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nLab > Latest Changes: Thom spectrum

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010
    • (edited Dec 21st 2010)

    wrote out parts of the proof of Ω • un≃π •MO\Omega^{un}_\bullet \simeq \pi_\bullet M O at Thom spectrum

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeDec 21st 2010

    By the way, is there any difference between a multiplicative spectrum and an E-infinity spectrum (the latter representing a multiplicative cohomology theory) ? I refer to the definition of multiplicative spectrum in Stong’s notes on cobordism theory.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    I’d think “multiplicative spectrum” is used synonymously to “ring spectrum” hence E ∞E_\infty-spectrum. But I haven’t checked with Stong’s notes.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeDec 21st 2010

    Thanks. Did you think how the notion of characteristic classes for cobordisms with (B,f)(B,f)-structure fit into your approach ? By (B r,f r)(B_r,f_r)-structure on a smooth bundle h:X→BO(r)h : X\to BO(r) over XX Stong means a lift of hh to B rB_r along f r:B r→BO(r)f_r : B_r\to BO(r). Then one looks at spectra (B,f)(B,f) made from (B r,f r)(B_r,f_r)-structures to define (B,f)(B,f)-structure on BO=colimBO(r)BO = colim BO(r)-bundles. Then for a multiplicative spectrum AA with a given map from Thom spectrum one defines a notion of characteristic class. Stong says that this was revealing historically.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    wait, I think I need to take that back, more later.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    Hi Zoram,

    sorry, I had been in the middle of a seminar and couldn’t reply further. By some weird coincidence, a little after I posted my first reply here we read page 6 here where the term “multiplicative” for a spectrum is used in a way that at least not evidently refers to the E ∞E_\infty-structure but just to a product structure.

    So I guess one should be careful.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    Do you have a link to Stong’s notes that you are looking at?

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeDec 21st 2010
    • (edited Dec 21st 2010)
    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    Thanks. I’ll have a look as soon as I am back on a stable connection.

    • CommentRowNumber10.
    • CommentAuthorzskoda
    • CommentTimeDec 21st 2010

    Regarding that there are several holidays in next couple of weeks, I can be more responsive than it was in recent months and supplement related discussion, if any. I like the topics.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    okay, that would be nice. I am not sure how much i will be able to find time to work in the next days, but we’ll see.

    • CommentRowNumber12.
    • CommentAuthorMike Shulman
    • CommentTimeDec 21st 2010

    I believe that historically (especially before the development of E ∞E_\infty things and good symmetric monoidal categories of spectra) the term “ring spectrum” was used to mean a monoid object in the stable homotopy category, i.e. a spectrum with a ring structure up to homotopy but without any coherent higher homotopies. I would guess that that is all you get from knowing that a cohomology theory has a product structure, too. Stong’s definition looks to be along those lines.

    • CommentRowNumber13.
    • CommentAuthorzskoda
    • CommentTimeDec 22nd 2010

    If I understood well, then we will have to distinguish then an incoherent and coherent version of multiplicativity of spectra in nlab I guess.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 2nd 2011

    added to Thom spectrum some basics of the Hopkins et al- theory of Thom specta for general line(spectral) ∞\infty-bundles.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2013
    • (edited Jun 19th 2013)
    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJul 8th 2013

    added pointers to more and original references for Thom spectrum – As a dual object.

    • CommentRowNumber17.
    • CommentAuthorDylan Wilson
    • CommentTimeJul 9th 2013
    • (edited Jul 9th 2013)

    In case anyone finds it useful here are some notes for a reading course on the ABGHR paper:

    Reading course

    nLab-ers might enjoy:

    My notes on the infty POV

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJul 9th 2013
    • (edited Jul 9th 2013)

    Thanks! That’s nice. I have added pointers to this to the References-sections at Thom spectrum and at (infinity,1)-module bundle .

    Maybe you can help me with the following: while the Pontryagin-Thom theory comes out very nice in homotopy theory sliced over EE-module spectra, for E=KUE = KU and for some purposes it comes out even nicer in KK-theory. Namely there we have the following statement (as mentioned at Poincare duality algebra):

    • the Thom construction on a manifold XX with twist χ\chi is passage to the dual object in the KK-category;

    • the Thom isomorphism is, if it exists for (X,χ)(X,\chi), an identification of an object with its dual;

    • the Umkehr of map is simply the dual morphism precomposed with this Thom isomorphism

    I would like to have a setup of homotopy theory over module spectra which reproduces this nice story. Of course it already does so pretty closely: given a map of manifolds X→YX \to Y ABG apply Spanier duality to get an Umkehr map DY→DXD Y \to D X, and then the Thom isomorphism serves to identify (in two stages, maybe) [DX,E]≃[X,E][D X, E] \simeq [X,E].

    So both pictures seem to match nicely. But I seem to be lacking just a little adjustment to the story. Not sure yet.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2013
    • (edited Jul 14th 2013)

    I have now added to Thom spectrum a brief paragraph (in the (∞,1)(\infty,1)-module-section) leading up to the characterization of the Thom space functor, hence the sections functor Γ:∞Grpd /RLine→RMod\Gamma : \infty Grpd_{/ R Line} \to R Mod.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2016

    I have added to Thom spectrum – For vector bundles a few more details. (More still necessary for a self-contained account.)

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2016

    I have added a bunch of further classical stuff to Thom spectrum – For vector bundles.

    In the course of this I gave G-structure a section In terms of (B,f)-structures.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 28th 2020

    added missing subscript in the first equation of this Def.

    diff, v64, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeDec 29th 2020

    Fixed a bunch of trivial issues in typesetting and wording.

    Also I noticed that the Idea-section is no good, lacking a decent lead-in. Have added an ellipsis to at least indicate, for the moment, the need to expand here. (No leisure right now, am on a plane…).

    diff, v65, current

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