Timeline for Criterion for alternation of the linking form
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 10, 2018 at 14:48 | answer | added | Michael Albanese | timeline score: 4 | |
| Feb 9, 2018 at 9:10 | answer | added | Yonatan Harpaz | timeline score: 5 | |
| Feb 7, 2018 at 14:12 | comment | added | Michael Albanese | In particular, spin implies the linking form is alternating, but as your example demonstrates, the converse is not true. | |
| Feb 7, 2018 at 14:11 | comment | added | Michael Albanese | You're absolutely right, I was being sloppy. What I should have said is that $w_2(x) = b(x, x)$ where $b$ is the linking form, $x \in H_2(M; \mathbb{Z})_{\text{tors}}$ and $w_2$ is viewed as a map $H_2(M; \mathbb{Z}) \to \mathbb{Z}_2$ via the isomorphism $H^2(M; \mathbb{Z}_2) \cong \operatorname{Hom}(H_2(M; \mathbb{Z}), \mathbb{Z}_2)$ by the Universal Coefficient Theorem (as $H_1(M; \mathbb{Z})$ is torsion-free, there is not $\operatorname{Ext}$ term). So the correct statement is that $b$ is alternating iff $w_2 : H_2(M; \mathbb{Z})_{\text{tors}} \to \mathbb{Z}_2$ is zero. | |
| Feb 6, 2018 at 23:42 | comment | added | user84144 | Is there something wrong with the following example? Consider $M = \mathbb{CP}^2 \times S^1$. Then $H^3(M; \mathbb{Z}) = \mathbb{Z}$, so the domain of the linking form is actually $0$, which I think qualifies it to be alternating. On the other hand, $w_2(M)$ is the reduction of $c_1(\mathbb{CP}^2)$, which is certainly not $0$. | |
| Feb 6, 2018 at 16:59 | comment | added | Michael Albanese | I think Wall's criterion ($w_2 = 0$) applies whenever $H_1$ is torsion-free, not just when the manifold simply connected. | |
| Jan 31, 2018 at 14:56 | history | asked | user84144 | CC BY-SA 3.0 |