Timeline for Is there a local simplicial formula for the Steenrod squares which commutes with the derivative on cochain level?
Current License: CC BY-SA 4.0
9 events
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| Jan 26, 2022 at 17:29 | comment | added | Denis Nardin | @AndiBauer Yes, the (∞-)category of spectra is symmetric monoidal and there is a lax symmetric monoidal functor $H$ from abelian groups to spectra sending $A$ to $HA$ (the spectrum representing ordinary homology with coefficients in $A$). But giving the whole story of stable homotopy theory is going to be hard in the comment boxes :). | |
| Jan 26, 2022 at 16:47 | comment | added | Andi Bauer | @DenisNardin: Stupid question again, but I thought $H\mathbb{Z}$ is a sequence of topological spaces. How does it define a ring, and how does it act on $H\mathbb{Z}_2$? Does $\mathbb{Z}_2$ being a $\mathbb{Z}$-module somehow lift to the corresponding Eilenberg-MacLane spectra? | |
| Jan 26, 2022 at 16:19 | comment | added | Denis Nardin | @AndiBauer Every stable cohomology operation $H^*(-;\mathbb{Z}/2)\to H^{*+k}(-;\mathbb{Z}/2)$ is induced by a map of spectra $H\mathbb{Z}/2\to \Sigma^k H\mathbb{Z}/2$. If the cohomology operation comes from a natural transformation of chain complexes this map is $H\mathbb{Z}$-linear for the standard $H\mathbb{Z}$-module structure on the spectrum $H\mathbb{Z}/2$. However, it is known that $\operatorname{Sq}^i$ is not $H\mathbb{Z}$-linear for $i>1$. | |
| Jan 26, 2022 at 15:47 | comment | added | Andi Bauer | @DenisNardin: Excuse the basic question, but by "$H\mathbb{Z}$-modules", do you mean the cohomology groups as $\mathbb{Z}$-modules? I'm a bit confused, I thought $Sq^i$ is a group homomorphism $H^k(X, \mathbb{Z_2})\rightarrow H^{k+i}(X, \mathbb{Z_2})$, and a group homomorphism is also a homomorphism of the corresponding $\mathbb{Z}$-modules? | |
| Jan 21, 2022 at 12:37 | comment | added | Denis Nardin | I am skeptical such a thing can exist. It is known that $\operatorname{Sq}^i$ is not a map of $H\mathbb{Z}$-modules for $i>1$, and this precludes it being given by a map of chain complexes. I guess there could be a non-additive map that commutes with the differential, but it feels unnatural... | |
| Jan 21, 2022 at 9:56 | history | edited | Andi Bauer | CC BY-SA 4.0 |
typo x instead of A; edited body
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| Jan 21, 2022 at 9:12 | comment | added | Andi Bauer | @TimCampion Thanks for the references! The second one explictly says they are using the original definitions via higher cup products. The first one sounds a bit like they are not considering different formulas for $p=2$ but only extending them to $p\neq 2$, but I'll have a closer look. | |
| Jan 20, 2022 at 23:23 | comment | added | Tim Campion | I'm not sure whether they achieve your precise conditions, but some of Anibal Medina and collaborators' work may be relevant, including this and this. | |
| Jan 20, 2022 at 23:16 | history | asked | Andi Bauer | CC BY-SA 4.0 |