Timeline for Homotopy type of transversal families of submanifolds through deformation
Current License: CC BY-SA 4.0
6 events
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| May 31, 2019 at 7:49 | history | edited | BrianT | CC BY-SA 4.0 |
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| May 31, 2019 at 7:08 | comment | added | BrianT | Thank you @Mike Miller. I will need to look at Ehresmann's fibration lemma, but the idea would be to write down explicitly a deformation, (for instance by means of the gradient flow of $f_s$ in point $2$. above). Moreover, it seems that for the point $1$. above, if we are in the case where $B_s = \rho^{-1}(s)$ is a family of regular level sets of some smooth function, then one can also perform gradient flow deformations to construct an isotopy. I will edit my question with these observations, and I will be happy to get your feedback in this regard. | |
| May 31, 2019 at 0:35 | review | Close votes | |||
| Jun 14, 2019 at 3:06 | |||||
| May 31, 2019 at 0:17 | comment | added | mme | $A$ and $B$ assumed compact. (In what follows $t$ will denote a generic element of the interval, and $s$ a specific one). The situation you have is that there is a 'parameterized intersection' $A_t \cap B_t \subset M \times [0,1]$. The fact that this is a submanifold of $M \times [0,1]$ follows because the two submanifolds $A_t \subset M \times [0,1]$ and $B_t \subset M \times [0,1]$ are transverse; furthermore the projection $A_t \cap B_t \to [0,1]$ is a submersion because each $A_s$ and $B_s$ intersect each other transversely. By Ehresmann's fibration lemma every fiber is diffeomorphic. | |
| May 30, 2019 at 21:36 | history | edited | Mike Shulman |
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| May 30, 2019 at 17:39 | history | asked | BrianT | CC BY-SA 4.0 |