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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 30th 2010
   • (edited Nov 30th 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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 $Spec tmf$. 1. 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 7th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have considerably expanded the [idea-section](http://ncatlab.org/nlab/show/tmf#idea) at _[[tmf]]_. Also I started some notes at _[Definition and construction -- Decomposition via Arithmetic fracture squares](http://ncatlab.org/nlab/show/tmf#DecomopositionViaArithmeticSquares)_, which is however very much stubby still.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 9th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexHave added to _[[tmf]]_ a section _[Maps to K-theory and to Tate K-theory](http://ncatlab.org/nlab/show/tmf#MapToTateKTheory)_. 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).

   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).

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 14th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a list of the [low degree homotopy groups of tmf](http://ncatlab.org/nlab/show/tmf#HomotopyGroups)

   added a list of the low degree homotopy groups of tmf

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 20th 2015
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexCorrected the indexing on the table in #4 (started at 1 instead of 0)

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

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeJun 1st 2020
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded the reference * [[David Gepner]], [[Lennart Meier]], _On equivariant topological modular forms_, ([arXiv:2004.10254](https://arxiv.org/abs/2004.10254)) <a href="https://ncatlab.org/nlab/revision/diff/tmf/56">diff</a>, <a href="https://ncatlab.org/nlab/revision/tmf/56">v56</a>, <a href="https://ncatlab.org/nlab/show/tmf">current</a>

   Added the reference

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

   diff, v56, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2020
   • (edited Sep 7th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded 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](https://bookstore.ams.org/surv-201)) <a href="https://ncatlab.org/nlab/revision/diff/tmf/57">diff</a>, <a href="https://ncatlab.org/nlab/revision/tmf/57">v57</a>, <a href="https://ncatlab.org/nlab/show/tmf">current</a>

   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)

   diff, v57, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2020
   • (edited Sep 7th 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave given the statement about the Boardman homomorphism for $tmf$ a little Properties-subsection ([here](https://ncatlab.org/nlab/show/tmf#BoardmanHomomorphism)) 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. <a href="https://ncatlab.org/nlab/revision/diff/tmf/58">diff</a>, <a href="https://ncatlab.org/nlab/revision/tmf/58">v58</a>, <a href="https://ncatlab.org/nlab/show/tmf">current</a>

   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.

   diff, v58, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeSep 7th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded DOI to * [[Michael Hopkins]], _Topological modular forms, the Witten Genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematics, Z&#252;rich 1994 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf), [doi:10.1007/978-3-0348-9078-6_49](https://doi.org/10.1007/978-3-0348-9078-6_49)) <a href="https://ncatlab.org/nlab/revision/diff/tmf/61">diff</a>, <a href="https://ncatlab.org/nlab/revision/tmf/61">v61</a>, <a href="https://ncatlab.org/nlab/show/tmf">current</a>

   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)

   diff, v61, current

10. • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeJan 18th 2022
    • PermaLink
    Author: Guest
    Format: MarkdownItexOnce 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."

    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.”

11. • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2022
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks for the heads-up. It looks like in the "Definition"-[Section 2](http://nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/tmf#definition) it's stated correctly, at least after the words "more precisely". Then section 3 is lacking the capitalization.

    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.