Bundles, Cohomology and Truncated Symmetric Polynomials

TL;DR

This paper investigates the algebraic and topological properties of truncated symmetric polynomials and their relation to the cohomology of classifying spaces, extending classical results to finite-dimensional projective spaces.

Contribution

It provides an explicit algebraic description of truncated symmetric polynomial ideals and characterizes the kernel of the induced cohomology homomorphism for finite-dimensional projective spaces.

Findings

Explicit generators for the ideal of truncated symmetric polynomials.

Description of the kernel of the cohomology map induced by (CP^d)^n -> BU(n).

Cohomology calculation of the homotopy fiber of the natural map.

Abstract

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).