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UBC Theses and Dissertations

Classifying spaces for topological Azumaya algebras Gant, William S.


In this thesis, we study a family of smooth varieties, whose members are denoted B_n^r(C), that bears a similar relationship to topological Azumaya algebras as the Grassmannians G_{n,r}(C) do to complex vector bundles. Specifically, we will show that the varieties B_n^r(C) form homotopical approximations to the classifying space BPGL_n(C). The varieties B_n^r(C) are obtained by first considering the variety of r-tuples of nxn complex matrices that generate the matrix algebra Mat_n(C), and then taking the quotient by an evidently free PGL_n(C)-action. The focus of this thesis is a computation of the singular cohomology groups of B_n^r(C) when n=2. We will show how these cohomological computations have applications in bounding the minimal number of generating sections of a topological Azumaya algebra over a paracompact space.

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