ar X iv : 0 90 6 . 47 99 v 1 [ m at h . A T ] 2 5 Ju n 20 09 BUNDLES , COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS

The cohomology of the classifying space BU(n) of the unitary groups can be identified with the the ring of symmetric polynomials on n variables by restricting to the cohomology of BT , where T ⊂ U(n) is a maximal torus. In this paper we explore the situation where BT = (CP) is replaced by a product of finite dimensional projective spaces (CP ), fitting into an associated bundle U(n)×T (S ) → (CP ) → BU(n). 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 (CP ) → BU(n). We also calculate the cohomology of the homotopy fiber of the natural map E Sn ×Sn(CP ) → BU(n).