added pointer to:

- Robert Stong, Chapter IV, Example 1, p. 40 of
*Notes on Cobordism theory*, Princeton University Press, 1968 (toc pdf, ISBN:9780691649016)

added pointer to:

- John Rognes,
*The sphere spectrum*, 2004 (pdf)

added pointer to:

- Ken-ichi Maruyama,
*$e$-invariants on the stable cohomotopy groups of Lie groups*, Osaka J. Math. Volume 25, Number 3 (1988), 581-589 (euclid:ojm/1200780982)

added pointer to

- Sławomir Nowak,
*Stable cohomotopy groups of compact spaces*, Fundamenta Mathematicae 180 (2003), 99-137 (doi:10.4064/fm180-2-1)

added pointer to:

- C. T. Stretch,
*Stable cohomotopy and cobordism of abelian groups*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 90, Issue 2 September 1981 , pp. 273-278 (doi:10.1017/S0305004100058734)

finally added pointer to

- Michael Barratt, Stewart Priddy,
*On the homology of non-connected monoids and their associated groups*, Commentarii Mathematici Helvetici, December 1972, Volume 47, Issue 1, pp 1–14 (doi:10.1007/BF02566785)

added pointer to

- Alain Connes, Caterina Consani,
*Absolute algebra and Segal’s Gamma sets*, Journal of Number Theory 162 (2016): 518-551 (arXiv:1502.05585)

with respect to understanding the sphere spectrum as $K(\mathbb{F}_1)$

]]>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.

]]>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 $GL_n(Z_{p^k})$.

]]>Thanks! I am having a look…

]]>There have been a few developments since the article I linked to as well, see here.

]]>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.

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?

]]>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.

]]>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.

]]>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 should have another look, maybe I missed it: what does the thesis achieve along these lines, beyond the vague comments?

]]>Maybe that Berman thesis I mentioned could count as an answer.

]]>have send a question to MO for more: *Stable Cohomotopy as KF_1*

added a subsection “Properties – As algebraic K-theory over the field with one element” (here)

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