Timeline for Classification of bundles, Postnikov towers, obstruction theory, local coefficients
Current License: CC BY-SA 4.0
8 events
when toggle format | what | by | license | comment | |
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Apr 14, 2019 at 18:33 | comment | added | Charles Rezk | The composite $\def\Aut{\mathrm{Aut}}$ $P_{n-1}\to B\Aut(K(A,n))\to B\Aut(A)$ factors through a map $P_1\to B\Aut(A)$. Since $P_1\approx B\pi_0G$, you can pullback everything along this map. That gives the final square I drew. | |
Apr 14, 2019 at 18:31 | comment | added | Charles Rezk | If a group $K$ acts on itself by left translation, then $K_{hK}\approx *$. In general there's an equivalence of the form $X_{h(K\rtimes H)}\approx (X_{hK})_{hH}$. | |
Apr 14, 2019 at 17:57 | comment | added | Overflowian | @CharlesRezk Thank you very much! Why $K(A,n)_{h\bigl(K(A,n)\rtimes Aut(A)\bigr)} \approx (*)_{h Aut(A)}$ ? I guess it is a formula valid in general for semidirect products but I only know $B(G\rtimes H)$ not the one for $E(B\rtimes H)$. Moreover, why we can consider just $\pi_0G $ instead of the whole $Aut(A)$ and everything still works? | |
Apr 14, 2019 at 16:37 | comment | added | Charles Rezk | Ultimately, I guess the way this is usually understood is in terms of minimal fibrations of simplicial sets: model the fibration as a minimal fibration of Kan complexes, then note that the simplicial automorphism monoid of a minimal model for $K(A,n)$ is isomorphic to a semi-direct product in simplicial groups, e.g., Chapter 25 of math.uchicago.edu/~may/BOOKS/Simp.djvu | |
Apr 14, 2019 at 16:12 | comment | added | Denis Nardin | Unfortunately the fact that they had to write the theory themselves makes me suspect that there's no elementary exposition yet. | |
Apr 14, 2019 at 15:52 | comment | added | Charles Rezk | @DenisNardin Thanks! Though I kinda want a reference appropriate for someone who is just now learning what a Postnikov tower is. | |
Apr 14, 2019 at 15:45 | comment | added | Denis Nardin | The reference I know is the one I gave in my old post, the paper by Blanc, Dwyer and Goerss The realization space of a ∏-algebra. | |
Apr 14, 2019 at 15:38 | history | answered | Charles Rezk | CC BY-SA 4.0 |