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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