Timeline for CW-structures and Morse functions: a reference request
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Sep 7, 2016 at 17:28 | comment | added | Ryan Budney | @algori: yes, it's the same, at least, up to a small chain homotopy. | |
| Jan 11, 2010 at 0:56 | comment | added | algori | Thanks again, Ryan! No, this was out of pure curiosity, I don't have any serious reasons to consider CW-structures rather than handle decompositions. If one performs the construction you described, does one get the same cellular chain complex as in the procedure described in Milnor's book? | |
| Jan 11, 2010 at 0:33 | comment | added | Ryan Budney | One way would be to follow the outline in my last paragraph, starting with "edit:". I imagine this has been discussed in the literature but I'm not sure where it would be. CW-structures on smooth manifolds aren't nearly as interesting or useful as handle decompositions so I think most people are happy to throw away CW-complexes for handle decompositions. Is this mostly a foundational issue for you or do you have a reason to favour CW-decompositions over handle decompositions of smooth manifolds? | |
| Jan 11, 2010 at 0:20 | comment | added | algori | Ryan, thanks! Yes, a metric is necessary. Will fix that. As to CW-structures on manifolds: how does one prove they exist without using, say, triangulations? I always assumed this can be done using Morse theory in some way ot another, but never bothered to find out, until now. As to how much tweaking should be allowed: I'm not sure myself; I was sort of hoping this had been discussed in the literature. | |
| Jan 10, 2010 at 23:48 | history | edited | Ryan Budney | CC BY-SA 2.5 |
add some further comments
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| Jan 10, 2010 at 23:29 | history | answered | Ryan Budney | CC BY-SA 2.5 |