Volume 258, 2000
Complex cobordism of Hilbert manifolds with some
applications to flag varieties of loop groups
We develop a version of Quillen's geometric cobordism theory for
infinite dimensional separable Hilbert manifolds. This cobordism theory has a
graded group structure under the topological union operation and has push-
forward maps for Fredholm maps. We discuss transverse approximations and
products, and the contravariant property of this cobordism theory. We define
Euler classes for finite dimensional complex vector bundles and describe some
applications to the complex cobordism of flag varieties of loop groups.
Quillen gave a geometric interpretation of complex cobordism groups
which suggests a way of defining the cobordism of separable Hilbert manifolds. In
order that such an extension be reasonable, it ought to reduce to his construction
for finite dimensional manifolds and also be capable of supporting calculations for
important types of infinite dimensional manifolds such as homogeneous spaces of
free loop groups of finite dimensional Lie groups and Grassmannians.
In this paper, we outline an extension of Quillen's work to separable Hilbert
manifolds and discuss its main properties. Although we are able to verify some
expected features, there appears to be a serious gap in the literature on infinite
dimensional transversality and without appropriate transverse approximations of
Fredholm and smooth maps we are unable to obtain contravariance or product
structure. However, covariance along Fredholm maps does hold as does contravari-
ance along submersions.
the relevant infinite dimensional transversality results
are indeed true then our version of Quillen's theory may be of wider interest. A
major motivation for the present work lay in the desire to generalize to loop groups
the finite dimensional results of Bressler & Evens [2, 3], and as a sample of appli-
cations, we describe some cobordism classes for flag varieties of loop groups and
related spaces which appeared in the second author's PhD thesis
We would like to thank Jack Morava for bringing the volume 'Global Analysis'
to our attention at a crucial moment.
2000 Mathematics Subject Classification. Primary 55N22 57R77 57R19 58B05; Secondary
Key words and phrases. cobordism, Fredholm map, Hilbert manifold, flag variety, loop group.
2000 American Mathematical Society