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wrote out parts of the proof of $\Omega^{un}_\bullet \simeq \pi_\bullet M O$ at Thom spectrum
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.
I’d think “multiplicative spectrum” is used synonymously to “ring spectrum” hence $E_\infty$-spectrum. But I haven’t checked with Stong’s notes.
Thanks. Did you think how the notion of characteristic classes for cobordisms with $(B,f)$-structure fit into your approach ? By $(B_r,f_r)$-structure on a smooth bundle $h : X\to BO(r)$ over $X$ Stong means a lift of $h$ to $B_r$ along $f_r : B_r\to BO(r)$. Then one looks at spectra $(B,f)$ made from $(B_r,f_r)$-structures to define $(B,f)$-structure on $BO = colim BO(r)$-bundles. Then for a multiplicative spectrum $A$ with a given map from Thom spectrum one defines a notion of characteristic class. Stong says that this was revealing historically.
wait, I think I need to take that back, more later.
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 $E_\infty$-structure but just to a product structure.
So I guess one should be careful.
Do you have a link to Stong’s notes that you are looking at?
Thanks. I’ll have a look as soon as I am back on a stable connection.
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.
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.
I believe that historically (especially before the development of $E_\infty$ 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.
If I understood well, then we will have to distinguish then an incoherent and coherent version of multiplicativity of spectra in nlab I guess.
added to Thom spectrum some basics of the Hopkins et al- theory of Thom specta for general line(spectral) $\infty$-bundles.
added two brief remarks to the Properties-section at Thom spectrum:
added pointers to more and original references for Thom spectrum – As a dual object.
In case anyone finds it useful here are some notes for a reading course on the ABGHR paper:
nLab-ers might enjoy:
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 $E$-module spectra, for $E = KU$ 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 $X$ with twist $\chi$ is passage to the dual object in the KK-category;
the Thom isomorphism is, if it exists for $(X,\chi)$, 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 $X \to Y$ ABG apply Spanier duality to get an Umkehr map $D Y \to D X$, and then the Thom isomorphism serves to identify (in two stages, maybe) $[D X, E] \simeq [X,E]$.
So both pictures seem to match nicely. But I seem to be lacking just a little adjustment to the story. Not sure yet.
I have now added to Thom spectrum a brief paragraph (in the $(\infty,1)$-module-section) leading up to the characterization of the Thom space functor, hence the sections functor $\Gamma : \infty Grpd_{/ R Line} \to R Mod$.
I have added to Thom spectrum – For vector bundles a few more details. (More still necessary for a self-contained account.)
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.
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