# Bundles, cohomology and truncated symmetric polynomials

### Alejandro Adem

Department of Mathematics Department of Mathematics University of British Columbia University of British Columbia Vancouver BC V6T 1Z2 Vancouver BC V6T 1Z2 Canada Canada### Zinovy Reichstein

Department of Mathematics Department of Mathematics University of British Columbia University of British Columbia Vancouver BC V6T 1Z2 Vancouver BC V6T 1Z2 Canada Canada

## Abstract

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 $E S_n\times_{S_n}(\ CP^d)^n\to BU(n)$.

## Cite this article

Alejandro Adem, Zinovy Reichstein, Bundles, cohomology and truncated symmetric polynomials. Doc. Math. 15 (2010), pp. 1029–1047

DOI 10.4171/DM/323