K(F_1) = sphere spectrum? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:58:03Z http://mathoverflow.net/feeds/question/1628 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum K(F_1) = sphere spectrum? Thomas Riepe 2009-10-21T11:02:49Z 2012-05-11T23:15:47Z <p>I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means? </p> http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum/1630#1630 Answer by Oscar Randal-Williams for K(F_1) = sphere spectrum? Oscar Randal-Williams 2009-10-21T11:21:44Z 2009-10-21T11:21:44Z <p>I understand that this is because GL<sub>n</sub>(F<sub>1</sub>) is supposed to be Sigma<sub>n</sub>, the symmetric group on n letters. Thus K(F<sub>1</sub>) = K(finite sets) which is the sphere spectrum by the Barratt-Priddy-Quillen-Segal theorem.</p> <p>But I have no idea why GL<sub>n</sub>(F<sub>1</sub>) should be Sigma<sub>n</sub>...</p> http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum/1644#1644 Answer by Peter Arndt for K(F_1) = sphere spectrum? Peter Arndt 2009-10-21T12:43:34Z 2009-10-21T12:43:34Z <p>Yes, taking GL _n(F _1) to be Sigma _n one can make sense both of the Q- and the +-construction and both yield the same answer as shown by Deitmar in <a href="http://arxiv.org/abs/math/0605429" rel="nofollow">http://arxiv.org/abs/math/0605429</a>.</p> <p>GL _n(F _1)=Sigma _n is suggested by several observations. One is that counting formulas for subspaces of n- dimensional vector spaces over F _q turn into counting formulas for subsets of n-element sets, if one sets q=1. So one could say that an n-dimensional vector space over F_1 is an n-element set and GL _n(F _1)=Aut(F _1^n)=Sigma _n. See Cohn's very nicely written <a href="http://arxiv.org/abs/math/0407093" rel="nofollow">http://arxiv.org/abs/math/0407093</a> for this.</p> <p>One gets another hint by looking at the Tits building for GL _n(F _q) (that is a simplicial complex where the group acts). There is a natural limit for q going to one 1 - what then remains is the so-called chamber of the building and the symmetry group of that is Sigma _n.</p> <p>Further hints that one should just drop addition (in comparison to the usual notion of module) come from arithmetic geometry, but that is maybe less convincing and a longer story...</p> http://mathoverflow.net/questions/1628/kf-1-sphere-spectrum/2383#2383 Answer by Benjamin Antieau for K(F_1) = sphere spectrum? Benjamin Antieau 2009-10-24T22:20:14Z 2012-05-11T23:15:47Z <p>Here is another heuristic, related to what Randal-Williams said above. The sphere spectrum is the unit object in nice categories of spectra. That is, ring spectra are algebras over the sphere spectrum. Now, to every scheme X you can associate a K-theory ring spectrum K(X), and this is contravariant. So, in the usual theory there is a morphism K(Z)->K(X) for all schemes X. So, finding F_1 also means finding something (its K-theory spectrum) that maps to the (homotopy) limit of all K-theory spectra. That this should be the unit object of the category of spectra doesn't seem very surprising.</p>