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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2014
   • (edited Apr 1st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexIt seems to me that there is a canonical map $$ Pic(KU)_{\leq 3} \longrightarrow GL_1(tmf) $$ (from the [[Picard 3-group]] of the $E_\infty$-ring [[KU]] to the [[infinity-group of units]] of [[tmf]]) and that it has a chance of being an iso on $\pi_{\bullet \leq 3}$. Does that sound plausible? So the reasoning for the map itself is this: Freed et al. have highlighted the observation that $Pic(KU)_{\leq 3} \simeq (\Omega ko)_{\leq 3}$ (e.g theorem 15.2 [here](http://ncatlab.org/nlab/files/FreedESI2012.pdf), see the discussion at [[super line 2-bundle]] where I am writing $|2\mathbf{sLine}|$ for $Pic(KU)_{\leq 3}$). But that would mean also that $Pic(KU)_{\leq 3}$ is also the homotopy fiber of $B String \to B O$. That in turn would yield the map $Pic(KU)_{\leq 3} \to GL_1(tmf)$ by the argument in section 8 of [Ando-Blumberg-Gepner 10](http://arxiv.org/abs/1002.3004). Their twist of $tmf$ would be the (delooping of the) restriction $B^2 U(1) \to Pic(KU) \to GL_1(tmf)$. (Thanks to Joost Nuiten for discussion of this point, but all remaining confusion is my fault.) Even though I have been staring at [Ando-Hopkins-Rezk 10](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf) for a bit, I am still not good at deciding what the image of that map in $GL_1(tmf)$ would be. But from what I am being told from an oracle I gather it has a good chance of being an iso in degree $\leq 3$. Does that sound plausible? If something like this holds, it makes me wonder that the following then seems suggestive: by the discussion at [[super 2-line bundle]] we also have that $|\mathbf{Br}(\mathbb{C})| \simeq Pic(KU)_{\leq 3}$, where now on the left I mean the geometric realization of the "Brauer 2-stack" of the ring of complex numbers, regarded in the context of $\mathbb{Z}_2$-graded algebras. So then we would have in this sense $$ Br(\mathbb{C}) \simeq Pic(KU)_{\leq 3} \stackrel{\simeq ?}{\to} GL_1(tmf)_{\leq 3} \,. $$ This looks moderately neat. In some precise sense $Pic(-)$ is the "non-connected delooping" of $GL_1$ and $Br(-)$ is in turn the "non-connected delooping" of $Pic$ (as made precise e.g. in [Szymik 11](http://ncatlab.org/nlab/show/Brauer%20group#Szymik11)).

   It seems to me that there is a canonical map

   $$Pic(KU)_{\leq 3} \longrightarrow GL_1(tmf)$$

   (from the Picard 3-group of the $E_\infty$-ring KU to the infinity-group of units of tmf)

   and that it has a chance of being an iso on $\pi_{\bullet \leq 3}$. Does that sound plausible?

   So the reasoning for the map itself is this: Freed et al. have highlighted the observation that $Pic(KU)_{\leq 3} \simeq (\Omega ko)_{\leq 3}$ (e.g theorem 15.2 here, see the discussion at super line 2-bundle where I am writing $|2\mathbf{sLine}|$ for $Pic(KU)_{\leq 3}$). But that would mean also that $Pic(KU)_{\leq 3}$ is also the homotopy fiber of $B String \to B O$. That in turn would yield the map $Pic(KU)_{\leq 3} \to GL_1(tmf)$ by the argument in section 8 of Ando-Blumberg-Gepner 10. Their twist of $tmf$ would be the (delooping of the) restriction $B^2 U(1) \to Pic(KU) \to GL_1(tmf)$.

   (Thanks to Joost Nuiten for discussion of this point, but all remaining confusion is my fault.)

   Even though I have been staring at Ando-Hopkins-Rezk 10 for a bit, I am still not good at deciding what the image of that map in $GL_1(tmf)$ would be. But from what I am being told from an oracle I gather it has a good chance of being an iso in degree $\leq 3$.

   Does that sound plausible?

   If something like this holds, it makes me wonder that the following then seems suggestive: by the discussion at super 2-line bundle we also have that $|\mathbf{Br}(\mathbb{C})| \simeq Pic(KU)_{\leq 3}$, where now on the left I mean the geometric realization of the “Brauer 2-stack” of the ring of complex numbers, regarded in the context of $\mathbb{Z}_2$-graded algebras.

   So then we would have in this sense

   $$Br(\mathbb{C}) \simeq Pic(KU)_{\leq 3} \stackrel{\simeq ?}{\to} GL_1(tmf)_{\leq 3} \,.$$

   This looks moderately neat. In some precise sense $Pic(-)$ is the “non-connected delooping” of $GL_1$ and $Br(-)$ is in turn the “non-connected delooping” of $Pic$ (as made precise e.g. in Szymik 11).