Timeline for Landweber Exact Functor Theorem for Cohomology
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 27, 2017 at 17:49 | comment | added | Denis Nardin | You can in fact get $H^*(X;\mathbb{Q})$ from $MU$ but it is more complicated and I do not know of a nice formula from it (it's some kind of limit over the values on the finite complexes, but it is complicated by the existence of $lim^1$ terms). However you can still see the whole theory of Chern classes, at least for finite complexes, so I do not understand your objection (hint: $H^*(BU;\mathbb{Q})=\lim_{n,k} H^*(Gr_k(\mathbb{C}^n);\mathbb{Q})$) | |
| Nov 27, 2017 at 17:36 | comment | added | Rene Recktenwald | Ups my bad :) So we can get $H^*(X;\mathbb{Q})$ but only for finite CW-Complexes? In particular I don't see any of the theory of chern classes here? This is confusing to me, because I would start with the Chern classes to make $H^*(X;\mathbb{Q})$ into a $MU^*$-module in the first place. | |
| Nov 27, 2017 at 17:26 | comment | added | Denis Nardin | I did say cohomological LEFT :). For finite spectra you just precompose with SW duality, and you get that $E^*X=MU^*X\otimes_{MU_*}E_*$. For non-finite spectra this breaks down since cofiltered limits do not commute with the tensor product (as you noticed) | |
| Nov 27, 2017 at 17:13 | comment | added | Rene Recktenwald | I am aware that they are coming from spectra, but I am not sure how to translate one into the other. Landweber writes in the original paper that it gives a homology theory on all CW-spectra and Rudyak does the same. So there are versions with non-finite complexes. Also surely one would want to be able to talk about $\mathbb{C} P^\infty$ which is an infinite complex | |
| Nov 27, 2017 at 17:09 | comment | added | Denis Nardin | In abelian groups finite products and finite coproducts are the same. There is a reason why the cohomological LEFT is always stated only for finite complexes (also, from your question seems that you're unaware that a homology theory and a cohomology theory are essentially the same amount of data, so retrieving one and retrieving the other are essentially the same thing) | |
| Nov 27, 2017 at 16:26 | history | asked | Rene Recktenwald | CC BY-SA 3.0 |