Talk:Cobordism

compactness[edit]

It should be supposed that the three manifolds M,N and W are compact ! Otherwise, any manifold M would be cobordant to the empty manifold via W:=Mx[0,1[, which would not be very interesting... —Preceding unsigned comment added by 162.38.126.112 (talk) 16:06, 6 March 2008 (UTC)[reply]

Pedagogy[edit]

This article seems to be written by an expert in the field, and as far as I can tell makes no mistakes.

But what I feel is almost totally lacking is an attempt to be generous to the reader, drawing them in with simple, clear explanations, at least in the initial part of the article. For example, I would not begin by saying that a cobordism is a "triple of manifolds . . .", but rather by saying that two n-manifolds are cobordant if there is an (n+1)-manifold whose boundary is their disjoint union. (I know this appears later in the article, but first impressions are important.)

I'd also mention a motivation for studying cobordism. (Since classification of smooth manifolds up to diffeomorphism is impossible thanks to Markov and the presentation problem for fundamental groups, cobordism is a coarser classification that is feasible.)

And why not mention the upshot of Thom's calculation, at least for the unoriented case.

The article should attract at least interested undergraduate math majors. In my opinion, as written it would repel them.Daqu (talk) 18:05, 2 February 2009 (UTC)[reply]

Weak and strong cobordism[edit]

I have not come across this terminology before: is there a reference? Unless I am misunderstanding the definition, oriented cobordism is strong, not weak as claimed. ranicki (talk) 14:12, 24 July 2009 (UTC)[reply]

Oops! My mistake – fixed!

—Nils von Barth (nbarth) (talk) 21:13, 26 July 2009 (UTC)[reply]

Definitions[edit]

I've just made the manifolds closed, but closed manifold says no boundary. I think we should check whether that article has a definition that is widespread ... Charles Matthews (talk) 16:40, 29 July 2009 (UTC)[reply]

The definition in the article is the one for cobordism of closed manifolds, which is the most basic and widespread. There's also a notion of cobordism between compact manifolds (ie: the manifolds which are cobordant are allowed to have boundary) but IMO it doesn't make sense to put it in the lead to the article as there are distracting technical issues -- do you consider the cobordism to be a manifold with corners? etc. Rybu (talk) 17:37, 4 August 2009 (UTC)[reply]

Problem with picture[edit]

I cannot quite make sense of the picture illustrating our article. What are M, N and W? The caption on Commons says "A sample cobordism, between a sphere and a torus." If I wanted to produce a cobordism between a sphere and a torus, I would simply take W to be the disjoint union of a solid ball and a solid torus, but clearly that's not what is pictured here. AxelBoldt (talk) 21:51, 13 February 2013 (UTC)[reply]

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Useful results needed[edit]

This page should discuss some of the results in https://web.archive.org/web/20170815201417/http://www.math.uiuc.edu/~cmalkiew/cobordism.pdf — Preceding unsigned comment added by 161.98.8.2 (talk) 20:16, 15 August 2017 (UTC)[reply]

Extended Cobordism[edit]

A section might give an introduction to the (infinity,n)-category of extended (fully local) Cobordism as explained in detail by Hopkins and Lurie. — Preceding unsigned comment added by 2A02:8109:8680:3241:45C4:83AF:6359:6AA9 (talk) 20:04, 17 January 2021 (UTC)[reply]