t)=p\ \forall t\in[0,1].$

Clearly $G$ is contractible, and then the question relates to showing that it is SDR-contractible. I know that this is not always the case (see comb space) but I'm hoping that the addition of the path-connected property makes this true.

I need to show this as part of a lemma in my masters project and would really appreciate any help. Thanks

Answer 1

I think the assumption you need is that $p$ is a non-degenerate basepoint of $G$, which is to say that the inclusion $\lbrace p\rbrace \hookrightarrow G$ is a cofibration, which is to say the pair $(G,\lbrace p\rbrace)$ has the homotopy extension property. It follows from this that the pair $$ (G,\lbrace p\rbrace)\times (I,\partial I) = (G\times I, \lbrace p\rbrace \times I \cup G\times \partial I) $$ also has the HEP. Now you can take your contracting homotopy $\sigma\colon\thinspace G\to PX$, or rather its adjoint $\tilde\sigma \colon\thinspace G\times I\to X$, and show that it is homotopic to a homotopy which fixes $p$. (You will be extending a homotopy $$ (\lbrace p \rbrace \times I \cup G\times \partial I)\times I \to X $$ given by $(x,t,s) \mapsto \tilde\sigma (x,(1-s)t)$.)

A good online reference for this might be the notes of Jesper Møller here, in particular Section 5.