Timeline for Atiyah duality without reference to an embedding
Current License: CC BY-SA 4.0
12 events
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| Jul 25, 2021 at 16:48 | comment | added | John Klein | @ConnorMalin yes, If M is compact, framed and with boundary, then stable sections with compact support coincides with the function space of stable maps $M/\partial M \to S%n$. Taking the adjoint the construction yields a stable duality map $M_+ \wedge M/\partial M \to S^n$. | |
| Jul 23, 2021 at 16:00 | comment | added | John Klein | like I wrote already, I don't have a reference, but I think the scanning result is due to Segal. I have my own approach to these problems which I wrote about 20 years ago. But my approach is different. | |
| Jul 23, 2021 at 15:52 | comment | added | Connor Malin | Well we can always pass to the interior of our manifold with boundary in that case. Would you happen to have a reference that treats the noncompact case? | |
| Jul 23, 2021 at 15:39 | comment | added | John Klein | No. I think not--that won't work. I think it's better to think of it as a map $M_+ \to \Gamma_{\text{cs}}(\tau_+)$, where the target is sections with compact support. This works for any finite dimensional manifold without boundary, not necessarily compact. In fact, you need this version of the map to prove the scanning result by induction, since locally any manifold looks like Euclidean space. | |
| Jul 23, 2021 at 15:35 | comment | added | Connor Malin | Thank you for the answer; I was wondering if your map $M_+ \rightarrow \Gamma(\tau^+)$ can be adapted to a map $M/\partial M \rightarrow \Gamma(\tau^+)$ if $M$ has a boundary. I am worried about continuity issues if one just naively extends by sending everything to the point at infinity. | |
| Jul 23, 2021 at 15:13 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 14:42 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 13:50 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 13:36 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 13:19 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 13:12 | history | edited | John Klein | CC BY-SA 4.0 |
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| Jul 23, 2021 at 13:02 | history | answered | John Klein | CC BY-SA 4.0 |