Timeline for Cobordism Theory of Topological Manifolds
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 5, 2022 at 12:52 | comment | added | The Amplitwist |
The bit.ly link in a comment above points to a diagram on tikzcd.yichuanshen.de. (Just in case the link breaks.)
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| Jul 16, 2019 at 3:20 | answer | added | Peter May | timeline score: 8 | |
| Jul 2, 2019 at 2:47 | comment | added | mme | @wonderich I was not aware of the trick to fix this that Moishe pointed out on your MSE post. It seems like if you want to understand that, you should read Thurston's book. But in the end all flavors of "PDiff bordism" will be the same as the flavors of "PL bordism", so depending on your goals this may not be a good use of time. I don't know anywhere the other flavors (spin etc) are worked out if they're not in the Madsen-Milgram book. You might start by browsing the Google scholar citations for that book. | |
| Jun 29, 2019 at 12:57 | comment | added | Robert Furber | Rather than a category, it seems that piecewise differentiable functions form a profunctor/distributor $\mathrm{PL}^{\mathrm{op}} \times \mathrm{DIFF} \rightarrow \mathbf{Set}$, in the sense that if $f$ is PL $W \rightarrow X$, $g$ PDIFF $X \rightarrow Y$, and $h$ is smooth $Y \rightarrow Z$ then $h \circ g \circ f$ is PDIFF $W \rightarrow Z$. | |
| Jun 29, 2019 at 12:54 | comment | added | Robert Furber | According to the book by Thurston cited by the Wikipedia article PDIFF, piecewise differentiable maps $\mathbb{R}^n \rightarrow \mathbb{R}^n$ are not closed under composition, which precludes there being a notion of "piecewise differentiable manifold". This is also noted in this answer by Goodwillie: mathoverflow.net/a/27673/61785 | |
| Jun 29, 2019 at 2:17 | comment | added | wonderich | @Mike Miller, if you know the mistake of Wikipedia page can you edit and point out where/which sentences have mistakes and how to modify them to correct? thank you! | |
| Jun 29, 2019 at 0:29 | comment | added | mme | (1) + (3) See the book by Madsen and Milgram. (2) I believe the wikipedia page is inaccurate, because I do not know a definition of a "piecewise differentiable manifold": the composition of piecewise smooth maps is not necessarily piecewise smooth. (Noted here.) But what confuses me is that if you did read and believe Wikipedia, they explain that "PDIFF manifolds" are the same notion as PL manifolds, which makes it clear that the answer to (2) is the same as the answer to (3). | |
| Jun 28, 2019 at 22:18 | comment | added | Arun Debray | The Manifold Atlas briefly discusses this question: map.mpim-bonn.mpg.de/…. I'd be very surprised if the spin and pin$^\pm$ PL/TOP bordism groups have been computed, but I'm not certain. | |
| Jun 28, 2019 at 22:03 | history | edited | wonderich | CC BY-SA 4.0 |
added 145 characters in body
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| Jun 28, 2019 at 22:03 | comment | added | user101010 | For smooth manifolds, topological concordance implies smooth concordance (see here). Judging from that question, I am guessing the relation is not so well understood. | |
| Jun 28, 2019 at 21:43 | history | asked | wonderich | CC BY-SA 4.0 |