Timeline for Handle decompositions using only 1-handles
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2017 at 6:33 | vote | accept | André Henriques | ||
| Aug 25, 2017 at 17:26 | answer | added | Allen Hatcher | timeline score: 10 | |
| Aug 25, 2017 at 13:33 | comment | added | André Henriques | @Danny Ruebman: It's really the 2nd statement I care about. That statement might be true in higher dimensions too (no need to ever introduce 0-handles and n-handles), but it's really the 2-dim case that I care about. Going to dim $\ge 3$ actually makes things easier. The following preprint has a proof of the result, under the assumption that $\partial_{in}$ and $\partial_{out}$ are connected: math.berkeley.edu/~katrin/papers/cerf.pdf. Unfortunately, that preprint explicitly excludes dimension $2$ (also, I don't want to assume that $\partial_{in}$ and $\partial_{out}$ are connected...) | |
| Aug 25, 2017 at 12:43 | comment | added | Danny Ruberman | If you are looking for a reference for a proof of the first statement, check an exposition of the h-cobordism theorem. Cancelling 0-handles and n-handles (for an n-dimensional cobordism) just relies on the boundary being non-empty. See for instance Rourke-Sanderson, Introduction to Piecewise Linear Topology, Lemma 6.13. It's also in Milnor's Lectures on the h-cobordism theorem. If the second is true in some generality, it might be in a basic text on 1-parameter Morse theory. Eg Cerf, Sur la stratification...s Pub Math IHES 39 (1970). | |
| Aug 25, 2017 at 12:34 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Aug 25, 2017 at 12:21 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Aug 24, 2017 at 14:11 | answer | added | Lee Mosher | timeline score: 4 | |
| Aug 23, 2017 at 14:53 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Aug 23, 2017 at 14:28 | comment | added | André Henriques | @Lee Mosher: Yes, the second $\partial_{in}\Sigma$ was a typo. I've added some clarification about what I mean by a relative handle decomposition (in particular, the situation of the handles with respect to the boundary). | |
| Aug 23, 2017 at 14:25 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Aug 23, 2017 at 14:05 | history | edited | André Henriques | CC BY-SA 3.0 |
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| Aug 23, 2017 at 13:01 | comment | added | Lee Mosher | And is there any requirement about how the 1-handles are situated with respect to the boundary decomposition? | |
| Aug 23, 2017 at 13:00 | comment | added | Lee Mosher | Should one of your $\partial_{in}\Sigma$'s be, perhaps, $\partial_{out}\Sigma$, or something else? | |
| Aug 23, 2017 at 12:44 | history | asked | André Henriques | CC BY-SA 3.0 |