In his beautiful paper On the cohomology and K theory of the general linear group over a finite field, Quillen constructs (if I understand correctly) an isomorphism on connected components of K-theory $K(\mathbb{F}_q)\mid_{\ge 1} \cong \text{Fib}(1-\Psi^q:ku\to ku).$ His construction, while very cool, seems to be based on a miracle: that the Brauer lift produces complex representations from mod-$p$ ones. It also uses the +-construction for defining K-theory (though perhaps it's not that hard to update his construction to work with Segal formalism)?

I'm curious whether there is a more modern perspective on this equivalence of spectra, maybe using cyclic structure on topological Hochschild homology.

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    $\begingroup$ For what it's worth, many years ago (2006?) I posed this question (more or less) to an expert on trace methods, and the response seemed to be that no such proof of Quillen's result is known. It's a little uncler whether Quillen was aware of Segal's K-theory construction and the Q-construction before he wrote the paper in question. My impression is that invented the plus-construction specifically for this calculation. One could probably determine the dates by looking through Quillen's notebooks. $\endgroup$ – Dan Ramras Aug 8 '16 at 18:42
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    $\begingroup$ Quillen's famous Letter to Segal (where he talks about "freeing myself from the shackes of the simplicial way of thinking" and discovering the category Q) is from July 1972 - he says he discovered Q in Spring '72 - and the Annals paper in question was published in November 1972. There was an earlier preprint of the Annals paper, though, which is on the K-theory archive (math.uiuc.edu/K-theory/1006), but it's undated. $\endgroup$ – Dan Ramras Aug 8 '16 at 18:43
  • $\begingroup$ Didn't Dwyer say something about this at the 2012 WCATSS? $\endgroup$ – Sean Tilson Aug 10 '16 at 12:17
  • $\begingroup$ @SeanTilson I don't recall anything about this from WCATSS. $\endgroup$ – Dan Ramras Aug 13 '16 at 16:48
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    $\begingroup$ The argument using Brauer lifting is a more complicated rearrangement to eliminate étale topology from his original very simple argument which appears in his 1970 ICM address, which proves the more general theorem that $BG(\mathbb F_q)^+[1/p]$ is the fiber of Frobenius on $BG[1/p]$. The cohomology of the connected algebraic group is the same as the compact group because the methods of computation (eg, Schubert cells) are all algebraic. $\endgroup$ – Ben Wieland Sep 22 '20 at 21:53

Firstly, there is a proof using the motivic spectral sequence (the Atiyah-Hirzerbruch style spectral sequence from motivic cohomology to algebraic $K$-theory). This is written in the master's thesis of Gabe Angelini-Knoll.

Gabe and Andrew Salch are also working to answer this question and a paper is apparently due. From Andrew's website:

"My student Gabe Angelini-Knoll and I have been working on the problem of computing the Waldhausen algebraic K-groups of the algebraic K-theory spectra of certain finite fields. This is an example of "iterated K-theory" and Rognes' redshift conjecture is not known in these cases. Thus far, Gabe and I have (with the aid of a new "THH-May" spectral sequence for computing topological Hochschild homology) computed the homotopy groups of THH(K(F_q)) smashed with the p-primary Smith-Toda complex V(1), for p > 3 and for many (but not all) prime powers q. Gabe is working on the computations of the homotopy groups of the C_p fixed points of this spectrum (this will probably be Gabe's thesis), with the goal of using trace methods to recover the K-groups of K(F_q).

We expect to post and submit our first two papers on this topic before the end of summer 2016. "

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    $\begingroup$ In case anyone is curious about the papers of Andrew Salch and I that are referred to here, they are now on the arXiv: link and link I should also mention that Eva Honing, a current student of Ausoni, has been working on THH of more general finite fields whereas I have restricted my attention to fields of order q where q is prime power that topologically generates the p-adic units. $\endgroup$ – Gabriel Angelini-Knoll Feb 15 '17 at 16:23

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