Characteristic classes detecting nontrivial fiberwise homotopy of sphere bundles

Question

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.

Answer 1

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 $\tilde K(BCokerJ)=0$.

Answer 2

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 $\mathbb{Z}\times B\operatorname{Aut}(S)$). The characteristic class map $KO \to h$ 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 $\psi^u - 1 : KO \to KO$ for $u$ a generator of $\mathbb{Z}_p^*/\pm 1$, where the map $h \to h'$ 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 $h \to h'$ 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 $KO \to h \to h'$ identifies with the boundary map in the fiber sequence $L_{K(1)}S \to KO \to KO$. So it's very computable.

Answer 3

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.

Answer 4

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.