Alejandro Adem and Zinovy Reichstein: Bundles, Cohomology and Truncated Symmetric Polynomials

Submission: 2009, Jun 25

The cohomology of the classifying space BU(n) of the unitary group can be identified with the ring of symmetric polynomials on n variables by restricting to the cohomology of BT, where T is a maximal torus in U(n). In this paper we explore the situation where BT = (CP^{infinity})^n is replaced by a product of finite dimensional projective spaces (CP^d)^n, fitting into an associated bundle U(n) x_T (S^{2d+1})^n -> (CP^d)^n -> 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^d)^n -> BU(n). We also calculate the cohomology of the homotopy fiber of the natural map ES_n x_{S_n} (CP^d)^n -> BU(n).

2000 Mathematics Subject Classification: 55R35; 05E95

Keywords and Phrases: Classifying space, bundle, cohomology, symmetric polynomial, regular sequence

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