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Here's another way to answer the original question. There is a theorem of Bredon and Kosinski (Annals, 1966) which says that if a manifold $M^n$ is stably parallelizable, then either $M^n$ is parallel …

Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one nonco …

The quotient group $Diff_1(M)/Diff_0(M)$ is a discrete group since $ Diff_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff …

For an interval $[a,b]\subset{\mathbb R}$ in which the height function $f:S\to {\mathbb R}$ has no critical values one obtains a product structure on $f^{-1}([a,b])$ by following flow lines of the gra …

For the first question, which concerns just the smooth category without reference to metrics, the answer depends on the dimension of $M$. Before explaining this a small clarification is needed. By "a …