Timeline for Homotopy type of TOP(4)/PL(4)
Current License: CC BY-SA 3.0
13 events
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| Feb 19, 2013 at 1:58 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| Feb 18, 2013 at 22:25 | comment | added | John Klein | @Ricardo: yes, that's what I meant. | |
| Feb 18, 2013 at 21:41 | comment | added | Ricardo Andrade | @Peter: Thank you for remarking that $\mathrm{PL}(n)$ is not defined as a subgroup of $\mathrm{TOP}(n)$. In fact, by $\mathrm{TOP}(n)/\mathrm{PL}(n)$ I meant the homotopy fibre of the map $B \mathrm{PL}(n)\to B \mathrm{TOP}(n)$, which I believe is the usual meaning. I edited the question to clarify that. @John: Are you by any chance referring to Michael Weiss' recent sequence of articles titled "Smooth maps to the plane and Pontryagin classes"? | |
| Feb 18, 2013 at 21:32 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
clarification
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| Feb 18, 2013 at 15:22 | comment | added | John Klein | @Peter: Michael Weiss has made some recent progress on understanding $\text{Top}(n)$. | |
| Feb 18, 2013 at 13:59 | comment | added | Peter May | PL(n) needs a different definition; subspace of Top(n) won't do; see for example 1.10 in Madsen and Milgram, "The classifying spaces for surgery and cobordism of manifolds". It is BTop(n) and BPL(n) that classify the appropriately defined bundles. As far as I know the question asked is still open. In general, BTop(n) and cognate spaces are not well understood; even their cohomology is not known, although the cohomology when one goes to the limit and considers BTop and BPL is completely understood. | |
| Feb 18, 2013 at 6:39 | history | edited | Ricardo Andrade | CC BY-SA 3.0 |
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| Sep 12, 2012 at 19:04 | comment | added | Ricardo Andrade | $\text{TOP}(n)$ is in fact the group of self-homeomorphisms of ${\mathbb R}^n$, as Qfwfq states. | |
| Sep 12, 2012 at 15:35 | comment | added | Sue | @Qfwfq : This is not really my area, but I'm pretty sure that $TOP(n)$ is suppposed to classify topological microbundles on a space. This means that it is something like the classifying space for something like the group of germs of homeomorphisms of $\mathbb{R}^n$ (or maybe the pseudogroup of homeomorphisms between open sets in $\mathbb{R}^n$). Similarly for $PL(n)$. I don't have it at hand, but there are proper definitions in a book by Madsen and Milgram. | |
| Sep 12, 2012 at 9:44 | comment | added | Qfwfq | @Steven Landsburg: From books.google.it/… I gather $TOP(n)$ is the group of self-homeomorphisms of $\mathbb{R}^n$ and $PL(n)$ the subgroup of piecewise linear ones. | |
| Sep 12, 2012 at 6:32 | comment | added | Benjamin Dickman | It was unknown 20 years ago, when this question was proposed as Conjecture 3.10 in 'Differential Topology, Foliations, and Group Actions' by Paul A. Schweitzer. Link: tinyurl.com/cddp8oq | |
| Sep 12, 2012 at 3:20 | comment | added | Steven Landsburg | What is $TOP(n)$? | |
| Sep 12, 2012 at 2:38 | history | asked | Ricardo Andrade | CC BY-SA 3.0 |