Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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-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 -structure sheaf has a tautological solution: take the underlying space to be Spectmf.

    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 -topos over the -site of formal duals of E-rings, the dual SpecMU of the Thom spectrum, is a well-supported object. the terminal morphism

      SpecMU*

      in the -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 Spectmf to SpecMU 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 𝒪Spectmf 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

    Added the reference

    diff, v56, current

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

    added pointer to:

    diff, v57, current

    • 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.

    diff, v58, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2020

    added DOI to

    diff, v61, current

    • CommentRowNumber10.
    • CommentAuthorGuest
    • CommentTimeJan 18th 2022

    Once nLab editing is open, someone should fix the mistake that (connective) tmf is defined as the global sections of a sheaf of E-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.”

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJan 18th 2022

    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.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 20th 2024

    Added

    • Jack Morgan Davies, Constructing and calculating Adams operations on dualisable topological modular forms [arXiv:2104.13407]

    diff, v71, current