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It is a theorem of A. Gramain from 1973 (Annales Sci. E.N.S.) that the diffeomorphism group of the projective plane has the homotopy type of $SO(3)$, the subgroup of isometries of the standard constan …

The special feature of $X$, a sphere with three or more punctures, that is being used here is that the space $E(X)$ of all homotopy equivalences $X\to X$ has $\pi_1 E(X)=0$. (Here we take the identity …

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 …

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 …

I wonder whether Igor Pak's "Lectures on Discrete and Polyhedral Geometry" might be appropriate as a textbook for an undergraduate geometry course. This is still in preliminary form, available on his …

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 …

The result you are looking for is Theorem 4.18 in "An Introduction to Morse Theory" by Yukio Matsumoto, published by AMS in 2002 (translated from Japanese). The connections between Morse functions, ha …