Timeline for Does every (co)homology functor (in particular, stable homotopy) factor through chain complexes?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 3, 2015 at 20:29 | comment | added | Peter May | There is a long ago published paper to the effect that the only homology theories computable by chain complexes are products of ordinary homology theories, and there is a follow up paper (or two?) by a different author that gives a modified notion of chain complex for which there are more theories, but I can't remember authors or dates. For sure these papers exist though! | |
| Oct 2, 2015 at 13:09 | vote | accept | Tim Campion | ||
| Oct 2, 2015 at 6:38 | answer | added | David C | timeline score: 15 | |
| Oct 2, 2015 at 6:35 | comment | added | Qiaochu Yuan | The relationship between taking homotopy groups of a spectrum and taking homology groups of a chain complex is given by the stable Dold-Kan correspondence, which identifies chain complexes of abelian groups with $H \mathbb{Z}$-module spectra (in a way that sends homology to homotopy). | |
| Oct 2, 2015 at 6:00 | answer | added | Fernando Muro | timeline score: 17 | |
| Oct 2, 2015 at 5:57 | comment | added | Tyler Lawson | The map ${\cal D}(Ch_{\Bbb Z}) \to GrAb$ has a section, sending a graded abelian group to the chain complex with zero differential, and so in a very weak sense the answer to your question is "yes". However, this doesn't respect any of the ambient structure (such as the triangulated structure in the derived category). | |
| Oct 2, 2015 at 3:46 | history | asked | Tim Campion | CC BY-SA 3.0 |