Timeline for Which cohomology classes are detected by tori?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Dec 16, 2019 at 11:18 | vote | accept | Jens Reinhold | ||
| Nov 19, 2019 at 18:39 | comment | added | David E Speyer | Basic observations: As in Neil Strickland's answer, I'll work in homology rather than cohomology and use his terms "basically toral" and "toral". We have a map $\lambda : H_n(X) \to \bigwedge^n H_1(X)$, dual to the cup product. If $\alpha \in H_n(X)$ is basically toral, then $\lambda(\alpha)$ is an elementary tensor so, if $\alpha$ is toral, then $\lambda(\alpha)$ is in the span of elementary tensors of $\lambda(H_n(X))$. In particular, if $\lambda$ is injective and its image is disjoint from the elementary tensors, there are no toral classes; this handles the case of curves of genus $\geq 2$. | |
| Nov 19, 2019 at 18:04 | history | became hot network question | |||
| Nov 19, 2019 at 12:19 | answer | added | Neil Strickland | timeline score: 17 | |
| Nov 19, 2019 at 10:18 | comment | added | Jens Reinhold | Ah, thanks, I got this wrong. I have edited the question accordingly. | |
| Nov 19, 2019 at 10:18 | history | edited | Jens Reinhold | CC BY-SA 4.0 |
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| Nov 19, 2019 at 10:10 | comment | added | Neil Strickland | Using the structure of $H^*(T^n)$, it seems to be equivalent to say that $\alpha\in T^*(X)$ iff there is a map $f\colon T^n\to X$ for some $n$ with $f^*(\alpha)\neq 0$. This means that $T^*(X)$ is the complement of an ideal $U^*(X)\leq H^*(X)$, not a subring. | |
| Nov 19, 2019 at 10:02 | history | edited | Jens Reinhold | CC BY-SA 4.0 |
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| Nov 19, 2019 at 9:55 | history | asked | Jens Reinhold | CC BY-SA 4.0 |