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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 …

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 …

$\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: $ …

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

[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 …

$\newcommand{\Ext}{\operatorname{Ext}}$$\newcommand{\To}{\longrightarrow}$$\newcommand{\dash}{\text{-}}$$\newcommand{\sSet}{\mathrm{sSet}}$$\newcommand{\ZZ}{\mathbb{Z}}$For convenience, I will denote …

$\newcommand{\into}{\hookrightarrow}$It seems that if $Z$ has the indiscrete topology, then the evaluation map $ev_0 : Z^I \to Z$ has the right lifting property with respect to any map. That provides …

[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{ …

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 …

$\newcommand{\RR}{\mathbb{R}} \newcommand{\To}{\longrightarrow} \newcommand{\id}{\mathrm{id}}$The example described in Tom Goodwillie's answer to a related mathoverflow question essentially solves thi …

[Edit: I have added some details and a more explicit example by Milnor.] I will present a couple of examples verifying the conditions required in the question.$\newcommand{\RR}{\mathbb{R}} \newcomman …