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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 30th 2010
• (edited Nov 30th 2010)

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

1. The problem of constructing $tmf$ as global sections of an $\infty$-structure sheaf has a tautological solution: take the underlying space to be $Spec tmf$.

2. 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.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 7th 2014

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 9th 2014

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).

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 14th 2014

added a list of the low degree homotopy groups of tmf

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeFeb 20th 2015

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

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJun 1st 2020

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 7th 2020
• (edited Sep 7th 2020)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 7th 2020
• (edited Sep 7th 2020)

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeSep 7th 2020