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 $V_0\cap V_1$ 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.