Bundles, cohomology and truncated symmetric polynomials

Abstract

The cohomology of the classifying space $BU(n)$ of the unitary group can be identified with the the ring of symmetric polynomials on $n$ variables by restricting to the cohomology of $BT$, where $T\subset U(n)$ is a maximal torus. In this paper we explore the situation where $BT=(C{P}^{\infty }{)}^n$ is replaced by a product of finite dimensional projective spaces $(C{P}^d{)}^n$, fitting into an associated bundle

We establish a purely algebraic version of this problem by exhibiting an explicit system of generators for the ideal of truncated symmetric polynomials. We use this algebraic result to give a precise descriptions of the kernel of the homomorphism in cohomology induced by the natural map $(C{P}^d{)}^n\to BU(n)$. We also calculate the cohomology of the homotopy fiber of the natural map $E{S}_n{\times }_{S_n}(C{P}^d{)}^n\to BU(n)$.