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.

]]>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.”

]]>added DOI to

- Michael Hopkins,
*Topological modular forms, the Witten Genus, and the theorem of the cube*, Proceedings of the International Congress of Mathematics, Zürich 1994 (pdf, doi:10.1007/978-3-0348-9078-6_49)

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 pointer to:

- Christopher Douglas, John Francis, André Henriques, Michael Hill,
*Topological Modular Forms*, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

Added the reference

- David Gepner, Lennart Meier,
*On equivariant topological modular forms*, (arXiv:2004.10254)

Corrected the indexing on the table in #4 (started at 1 instead of 0)

]]>added a list of the low degree homotopy groups of tmf

]]>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).

]]>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.

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.

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