tea.mathoverflow.net - Discussion Feed (Would some differential geometers/surgery theorists check over my math.SE answer?)

David Speyer comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14420)

David Speyer — Fri, 06 May 2011 07:38:16 -0700

Thank you to everyone who looked at this!

Will Jagy comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14416)

Will Jagy — Thu, 05 May 2011 16:58:15 -0700

I spoke with Robert Bryant for a while today, he agreed that Dick Palais would know. I showed him this thread and the original question on M.SE. He went through the answers on MSE (prior to Palais) and agreed with a number of points. The final idea he had was a five-dimensional example, due largely to Gompf, where the fibers were homeomorphic but not diffeomorphic. I suggested he register on Meta and put an answer here. I don't think he will do this today or even tomorrow. I also explained that there was no knowing the background of an anonymous user on MO or MSE.

deane.yang comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14415)

deane.yang — Thu, 05 May 2011 15:50:51 -0700

Dick Palais has posted an answer on math.stackexchange.com

deane.yang comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14413)

deane.yang — Thu, 05 May 2011 12:39:03 -0700

Sorry, but this is beyond my expertise (but I am beyond flattered that Will mentioned me first). The answer certainly looks plausible to me. Palais would definitely know. Is there perhaps a way to do this using the tangent spaces of the fibers, perhaps also using a Riemannian metric?

Andrew Stacey comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14407)

Andrew Stacey — Wed, 04 May 2011 23:57:08 -0700

Looks good to me (and I just left a comment there saying so).

Will Jagy comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14406)

Will Jagy — Wed, 04 May 2011 20:25:50 -0700

For what it's worth, I think Deane Yang reads meta, but I think Dick Palais and Robert Bryant may not. So maybe making a question on main MO is the way to go.

David Speyer comments on "Would some differential geometers/surgery theorists check over my math.SE answer?" (14405)

David Speyer — Wed, 04 May 2011 19:41:16 -0700

Hi all. MBL asked a question on math.SE about why they were having trouble finding things that looked like vector bundles but weren't. See http://math.stackexchange.com/questions/36793/an-example-of-a-triple-e-pi-m-which-is-not-a-vector-bundle/37086#37086 . His or her question was phrased imperfectly, but it seemed to me like a large part of what he or she was seeing is that, if you have a map of smooth manifolds pi: E --> M which is locally diffeomorphic to U \times R^n \to U, then it automatically is diffeomorphic to a vector bundle. As I understand it, this is true, and the reason is that the Diffeomorphism group of R^n is homotopy equivalent to GL_n(R).

This is pretty far from my strength, so I'd rather that a real expert on this sort of thing checked over my answer. But I wasn't sure that any would show up on math.SE, so I wrote one out.

Would it be acceptable to post a request for such an expert on MO? Or is there an expert already reading this post who would look it over?

On a side note, what is the correct name for the field of mathematics that studies the homotopy types of diffeomorphism/PL-automorphism/homomorphism groups of manifolds? As you can see from the title of this thread, I was not sure whom to address this to.