Timeline for Mayer-Vietoris sequence in group cohomology for arbitrary pushout squares of groups?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2022 at 5:50 | comment | added | Marc Hoyois | It's a classical result that the classifying space functor B induces an equivalence between ∞-groups and pointed connected spaces (the inverse is the loop space). A modern reference is Theorem 5.2.6.10 in Lurie's Higher Algebra. It follows that the classifying space functor from ∞-groups to spaces commutes with pushouts, since pushouts preserve pointed connected spaces. | |
| Aug 25, 2022 at 21:25 | comment | added | ಠ_ಠ | @MarcHoyois Thanks! Do you happen to know of a reference where this is discussed? | |
| Aug 25, 2022 at 16:17 | comment | added | Marc Hoyois | You always get a Mayer-Vietoris sequence if you compute the pushout in ∞-groups/E_1-groups: the square of classifying spaces is then a pushout square. Hence you get Mayer-Vietoris for the pushout in groups whenever the pushout in ∞-groups is again a group (i.e., 0-truncated). This is true if both maps are injective, but that's not necessary. | |
| Aug 25, 2022 at 13:27 | answer | added | Donu Arapura | timeline score: 3 | |
| Aug 25, 2022 at 12:36 | history | asked | ಠ_ಠ | CC BY-SA 4.0 |