2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99832In the present paper we study some homotopy invariants which can be defined by means of bundles with fiber a matrix algebra. We also introduce some generalization of the Brauer group in the topological context and show that any its element can be represented as a locally trivial bundle with a group of invertible operators in a Hilbert space as the structure group. Finally, we discuss its possible applications in the twisted $K$-theory.34 pages. v5: The part concerning the generalized Brauer group has been completely rewritten. An application to twisted $K$-theory is addedAlgebraic TopologyK-Theory and Homologytext