Timeline for Maslov index equal to $2$ implies that the disk is not multiply covered
Current License: CC BY-SA 4.0
9 events
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| Jul 4, 2021 at 10:38 | comment | added | user174565 | Sorry to bother you , you have been very useful for understanding the missing details of the paper I am trying to read , but I am missing one last one if you could take a look at this question mathoverflow.net/questions/396147/… I would appreciate it. @Jonny Evans | |
| Jul 3, 2021 at 20:52 | history | bounty ended | CommunityBot | ||
| Jul 3, 2021 at 13:56 | comment | added | Jonny Evans | The proof is identical to what you might see in Milnor's "Topology from the differentiable viewpoint": if you have a homotopy which avoids the boundary them the usual argument produces for you a cobordism of intersections whose boundaries are what you started with and what you finished with. If your homotopy crosses the boundary them your cobordism might pick up extra boundary components (intersections escape to the boundary). | |
| Jul 3, 2021 at 13:26 | comment | added | user174565 | Alright thank you. For proving that in fact this is an homotopy invariant on the maps , i.e., if $w$ and $w'$ are homotopic then their intersection number at $p$ is the same we need to use differetial topology techiques similar to what's done for degree right ? I am asking since the degree is defined for compact manifolds without boundary , and here we are working with $D^2$, so I would think some minor changes would be needed. | |
| Jul 3, 2021 at 11:55 | comment | added | Jonny Evans | Yes, the dimensions are complementary (2+0=2) so you get an intersection number by counting points weighted by signs/multiplicities. It's only homotopy invariant (moving the point) if your point stays in the complement of L. | |
| Jul 3, 2021 at 7:27 | comment | added | user174565 | If so I can see why we have positivity since the disk is holomorphic , otherwise I am not sure. | |
| Jul 3, 2021 at 7:19 | vote | accept | CommunityBot | ||
| Jul 3, 2021 at 7:07 | comment | added | user174565 | Thanks for the answer. Could you just be a little more specific by what you mean as the intersection number between $w$ and $p$ using the intersection pairing between $H_2(S^2,L)$ and $H_0(S^2 -L)$? , i.e., how it's defined? Can I think of it as in the context of differential topology? | |
| Jul 2, 2021 at 21:31 | history | answered | Jonny Evans | CC BY-SA 4.0 |