Talk:Thom space

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)

The Thom isomorphism.

The article attempts to explain the meaning of the Thom isomorphism as follows:

We can loosely interpret the theorem in the following geometric sense. Since E is a vector bundle it retracts onto the base B. So we might suppose that E would be cohomologically equivalent to B. In a way, the theorem bears out this expectation

This is not correct. In a vector bundle the total space $E$ is homotopy equivalent to the base space $B$, so these two spaces always have isomorphic cohomology groups. The Thom isomorphism, on the other hand, can be interpreted as a generalization of the suspension isomorphism. For a space $B$ the the classical suspension isomorphisms is the isomorphisms $H^{i}(B)\cong {\tilde {H}}^{n+1}(\Sigma ^{n}B)$ where $\Sigma ^{n}B$ is the $n$-fold suspension of $B$. For an $n$-dimensional vector bundle $E\to B$ its Thom space can be understood as an $n$-fold suspension of $B$, which is however twisted by the structure of the bundle. The Thom isomorphism says that if the bundle is nice enough the twist is not seen by the cohomology theory and the suspension isomorphism still holds. The meaning of "nice" depends here on the cohomology theory at hand. For the singular cohomology with ${\mathbb {Z} }/2$-coefficients every vector bundle is "nice", for the integral cohomology theory one needs to assume that bundles are orientable etc. --69.204.54.113 (talk) 05:28, 22 August 2008 (UTC)

Funnily enough, I was just reading it now, came across that passage, and wondered, what on earth? Please include your comments into the article itself. There is no need to be timid. --C S (talk) 12:24, 23 August 2008 (UTC)

Base points

In the name of precision, I just want to point out that the Thom space of the trival bundle over B is the suspension of B+, B plus a disjoint base-point. I added this to the article. Djcrowley 15:20 6th May 2009 —Preceding undated comment added 13:19, 6 May 2009 (UTC).

Relation between Thom class and Euler class

The pullback of Thom class by zero section is Euler class.Yunzhi (talk) 13:32, 9 October 2009 (UTC)

Notational problem

Not such a great idea to use the same notation for the Thom space — T(M) — that is used for the tangent bundle of the manifold M. Especially in an article like this one where there is constant danger of confusion. Why not Th(M) or Thom(M) or something?2600:1700:E1C0:F340:24A9:18FE:5F8:B9F8 (talk) 22:24, 25 September 2019 (UTC)

I agree and will go ahead and change the notation. —- Taku (talk) 23:22, 26 September 2019 (UTC)