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

In this paper, we describe the total space $E_{com} U(3)$ of the principal $U(3)$-bundle associated with the classifying space for commutativity $B_{com} U(3)$ as a homotopy colimit of a diagram of spaces and offer a computation of the mod $2$ and mod $3$ cohomologies of $E_{com} U(3)$ by utilizing the spectral sequence associated with a homotopy colimit. We investigate the cohomology of different spaces in the homotopy colimit diagram. These spaces are intriguing in their own right and contribute to the overall fascination of the analysis. We also present the ring structure of the rational cohomology of $E_{com} U(3)$.