Bundles, Cohomology and Truncated Symmetric Polynomials
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).
Keywords
Cite
@article{arxiv.0906.4799,
title = {Bundles, Cohomology and Truncated Symmetric Polynomials},
author = {Alejandro Adem and Zinovy Reichstein},
journal= {arXiv preprint arXiv:0906.4799},
year = {2017}
}
Comments
Some arguments have been streamlined. To appear in Documenta Math