Bundles, Cohomology and Truncated Symmetric Polynomials

Details Bundles, Cohomology and Truncated Symmetric Polynomials Bundles, Cohomology and Truncated Symmetric Polynomials

2009-06-25

arxiv.org/abs/0906.4799v2

Mathematics

Algebraic Topology

Some arguments have been streamlined. To appear in Documenta Math

Scientific paper

The cohomology of the classifying space BU(n) of the unitary groups 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).

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