Question

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

Answer 1

I understand that this is because GL_n(F₁) is supposed to be Sigma_n, the symmetric group on n letters. Thus K(F₁) = K(finite sets) which is the sphere spectrum by the Barratt-Priddy-Quillen-Segal theorem.

But I have no idea why GL_n(F₁) should be Sigma_n...

Answer 2

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.

Answer 3

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 http://arxiv.org/abs/math/0605429.

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 http://arxiv.org/abs/math/0407093 for this.

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.

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...