Homotopy theory of bundles with fiber matrix algebra

TL;DR

This paper develops a homotopy-theoretic framework for bundles with matrix algebra fibers, providing a geometric understanding of the H-space structure on classifying spaces related to tensor products of special unitary bundles.

Contribution

It introduces a stable theory for bundles with matrix algebra fibers and describes the H-space structure on SU in geometric terms.

Findings

Defined an equivalence relation on such bundles over finite CW-complexes.

Provided a geometric description of the SU_taplus structure.

Established connections between bundle theory and homotopy-theoretic structures.

Abstract

In the present paper we consider a special class of locally trivial bundles with fiber a matrix algebra. On the set of such bundles over a finite $CW$-complex we define a relevant equivalence relation. The obtained stable theory gives us a geometric description of the H-space structure $\BSU_\otimes$ on $\BSU$ related to the tensor product of virtual $\SU$-bundles of virtual dimension 1.