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added to tmf a section that gives an outline of the proof strategy for how to compute the homotopy groups of the $tmf$-spectrum from global sections of the $E_\infty$-structure sheaf on the moduli stack of elliptic curves.
A point which I wanted to emphasize is that
The problem of constructing $tmf$ as global sections of an $\infty$-structure sheaf has a tautological solution: take the underlying space to be $Spec tmf$.
From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact:
In the $\infty$-topos over the $\infty$-site of formal duals of $E_\infty$-rings, the dual $Spec M U$ of the Thom spectrum, is a well-supported object. the terminal morphism
$Spec M U \to *$in the $\infty$-topos is an effective epimorphism, hence a covering of the point.
Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of $Spec tmf$ to $Spec M U$ is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute $\mathcal{O} Spec tmf$ on that.
I have considerably expanded the idea-section at tmf. Also I started some notes at Definition and construction – Decomposition via Arithmetic fracture squares, which is however very much stubby still.
Have added to tmf a section Maps to K-theory and to Tate K-theory.
Also I have split the “Definition and Construction”-section into a Definition-section and a Construcion-section and added some actual (though basic) content to the Definition section (the Construction-section remains very piecemeal, naturally but nevertheless woefully).
added a list of the low degree homotopy groups of tmf
Corrected the indexing on the table in #4 (started at 1 instead of 0)
Added the reference
added pointer to:
have given the statement about the Boardman homomorphism for $tmf$ a little Properties-subsection (here) of its own (splitting it off from the subsection on stable homotopy groups).
Will also give this a stand-alone entry: Boardman homomorphism in tmf, for ease of hyperlinking from elsewhere.
added DOI to
Once nLab editing is open, someone should fix the mistake that (connective) tmf is defined as the global sections of a sheaf of $E_{\infty}$-rings. That’s not true - it’s only known definition is as a connective cover of Tmf. For instance, see Behrens’ survey article in the Handbook. To quote the Hill-Lawson paper (p. 6): “Finally, the construction of the object tmf by connective cover remains wholly unsatisfactory, and this is even more true when considering level structure. In an ideal world, tmf should be a functor on a category of Weierstrass curves equipped with some form of extra structure. We await the enlightenment following discovery of what exact form this structure should take.”
Thanks for the heads-up.
It looks like in the “Definition”-Section 2 it’s stated correctly, at least after the words “more precisely”. Then section 3 is lacking the capitalization.
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