On the Classifying Space for Commutativity in $U(3)$

Santanil Jana

Published 2021-06-17Version 1

For a Lie group $G$, let $B_{com} G$ be the classifying space for commutativity. Let $E_{com} G$ be the total space of the principal $G$-bundle associated to $B_{com} G$. In this article, we present a computation of the cohomology of $E_{com} U(3)$ using the spectral sequence associated to a homotopy colimit. As a part of our computation we will also compute the integral cohomology of $U(3)/N(T)$ and $(U(3)/T) \times_{\Sigma_3} (U(3)/T)$ where $T$ is a maximal torus of $U(3)$ with normalizer $N(T)$.