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It is well known that the diffeomorphism group of there sphere $\operatorname{Diff}(S^n)$ has the homotopy type of a product $X:=O(n+1)\times \operatorname{Diff}_{\partial D^n}(D^n)$ of the orthogonal …

Here is another obstruction. Suppose $M^n$ is a closed simply connected manifold which admits only trivial vector bundles. Then $M$ cannot be a $\mathbb{Z}/2\mathbb{Z}$-homology sphere, unless $n=3$ …

Suppose $M = S^n$ is a sphere with $n$ odd and at least $5$. Pick your favorite Lie group $G$ for which $\pi_{n-1}(G)$ is non-trivial. (Many examples may be found here. For example, for any $n > 3$ …

As Igor shows, every endomorphism of a simple Lie group $G$ has degree $\in\{0,\pm 1\}$. On the other hand, every compact Lie group admits self maps of other degrees. Namely, the $k$-th power map $g\ …

I'm way late to the party, but with a bit more work, one can understand $SU(4)/SU(2)\times SU(2)$ even more precisely. Namely $SU(4)/SU(2)\times SU(2)$ is diffeomorphic to $T^1 S^5$, the unit tan …

The same technique Allen mentioned also shows that $\mathbb{H}P^{2n}$ doesn't admit any orientation reversing diffeomorphisms. However, it's also true that $\mathbb{H}P^{2n+1}$ doesn't admit any orie …

My question, in its most general form is this: Given a fiber bundle $F\rightarrow E\rightarrow B$, when is there a fiber bundle $B\rightarrow E\rightarrow F$? Here, F,E, and B can lie in whichev …

Even if you ask that $f$ induces trivial maps on all (singular) homology and cohomology groups, there are still easy manifold examples. (This actually arises as an exercise in Hatcher's AT). For ins …

I think (but I'm not sure) that you CAN define characteristic classes from $BSO(n)$ with $\mathbb{Z}$ coefficients, but it's not nearly as easy to work with them or compute them. One of the big probl …

This question is motivated by What manifolds are bounded by RP^odd? (as well as a question a fellow grad student asked me) but I can't seem to generalize any of the provided answers to this setting. …

For 1 and 2, consider the $S^2$ bundles over $S^4$ (with structure group SO(3)). Using clutching functions, one can see that there is a $\mathbb{Z}$s worth of such bundles indexed by, say, k. Using …

I don't know of anything as bare hands as the proof that $\pi_1(G)$ must be abelian, but here's a sketch proof I know (which can be found in Milnor's Morse Theory book. Plus, as an added bonus, one l …

I'm trying to understand Milnor's proof of the existence of exotic 7-spheres. Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be descr …

It's my understanding that the octonians aren't associative enough to have a projective plane in the usual sense. That is, you'd want to define $\mathbb{O}P^{2}$ as the collection of Cayley lines in …

This doesn't directly address your question, but it does give you a way of thinking about torsion in the cohomology of Lie groups in general. (This is all coming from Borel and Serre's Sur certains s …