I present here a reference for the comment stated by Peter May in a comment to Tom Goodwillie's answer in this thread. It also makes precise the comment by John Klein below the question.

Corollary 2 in the article

Ross Geoghegan and William Haver, On the space of piecewise linear homeomorphisms of a manifold, Proceedings of the A.M.S., volume 55, number 1, 1976, pages 145-151

contains the following result:

Let $M$ be a compact PL manifold (without boundary) of dimension different from 4. Let $PLH(M)$ denote the space of PL homeomorphisms of $M$ with the compact-open topology. Further, let $H^\ast(M)$ denote the space of homeomorphisms of $M$ which are isotopic to PL homeomorphisms, equipped with the compact-open topology. In other words, $H^\ast(M)$ is the union of all the path components of the space of homeomorphisms of $M$ which contain a PL homeomorphism. Then the inclusion of $PLH(M)$ into $H^\ast(M)$ is a weak equivalence.

Unfortunately, I am not in a position to say anything about the proof of the above result. Perhaps someone else can provide more information.

I present here a reference for Peter May's comment to Tom Goodwillie's answer in this thread. It also corroborates the comment by John Klein below the question stating that there is no obvious topology on the set of PL homeomorphisms which recovers the correct homotopy type.

Corollary 2 in the article

Ross Geoghegan and William Haver, On the space of piecewise linear homeomorphisms of a manifold, Proceedings of the A.M.S., volume 55, number 1, 1976, pages 145-151

contains the following result:

Let $M$ be a compact PL manifold (without boundary) of dimension different from 4. Let $PLH(M)$ denote the space of PL homeomorphisms of $M$ with the compact-open topology. Further, let $H^\ast(M)$ denote the space of homeomorphisms of $M$ which are isotopic to PL homeomorphisms, equipped with the compact-open topology. In other words, $H^\ast(M)$ is the union of all the path components of the space of homeomorphisms of $M$ which contain a PL homeomorphism. Then the inclusion of $PLH(M)$ into $H^\ast(M)$ is a weak equivalence.

Unfortunately, I am not in a position to say anything about the proof of the above result. Perhaps someone else can provide more information.

Corollary 2 in the article

Ross Geoghegan and William Haver, On the space of piecewise linear homeomorphisms of a manifold, Proceedings of the A.M.S., volume 55, number 1, 1976, pages 145-151

contains the following result:

Let $M^n$ be a compact PL manifold without boundary which has dimension $n \neq 4$. Let $PLH(M)$ denote the space of PL homeomorphisms of $M$ with the compact-open topology. Let also $H^\ast(M)$ denote the space of homeomorphisms of $M$ which are isotopic to PL homeomorphisms, equipped with the compact-open topology. In other words, $H^\ast(M)$ is the union of all the path components of the space of homeomorphisms of $M$ which contain a PL homeomorphism. Then the inclusion of $PLH(M)$ into $H^\ast(M)$ is a weak equivalence.

This result generalizes the one stated by Peter May in a comment to Tom Goodwillie's answer here. It also makes precise the comment by John Klein below the question.

Unfortunately, I am not in a position to say anything about the proof of the above result. Perhaps someone else can provide some information.

I present here a reference for the comment stated by Peter May in a comment to Tom Goodwillie's answer in this thread. It also makes precise the comment by John Klein below the question.

Corollary 2 in the article

Ross Geoghegan and William Haver, On the space of piecewise linear homeomorphisms of a manifold, Proceedings of the A.M.S., volume 55, number 1, 1976, pages 145-151

contains the following result:

Let $M$ be a compact PL manifold (without boundary) of dimension different from 4. Let $PLH(M)$ denote the space of PL homeomorphisms of $M$ with the compact-open topology. Further, let $H^\ast(M)$ denote the space of homeomorphisms of $M$ which are isotopic to PL homeomorphisms, equipped with the compact-open topology. In other words, $H^\ast(M)$ is the union of all the path components of the space of homeomorphisms of $M$ which contain a PL homeomorphism. Then the inclusion of $PLH(M)$ into $H^\ast(M)$ is a weak equivalence.

Unfortunately, I am not in a position to say anything about the proof of the above result. Perhaps someone else can provide more information.