Timeline for $\mathrm{String}/\mathbb{CP}^{\infty}=\mathrm{Spin}$ or a correction to this quotient group relation
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2023 at 18:39 | comment | added | Dmitri Pavlov | @SebastianGoette: I am not aware of any written accounts. Here is my guess how it could work. Working in the setting of presheaves of simplicial groups on the site of smooth manifolds, we can construct String(H) as an extension of Spin by the smooth ∞-group U(H)//U(1), where // denotes the stacky quotient (i.e., homotopy colimit). Then we can take π_0 of the resulting fiber sequence of presheaves of simplicial groups, which produces an exact sequence PU(H)→π_0(String(H))→Spin of sheaves of groups on the site of smooth manifolds. | |
| Nov 24, 2023 at 13:33 | comment | added | Konrad Waldorf | @SebastianGoette: not that I know. The group $Gau(P)$ from my answer below maps via evaluation at some point of $P$ to $PU(H)$, establishing the homotopy equivalence. The total space of $P$ is homotopy equivalent to $String$ but does not seem to allow a group structure. | |
| Nov 24, 2023 at 8:55 | comment | added | Sebastian Goette | I believe $PU$ is a model for $B^2\mathbf Z$. Is it possible to represent String by an honest group containing $PU$ such that the quotient is Spin? This might not be the natural thing to do, but is it possible? | |
| Nov 23, 2023 at 18:52 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |