UBC Theses and Dissertations

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UBC Theses and Dissertations

Classifying space for commutativity and unordered flag manifolds Jana, Santanil


The first part of this thesis is dedicated to the computation of the cohomology of the total space of the principal U(3)-bundle of the classifying space for commutativity for U(3). The second part delves into examining the stable and unstable cohomology rings of the unordered flag manifolds. Classifying space for commutativity in U(3). The total space of the principal G-bundle associated with the classifying space for commutativity BcomG is denoted by EcomG. In Chapter 2, we describe EcomU(3) as a homotopy colimit of a diagram of spaces and detail a method of computation of the mod p cohomology of EcomU(3) by utilizing the spectral sequence associated with a homotopy colimit. In order to perform this computation, it is important to determine the cohomology of various spaces that are present in the homotopy colimit diagram. We present some of these computations, which are fascinating on their own, and delve into intriguing topics that we explore further in Chapters 3 and 4. We also present the ring structure of the rational cohomology of EcomU(3). Cohomology of the unordered flag manifolds. Unordered flag manifolds are the orbit spaces of the natural action of the symmetric group on the complete flag manifolds. The complex unordered flag manifold of order n can be defined as the total space of a fiber sequence, wherein the base is the extended power of U(1), and the fiber is U(n). In Chapter 3, we establish the Hopf ring structure of the cohomology of extended power of a space. Utilizing this description, we prove the homological stability of the complex unordered flag manifolds and describe their stable cohomology. In Chapter 4, we demonstrate a pullback formula and present an algorithmic approach to computing the unstable cohomology of the unordered flag manifolds using spectral sequences. We also detail a few low-dimensional cohomology computations with mod 2 coefficients and offer an infinite family of examples, namely the mod p cohomology of the unordered flag manifolds of order p.

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