Timeline for Examples of non-simply connected manifolds with trivial H^1
Current License: CC BY-SA 3.0
11 events
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| Jul 9, 2011 at 20:47 | comment | added | J.C. Ottem | Some of these answers have $H_1=0$ also. | |
| Jul 9, 2011 at 20:18 | comment | added | user16335 | Ops, I have just seen the mistake in the title. Sorry, in fact I mean H_1 and not H^1... . I would like to thank you all for your quick answers and useful examples. | |
| Jul 8, 2011 at 9:48 | comment | added | Tom Mrowka | To explain Joel Fine's answer. By transversality you can make a loop in the regular neighborhood miss the 2-complex and hence a smaller regular neighborhood. Then the loop is homotopic into the boundary. If a loop in the boundary is null homotopic in the regular neighborhood the disk it bounds in the regular neighbor can similarly be made to miss the 2-complex and hence is homotopic into the boundary. Thus the two complex and the boundary of the regular neighborhood have the same fundamental group. – Tom Mrowka 11 hours ago | |
| Jul 7, 2011 at 19:19 | comment | added | Neil Strickland | @Joel: why is the fundamental group of the boundary of a regular neighbourhood the same as that of the original complex? | |
| Jul 7, 2011 at 14:27 | comment | added | Joel Fine | You've almost answered your own question. You can realise any finitely presented group as the fundamental group of a closed 4-manifold. You take your 2-dimensional CW complex, embed it in R^5 (dim 5 is necessary so you can separate all the 2-cells) and then take the boundary of a tubular neighbourhood (and perhaps smooth off any corners). The result is a closed 4-manifold with the same fundamental group as the CW complex, unless I'm mistaken. | |
| Jul 7, 2011 at 14:12 | answer | added | J.C. Ottem | timeline score: 9 | |
| Jul 7, 2011 at 12:26 | answer | added | Neil Strickland | timeline score: 5 | |
| Jul 7, 2011 at 12:16 | answer | added | Francesco Polizzi | timeline score: 4 | |
| Jul 7, 2011 at 12:08 | history | edited | agt |
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| Jul 7, 2011 at 12:06 | answer | added | Matthew Kahle | timeline score: 12 | |
| Jul 7, 2011 at 11:39 | history | asked | Luciano Mari | CC BY-SA 3.0 |