Timeline for Does the Gysin map in $K$-theory respect bordism?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Sep 4, 2020 at 4:33 | vote | accept | geometricK | ||
| Aug 20, 2020 at 22:43 | answer | added | Bad English | timeline score: 2 | |
| Aug 18, 2020 at 2:11 | comment | added | geometricK | @PaulSiegel Could you elaborate how the Gysin maps factor through the Mayer-Vietoris boundary maps on $W'$ and why this implies the result in $K$-homology? I would like to adapt your argument to $K$-theory. | |
| Aug 17, 2020 at 1:33 | history | edited | geometricK | CC BY-SA 4.0 |
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| Aug 16, 2020 at 20:48 | history | edited | geometricK | CC BY-SA 4.0 |
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| Aug 16, 2020 at 20:09 | answer | added | Nicholas Kuhn | timeline score: 4 | |
| Aug 16, 2020 at 17:40 | comment | added | Paul Siegel | I know how to do this in K-homology, which gives me confidence that it also works for K-theory, but I haven't worked it out. In K-homology you can attach cylindrical ends $X_1 \times [0, \infty)$ and $X_2 \times [0, \infty)$ to $W$ and argue that the Gysin maps factor through the Mayer-Vietoris boundary maps on the new object $W'$. You then lift the K-homology class on W to a class in a certain K-homology-like group for W'; this requires care since W' is non-compact. (One way to do it involves C*-algebras and coarse geometry.) Functoriality of Mayer-Vietoris completes the proof. | |
| Aug 16, 2020 at 13:52 | history | asked | geometricK | CC BY-SA 4.0 |