Bundles, Cohomology and Truncated Symmetric Polynomials

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 $BT = (\ CP^{\infty})^n$ is replaced by a product of finite dimensional projective spaces $(\ CP^d)^n$, fitting into an associated bundle $$U(n)\times_T (\ S^{2d+1})^n\to (\ CP^d)^n\to BU(n).$$ 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 $(\ CP^d)^n\to BU(n)$. We also calculate the cohomology of the homotopy fiber of the natural map $E S_n\times_{S_n}(\ CP^d)^n\to BU(n)$.

2010 Mathematics Subject Classification: 55R35, 05E05

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

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