Timeline for Finding cocycles that square to zero
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
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| Jul 11, 2021 at 3:31 | comment | added | Fernando Muro | However, the comment and answers of @mme are probably more useful to you since in your question you only mention DGAs, so maybe you don't care about the E-infinity structure. Actually, that answer is complete, I think you should accept it. | |
| Jul 11, 2021 at 3:30 | comment | added | Fernando Muro | Moreover, if there is a chain e-infinity operad whose category of algebras happens to be left proper, then it is equivalent to ask whether the sub-E-infinity algebra of $a$ generated by a representative of $x$ is rectifiable. if there is not, this is weaker. all this implies that all steenrod operations, massey products etc. vanish on $x$, and in general all primary, secondary and higher order operations on $x$, but even that wouldn't suffice. | |
| Jul 11, 2021 at 3:28 | comment | added | Fernando Muro | Since cochains are an E-infinity algebra, $x$ has odd degree and you're happy about quasi-iso replacements, your question is kind of equivalent to the following one: given an E-infinity álgebra $A$ and an odd cycle $x\in H_*(A)$, is there a quasi-isomorphic E-Infinity algebra $B$ such that $x$ is represented by a cochain $y\in B$ such that the sub-E-infinity algebra generated by $y$ is strictly commutative? | |
| Jul 10, 2021 at 18:56 | answer | added | mme | timeline score: 5 | |
| May 11, 2021 at 2:25 | comment | added | mme | If you are willing to weaken your ask to "There exists an $A_\infty$-homomorphism from the exterior algebra on a degree 3 generator $f: \Bbb Z[3] \to C^*(X;\Bbb Z)$ with $[f(1)] = x_3$ for $[x_3]$ a chosen cohomology class", then this does exist for $SU(n)$. The data of an $A_\infty$-homomorphism is more or less precisely what the vanishing of the higher "Massey powers" guarantees: first an element $x_5 \in C^*$ with $dx_5 = x_3^2$, then an element $x_7 \in C^*$ with $dx_7 = x_3 x_5 + x_5 x_3$... You probably already know all of this but perhaps this weaker answer is enough for your needs. | |
| Jan 7, 2010 at 14:04 | comment | added | Tim Perutz | No, I haven't - only vanishing of the Massey powers. A nice test example would be the space $SU(N)$, whose cohomology is an exterior algebra on generators in degrees $3,5,\dots,2N-1$. A surprising little paper of Karoubi ("Stabilizing and commuting cochains", C. R. Acad. Sci. Paris Ser. I Math. 333 (2001), no. 8, 769-771) shows that there's a functorial DGA model for cohomology in which one can find commuting representatives for a countable set of commuting cohomology classes. | |
| Jan 3, 2010 at 23:21 | comment | added | Mariano Suárez-Álvarez | Tim, have you found more stringent necessary conditions? I'd love to know... | |
| Dec 12, 2009 at 19:02 | answer | added | Mariano Suárez-Álvarez | timeline score: 5 | |
| Dec 12, 2009 at 18:44 | history | asked | Tim Perutz | CC BY-SA 2.5 |