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have send a question to MO for more: Stable Cohomotopy as KF_1
Maybe that Berman thesis I mentioned could count as an answer.
I should have another look, maybe I missed it: what does the thesis achieve along these lines, beyond the vague comments?
Well, yes, only vague comments. He thanks Jack Morava for “suggesting that my results are related to the field with one element”.
But what kind of “further developments” were you hoping for?
I was looking for more like Guillot’s result, strengthening the analogy between stable cohomotopy and K-theory. How about Chern-characters, Todd classes, A-roof genus, Riemann-Roch theorem over F_1? How about the comparison map between all structures, as we “extend scalars” from F_1 to C. This last question is related to what we are discussing in the thread on Burnside ring.
Riemann-Roch theorem over F_1
This is a major open problem, it is one of the main obstructions to a proof of the Riemann hypothesis via algebraic geometry over $\mathbb{F}_{1}$ (although for me what the analogue of the Frobenius should be is the main conceptual gap). I think Alain Connes may have some kind of results around Riemann-Roch over $\mathbb{F}_{1}$ in some setting, though, albeit not strong enough for the applications to the Riemann hypothesis.
This is a major open problem, it is one of the main obstructions to a proof of the Riemann hypothesis via algebraic geometry over $\mathbb{F}_1$
Thanks for saying this, I didn’t know. What would be a good source to read up on this?
Though I have never focused on it in the same way as I have on other things, I’ve been ruminating off and on upon $\mathbb{F}_{1}$ for quite a long time, and mostly do not remember where I picked up things! However, perhaps the highlighted text towards the bottom of pg. 20 here would be the kind of thing you are looking for?
I am not very familiar with Connes’ work, but it seems (regarding my remark about the Frobenius in #7) that he does have an idea of what the analogue of the Frobenius should be, so that the Riemann-Roch theorem actually is the principal obstruction in his setting.
There have been a few developments since the article I linked to as well, see here.
Thanks! I am having a look…
Riemann-Roch theorem over F_1
In the process of some pondering on the place of $\mathbb{F}_1$ in Sylow $p$-group theory, John Baez pointed me to this paper by Kapranov and Smirnov which speaks of Riemann-Roch over $\mathbb{F}_{1^n}$ as counting residues mod $n$ of the number of integer points of some polyhedron.
Hmm, so what’s $K \mathbb{F}_{1^n}$?
The Sylow thought, by the way, is that the $p$-Sylow subgroup of any $GL_n(Z_{p^k})$ is the maximal unipotent subgroup, and any group embeds in $S_n$ which embeds in $GL_n(Z_{p^k})$.
I haven’t seen a discussion of $\mathbb{F}_{1^n}$ in Connes-Consani; they have apparently proved the only finite semi-field that’s not a field is $(\{0,1\},max,+)$, and they take this to be the “prime field” of characteristic 1.
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with respect to understanding the sphere spectrum as $K(\mathbb{F}_1)$
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