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Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

10 votes

Classification problem for non-compact manifolds

I just wanted to mention that the fact that the word problem is "hard" for groups doesn't mean that there can be no classification for manifolds of dimension $\geq$ 4. The problem is that the word "h …
Jason DeVito - on hiatus's user avatar
8 votes

When does a submersion have connected fibers?

If $M$ and $N$ are both compact, then the submersion $F$ can be thought of as a fiber bundle map with fiber $F^{-1}(p)$ for any $p\in N$. Then one can apply the long exact sequence of homotopy groups …
Jason DeVito - on hiatus's user avatar
24 votes
1 answer
3k views

Characteristic classes of sphere bundles over spheres in terms of clutching functions

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 …
Jason DeVito - on hiatus's user avatar