Common subbundles and intersections of divisors

Abstract

Let V 0 and V 1 be complex vector bundles over a space X . We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when V 0 and V 1 can be embedded in a bundle U in such a way that V 0 ∩ V 1 has dimension at least k everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory.

An equivariant generalization of the Miller splitting theorem

“…Let V and W be Hermitian spaces, ie complex vector spaces equipped with Hermitian inner products, such that dim.V / 6 dim.W /. We refer the reader to an overview given by Strickland in [12, Appendix A] for a review of the original theory of functional calculus and we take as given knowledge of all results and statements made in [12]. We also follow the conventions taken in the referenced paper, though we make three notational changeswe use | instead of for adjoint, we use s.V / rather than w.V / for the space of self-adjoint endomorphisms of V and if˛2 s.V / we denote the eigenvalues of alpha (which are real numbers as standard) by e 0 .˛/ 6 e 1 .˛/ 6 ordered by the standard 6 ordering on R. Our norms on spaces of linear maps are assumed to be operator norms.…”

On Morava K-theories of an S-arithmetic Group

“…Proof of 1.1. The proof of the main theorem is based on Strickland's analysis of unitary bundles in [11] applied to the homotopy fibre square 3.1. Let V be a complex vector bundle over a space X and write P V for the associated bundle of projective spaces and U (V ) for the associated bundle of unitary groups U (V ) = {(x, g)|x ∈ X and g ∈ U (V x )} Let EU (V ) denote the geometric realization of the simplicial space {U (V ) n+1 } n≥0 and put BU (V ) = EU (V )/U (V ) the usual simplicial model for the classifying space of U (V ).…”