Timeline for Unoriented bordism with twisted orientation
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2022 at 11:30 | comment | added | mjungmath | Ah, so it's only applicable in this special situation. I thought there was some bigger machinery running in the background such as a UCT with local coefficients. Thank you for the quick response! | |
| Jun 23, 2022 at 11:19 | comment | added | Oscar Randal-Williams | @YoungMath: As $\mathbb{R}^d$ is simply-connected, the local coefficient system $\mathcal{O}_{\mathbb{R}^d}$ must in fact be constant, so one can identify it (canonically) with the constant coefficient system given by its stalk at $0 \in \mathbb{R}^d$, which by definition is $H_d(\mathbb{R}^d, \mathbb{R}^d \setminus \{0\};\mathbb{Z})$. One then applies the usual UCT. | |
| Jun 23, 2022 at 9:48 | comment | added | mjungmath | Sorry for excavate this thread, but could you elaborate the step where you use UCT? I couldn't find any evidence that UCT still holds in the case of local coefficients. The literature is unfortunately very sparse on this. | |
| May 8, 2019 at 7:09 | comment | added | Oscar Randal-Williams | From the point of view taken here, I would say that an orientation is a choice of trivialisation of the local system $\mathcal{O}_M$. This is equivalent to a class in $H_d(M; Z)$ which restricts to a generator of the local homology of each point, as this is an invertible global section of $\mathcal{O}_M$. | |
| May 8, 2019 at 5:19 | vote | accept | Yuji Tachikawa | ||
| May 8, 2019 at 2:43 | comment | added | Yuji Tachikawa | Thank you for the clarification. I guess I was confused even in the orientable and oriented case: the orientation is not the choice of the fundamental class (which is a canonical generator of $H_d(M,\mathcal{O}_M)$) but the choice of a generator of $H_d(M,\mathbb{Z})$ (which does not have a canonical choice.) Am I right? | |
| May 7, 2019 at 20:21 | comment | added | Oscar Randal-Williams | @ArunDebray: Well, that is what it should be. But I expect that some might mean something like fixing a basepoint $m \in M$ and considering local systems as $\mathbb{Z}[\pi_1(M,m)]$-modules, where this one is given by the first Stiefel--Whitney class as a $\pi_1(M,m) \to \mathbb{Z}^\times$. | |
| May 7, 2019 at 20:16 | history | edited | Oscar Randal-Williams | CC BY-SA 4.0 |
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| May 7, 2019 at 19:37 | comment | added | Arun Debray | I always thought $\mathbb Z^w$ was defined to be the orientation local system that you called $\mathcal O_M$, not just the isomorphism class of that system. Perhaps I've been sloppy with notation. | |
| May 7, 2019 at 19:17 | history | answered | Oscar Randal-Williams | CC BY-SA 4.0 |