...where $M$ is a smooth manifold and $p, q \in M$. Does anyone know of any slick or accessible proofs of this? I was referred to Milnor's "On Spaces Having the Homotopy Type of a CW Complex" which is a bit over my head. I was wondering if anything simpler had been discovered in the intervening decades.
Edit: It looks like these are the steps Milnor takes (for proofs of which he cites papers I can't find or papers in languages i don't read):
Lemma 1: Let $X$ be a topological space. $X$ has the homotopy type of an absolute neighborhood retract (ANR) iff $X$ has the homotopy type of a countable CW complex
Lemma 2: If $Y$ is compact metric and $X$ is an ANR then the space of maps $Y\rightarrow X$ is an ANR.
From these lemmas I can prove that $\{\gamma \in C^0([0,1],M)\}$ has the homotopy type of a CW-complex since I can prove $M$ has the homotopy type of a countable CW-complex. But I don't know how to get from there to the homotopy type of a subspace where the endpoints are fixed. And I don't know how to prove the lemmas.