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I am looking for characteristic classes of vector bundles (either real or complex) with values in generalized multiplicative cohomology theories such that:

i) they vanish if the bundle of unit spheres $S(E)$ of the vector bundle $E$ (viewed as a real vector bundle if $E$ is complex) is stably fiberwise-homotopically trivial.

ii) the are well behaved under direct image(umkehr) homomorphism, for bundles oriented over the corresponding cohomology theory e.g., skew functoriality etc...

Other than Stiefel-Whitney and Wu classes in ordinary cohomology I know only Bott's cannibalistic classes in $K$-theory. But I don't know how Bott classes behave with regard to ii). Moreover my feeling is that, precisely because of ii), the right place for the characteristic classes which I need should be the complex cobordism.

Any reference to related topics would be more than welcome.

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Could you expand a little more on what you mean by "well behaved under direct image (umkehr) homomorphism..."? –  Mark Grant Apr 26 '12 at 12:07
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I don't understand ii), but i), for your multiplicative cohomology theory $h$, is given by any element in the $h^{\ast}(G/O)$. –  John Klein Apr 26 '12 at 22:39
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4 Answers

up vote 3 down vote accepted

An evident general construction is to take any multiplicative cohomology theory $E^*$ and a cohomology operation $\psi$ in that theory, say taking $E^q$ to $E^{q+n}$. For an $E$-oriented real $q$-bundle $\xi$ over $X$, $\theta^{-1}(\psi(\mu))$ gives the value of a characteristic class, where $\mu$ is the Thom class of $\xi$ and $\theta$ is the Thom isomorphism. For stable $\psi$, defined for all $q$, this will give a characteristic class on all $E^*$-oriented bundles of the sort wanted, modulo precision about condition (ii). [Senior moment nonsense eliminated]. You are studying the J-map $BO\to BF$ (or $BU \to BSF$), where $BF$ classifies stable spherical fibrations (oriented for $BSF$). A lot more is known than Adams knew. In particular, he didn't yet have the Adams conjecture. Rationally, $BF$ is a point. At an odd prime $p$, $BF$ factors as $BJ\times BCokerJ$, and at $2$ there is a non-split fiber sequence $BCokerJ \to BSF \to BJ$. The $J$-map at $p$ is best thought of as a map $BSpin\to BSF$ ($BO\simeq BSpin$ at $p>2$). By the Atiyah-Bott-Shapiro orientation, the $J$-map $BSpin\to BSF$ factors through the classifying space $B(SF;kO)$ for $kO$-oriented spherical fibrations and, at any $p$, $B(SF;kO)$ splits as $BSpin\times BCokerJ$ (but BSpin is seen in two pieces, one carrying the Wu classes, the other the rest of the Adams splitting). The intuition is that $BCokerJ$ and thus the unknown parts of the stable homotopy groups of spheres can be ignored, leaving the focus on the quite computable composite $BSpin \to BSF \to BJ$. This is too fast, and details are in Chapter V of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra''. In ordinary mod $p$ cohomology, calculations are thoroughly understood but don't shed light on your question. They are also understood for $K$-theory, by work of Hodgkin and Snaith, and here the intuition that $Coker J$ can be ignored is made precise by Hodgkin's result that $\tilde K(BCokerJ)=0$.

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The rational Pontryagin classes do not satisfy (i)! –  Tom Goodwillie Apr 26 '12 at 14:09
    
Whoops, total nonsense of course; nothing rational satisfies (i) in ordinary cohomology. –  Peter May Apr 27 '12 at 3:10
    
@P.May: I will give a look to the chapter V after your remarks. I am trying to calculate the image under J-homomorphism of the index bundle of a family of pseudodifferential operators from its symbol class. The index bundle = $\pi !(symb. cl.)$ in $K$ theory. Alternatively, I can look at characteristic classes in some multiplicative theory $h$, obstructing the vanishing of J, which are computable from the symbol. I definitely need some geometric description of such an $h.$ Is there any geometric description of BSF, in the spirit of Quilen's approach to complex cobordism? Any reference? –  Jacobo Pejsachowicz Apr 30 '12 at 12:02
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You might have luck with the "universal" choice of h, namely the cohomology theory given by the connective spectrum corresponding to the E_\infty-space of stable spheres under smash product (as a plain space, this is Z x BAut(S)). The characteristic class map KO --> h you want is then essentially the real J-homomorphism, studied extensively by Adams in a series of papers.

This h isn't very computable, though, since for instance its coefficient groups in degrees > 1 are the stable homotopy groups of spheres in degrees > 0. For a more computable variant I would suggest working completed a prime p and taking h' = suspension of the K(1)-local sphere = suspension of fiber of \psi^u - 1 : KO --> KO for u a generator of Z_p^*/+-1, where the map h --> h' is gotten from Rezk's logarithm in the K(1)-local case. Adams's results more-or-less imply that this h' detects a lot of the same information as h (on coefficient groups, the map h --> h' is Adams's e-invariant, up to normalization), but h' is a much more tractable cohomology theory.

Edit: Oh, I should say, up to a unit factor (at least at an odd prime), the composite map KO --> h --> h' identifies with the boundary map in the fiber sequence L_K(1)S --> KO ---> KO. So it's very computable.

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Sorry, I should've used connective K-theory for the map KO --> h. Of course for evaluating on spaces in degree zero it doesn't matter. –  Dustin Clausen Apr 26 '12 at 21:23
    
Thanks. I "know" the Adams papers. Could you suggest please some other reference as well. –  Jacobo Pejsachowicz Apr 27 '12 at 8:23
    
You can read about the logarithm in Rezk's paper math.uiuc.edu/~rezk/units-paper.pdf, see esp. Theorem 1.9. I also wrote a bit about this in the paper here: arxiv.org/pdf/1110.5851.pdf, see esp. the first part of the proof of Theorem 4.4. –  Dustin Clausen Apr 27 '12 at 18:22
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Jacobo, I don't know where to put answers to questions asked after answers, and I don't have much of an answer. I do know how to manufacture the localization of $BSF$ at a prime $p$ out of symmetric groups (a multiplicative version of Barratt-Priddy-Quillen), also in the old $E_{\infty}$ ring book. However, I think by geometric you mean algebro-geometric, interpreting Quillen's relationship with formal groups that way. For sure I know nothing along those lines and don't expect anything.

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You could edit your answer and make it clear that you are answering a comment. I've seen other people do that. –  David Roberts May 1 '12 at 5:34
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I can hardly give an exact meaning to ii). The characteristic class $c$ has to be stable and hence a natural transformation from (say) $K^*$ to $ h^*$. The best one can hope is that If $\pi \colon M \to N$ is a map between manifolds that is orientable both for $K^*$ and $h^*$ then there is a correction term such as the Todd class which allows to relate the umkher $\pi_*$ in $h^*$ with the umkher map $\pi!$ in $K^*$. But maybe this is not compatible with i). The less I need is to be able to compute $c[\pi!(\xi)]$ from $c[\xi]$ in some way.

In section 4 of arXiv:1005.3246 I carried out the above computation in cohomology with $Z_p$ coefficients by relating the reduction mod $p$ of the integral Pontriagin classes which are well behaved under umkher with the characteristic classes for sphere bundles defined by Wu, which are constructed according to the scheme in Answer 1 and hence verify i). However this computation gives nontrivial results only in few dimensions and the same happens with Stiefel-Whitney classes. My hope is to obtain better results using generalized cohomology theories.

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