Timeline for Applications of the Dold-Kan correspondence
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 28, 2022 at 3:01 | comment | added | Dmitri Pavlov | @Dora: So, to be clear, “all of this” is not proved in Hatcher: Hatcher does not construct functorial Eilenberg–MacLane spaces. He also does not construct an explicit natural isomorphism between simplicial chains and maps to the Eilenberg–MacLane space. | |
| Dec 28, 2022 at 2:50 | comment | added | Dmitri Pavlov | @Dora: One advantage is that the proofs are much shorter. In my answer, I gave a construction of (generalized) Eilenberg–MacLane spaces, proved that singular cohomology is representable, and proved that cohomology operations are classified by maps between Eilenberg–MacLane spaces. All of this is proved by Hatcher, but (say) the construction of Eilenberg–MacLane spaces alone occupies 3 pages (365–367), and it's not even functorial. Many more pages are used to show representability of cohomology etc. And the other applications are not present in Hatcher at all. | |
| Dec 28, 2022 at 2:44 | comment | added | Dmitri Pavlov | @Z.M: Yes, I should have remembered this, since it follows from a theorem in one of my papers. It's E_∞ on both sides of the correspondence. | |
| Dec 28, 2022 at 2:43 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
edited body
|
| Dec 28, 2022 at 2:17 | comment | added | Z. M | The last claim seems to be false — simplicial commutative $k$-algebras are not equivalent to $E_\infty$-$k$-algebras. The free objects in the former are given by polynomial algebras, but the free objects in the later are given by $\bigoplus_{n\in\mathbb N}k[S^{\times n}]_{h\Sigma_n}$, which has nontrivial group homology in positive characteristic. | |
| Dec 28, 2022 at 2:10 | comment | added | Dora | simplify the computation of cohomology operations (say, that the stable ones are given by reduced power operations?). That is still kind of a painful computation, at least as I understand it. | |
| Dec 28, 2022 at 2:10 | comment | added | Dora | What is the advantage of using DK to handle cohomology operations? The results in your first few paragraphs are all in eg Hatcher, and the proofs are not hard. My favorite approach to representability is to use Brown to see that they are representable, and then evaluate on spheres to see that the representing spaces are Eilenberg-MacLane spaces. Once you have that, it’s immediate that cohomology operations are in bijection with appropriate cohomology groups of Eilenberg MacLane spaces. Do the models from DK maybe (continued) | |
| Dec 28, 2022 at 1:16 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 920 characters in body
|
| Dec 28, 2022 at 1:10 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |