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3
votes
0answers
99 views

Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose negative answer I showed by a dimension-increasing map.) Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...
2
votes
0answers
225 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed 4-manifolds. Are there any closed 4-manifolds with uncountably many ...
17
votes
0answers
457 views

Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds? Related: if such a ...
8
votes
1answer
344 views

Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...
11
votes
1answer
535 views

Are there non-compact, non-smoothable manifolds?

There do exist manifolds which do not admit any smooth structure at all. But the only examples I've heard of are all compact. Are there any non-compact, non-smoothable manifolds?
4
votes
1answer
184 views

smooth homotopy on exotic R^4

Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric. Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to ...
8
votes
5answers
937 views

Is there a relation between 4-dimentional general relativity and exotic smooth structures on $\mathbb{R}^4$?

Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemanniann manifolds of signature $(n-1,1)$. I've heard more than once people say that ...