Talk:Thom space

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Is it not firstly the bundle of one-point compactifications, though? I don't see how T(p) is defined.

Charles Matthews 09:21, 5 Dec 2004 (UTC)

I think the answer to your question is yes. If E is a k-vector bundle then T(E) is a k-sphere bundle. So we are taking the one-point compactification of each fiber, remembering that we've only added one new point. About T(p): maybe we need more hypotheses for this. Here is my reasoning. Since E is locally compact Hausdorff it is dense in T(E), and since p: E -> B is continuous there is a unique map T(p) extending p to all of T(E). Am I missing something?

Alodyne 23:53, 5 Dec 2004 (UTC)

Bear with me a moment. Take B a circle and E the trivial bundle BxR. Then what we do is take Bx[0,1] inside E; think of this as an annulus. The usual thing is then to say, take both boundary circles of the annulus and identify them to one point P. We can get this in stages: identify (b,0) first with (b,1), and that's a 2-torus T. Then we collapse a circle on T to the point P. No problem about projecting T to B; but when we collapse the circle? There would have to be some symmetry breaking going on.

Charles Matthews 17:02, 7 Dec 2004 (UTC)

You are right and see also the below comment. Alodyne 21:21, 7 Dec 2004 (UTC)

Forgive me if I'm being stupid, but I don't see how T(p) : T(E) → B is a sphere bundle. To which point in the base does the "new point" project? In order to get a sphere bundle it seems like it would have to project everywhere (it lies in the "fiber" above every point). -- Fropuff 19:26, 2004 Dec 7 (UTC)

I am the one who is stupid; you're absolutely right. So can we agree that T(E) is not so much a bundle over anything (not sure why I ever thought so, at this point...)? And that all references to such should be removed? Alodyne 21:21, 7 Dec 2004 (UTC)

OK, I see what my problem was. If we do as Charles has suggested above and first take the bundle of one-point compactifications, we do get a sphere bundle. But the Thom complex is the quotient of the total space of the sphere bundle obtained by gluing all the points at infinity together. I edited the definition to reflect this. Thanks both of you for your helpful and timely corrections!

Alodyne 02:19, 13 Dec 2004 (UTC)

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