Timeline for Special Case of the Toral Rank/Halperin-Carlsson Conjecture
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 29, 2015 at 9:28 | history | edited | Mark Grant | CC BY-SA 3.0 |
added comments from Greg Lupton and link to notes of Vicente Munoz
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| Jun 29, 2015 at 8:02 | comment | added | Mark Grant | BTW, I'm not really an expert in the TRC, but I know a couple...I'll email them to alert them of your question. | |
| Jun 29, 2015 at 8:01 | comment | added | Mark Grant | Looking at the proof of 7.17, it seems that if $X'$ can be chosen to be a compact manifold, then the action is smooth and free by construction ($X'$ is the total space of a smooth principal $T^n$-bundle). However, I'm not sure we can always find a manifold in the rational homotopy type. There is a well-developed rational surgery theory for answering this type of question (see Theorem 3.2 in the book, or here: mathoverflow.net/questions/115911/…) | |
| Jun 29, 2015 at 3:47 | comment | added | anon271 | Basically, suppose that I can show: if $X$ is a compact manifold with a smooth, free action of $T^n$, then the sum of the Betti numbers of $X$ is at least $2^n$. Will this suffice to show the general conjecture? Is it considered to be an interesting case? (Sorry for the repeated questions!) | |
| Jun 29, 2015 at 3:39 | comment | added | anon271 | Thanks for the enlightening answer! I have looked briefly at the book, although unfortunately I am completely new to minimal models. I had a question about Proposition 7.17, however. The proposition mentions being able to choose X' such that it is a compact manifold in certain cases. Am I correct in understanding that the free torus action in these instances can be chosen to be smooth as well as free? (After all, the original torus action is only a continuous action.) I looked for the reference provided in the proposition but found nothing. | |
| Jun 20, 2015 at 17:00 | vote | accept | anon271 | ||
| Jun 19, 2015 at 7:40 | history | answered | Mark Grant | CC BY-SA 3.0 |