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Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ fo …

$\newcommand{\real}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Sing}[1]{\operatorname{Sing}(#1)}$$\newcommand{\counit}{\epsilon}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\proj}{\mathrm{proj} …

[This should probably be a comment, since it is so short. Nevertheless, it is an answer to the question.] The result you ask for is a consequence of proposition 2.2.15 in Sam Isaacson's Ph.D. thesis …

Observe that any retract of $\newcommand{\RR}{\mathbb{R}} \RR^n$ is necessarily a closed subspace of $\RR^n$. Assuming this necessary condition, the answer to the question is affirmative. More precise …

[This answer is mostly a long comment to Peter May's answer.] Edit: I have corrected some arrows which were pointing the wrong way.$\newcommand{\real}[1]{\lvert #1\rvert} \newcommand{\Map}{\operatorn …

I have no idea at the moment where to find a reference for the specific result you seek. However, it can be deduced from the following fact: topological manifolds (paracompact and Hausdorff) are absol …

I will present an example involving only (non-compact) manifolds without boundary. As far as I know, the analogous problem for closed manifolds is wide open. Nevertheless, the article Configuration sp …

[Since my comment above appears to have been helpful, I am repeating it here.] I must admit I am unfamiliar with L-theory. Nevertheless, I came across a recent article on the arXiv which is related: …

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 topolog …

[Edit: Allen Hatcher posted an answer while I was writing this one. Both answers seem to use similar ideas. I will leave my answer here anyway.] $\newcommand{\Diff}{\operatorname{Diff}}$$\newcommand{ …

$\newcommand{\ZZ}{\mathbb{Z}}$$\newcommand{\CC}{\mathbb{C}}$A simple counter-example is given by $M = \CC P^3$. Recall first that the cohomology ring of $\CC P^3$ is a truncated polynomial algebra: $ …

I am looking for an example of a well-pointed space in which no (sufficiently small) neighbourhood of the base-point is contractible. As usual, a well-pointed space is a pointed space in which the inc …

What follows is merely a reference to the excellent answer and comment by Karol Szumiło in this mathoverflow question asked by Mike Shulman. There, Karol provides arguments and bibliographic sources w …

[As requested by Vidit Nanda, I am reposting a slightly edited version of my comment above as an answer. Nevertheless, I hope someone will eventually give a satisfactory answer to this question.] The …

$\newcommand{\set}[1]{\lbrace #1 \rbrace}$I will assume that the notation $\Sigma X$ in the question denotes the unreduced suspension of the space $X$. Quick answer: The notion of homotopy equivalenc …