Alejandro Adem

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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In this thesis, we establish decomposition theorems for topological Azumaya algebras, and topological Azumaya algebras with involutions of the first kind. Decomposition of topological Azumaya algebrasLet A be a topological Azumaya algebra of degree mn over a CW complex X. We prove that if m and n are relatively prime, m
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The present thesis consists of 2 parts. Chapter 1 is about applications of twisted equivariant K-theory to condensed matter. We consider non-interacting electrons on a half-crystal (a crystal with a boundary), with a gapped bulk condition, under quasi-adiabatic evolution. In A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 we found that Fermi surfaces for these systems under quasi-adiabatic evolution are classified by the K⁻¹-group of the surface Brillouin zone Td⁻¹. Systems with time-reversal and particle-hole symmetry were also considered and we obtained different KR-groups for the different cases. In Chapter 1 I rewrite A. Adem, O. Antolin, G. Semenoff and D. Sheinbaum JHEP, 2016 in a more function-analytic language and further solve technical issues to extend it to include crystallographic symmetries on the directions parallel to the boundary. In Chapter 2 I reproduce the relevant parts of my joint work with C. Okay (C. Okay and D. Sheinbaum arXiv:1905.07723). There we explored a connection between twisted equivariant K-theory to contextuality in quantum mechanics. We also reformulated the sheaf-theoretic framework of S. Abramsky and A. Brandenburger New Journal of Physics, 2011 for contextuality and connect it to another one employing a group cohomology approach of C. Okay, S. Roberts, S.D Bartlett, and R. Raussendorf Quantum Information and Computation, 2017. This leads to the construction of a classifying space for contextuality, from which Wigner functions are classes in its twisted K-theory.

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Let G be a real linear algebraic group and L a finitely generated cosimplicial group. We prove that the space of homomorphisms Hom(Ln;G) has a homotopy stable decomposition for each n ≥ 1. When G is a compact Lie group, we show that the decomposition is G-equivariant with respect to the induced action of conjugation by elements of G. In particular, under these hypotheses on G, we obtain stable decompositions for Hom(Fqn;G) and Rep(Fqn;G) respectively, where Fqn are the finitely generated free nilpotent groups of nilpotency class q-1. The spaces Hom(Ln;G) assemble into a simplicial space Hom(L;G). When G=U we show that its geometric realization B(L;U) has a non-unital E-infinity-ring space structure whenever Hom(L0;U(m)) is path connected for all m ≥ 0.We describe the connected components of Hom(Fqn;SU(2)) arising fromnon-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalizedquaternion groups Q2q in SU(2), of order 2^q. Using this result, we exhibit thehomotopy type of SHom(Fqn;SU(2)) and a homotopy description of the classifying spaces B(q;SU(2)) of transitionally q-nilpotent principal SU(2)-bundles. The above computations are also done for SO(3) and U(2).Finally, for q = 2, the space B(2;G) is denoted BcomG, and we compute theintegral cohomology ring for the Lie groups G = SU(2) andU(2). We also includecohomology calculations for the spaces BcomQ2q .

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This dissertation studies semidirect products of a torus by a finite group from the representation theory point of view. The finite group of greatest interest is the cyclic group of prime order. Such semidirect products occur in nature as isotropy groups of Lie groups acting on themselves by conjugation and as normalizers of maximal tori in reductive linear algebraic groups. The main results of this dissertation are: a) the calculation of the representation ring of such semidirect products as an algebra over the integers for certain special cases, b) the adaptation of an algorithm from invariant theory to find finite presentations of representation rings, c) the computation of the topological K-theory of the classifying space of certain semidirect products, d) the demonstration that the equivariant K-theory of the projective unitary group of degree 2 acting on itself by conjugation is not a free module over its representation ring.

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The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G) using the descending central series of free groups. If G is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this thesis we introduce natural subspaces B(q,G)_p of B(q,G) defined for a fixed prime p. We show that B(q,G) is stably homotopy equivalent to a wedge sum of B(q,G)_p as p runs over the primes dividing the order of G. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial p-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial p-groups of rank at least 4 the space B(2,G) does not have the homotopy type of a K(π,1) space. Furthermore, we give a group theoretic condition, applicable to symmetric groups and general linear groups, which implies the space B(2,G) not having the homotopy type of a K(π,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.

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In this work we study the space of group homomorphisms Hom (π,G) forspecial choices of π and G. In the first part of this thesis, we enumerate anddescribe the path components for the spaces of ordered commuting k-tuplesof orthogonal and special orthogonal matrices respectively. This correspondsto choosing π = ℤk and G = O(n); SO(n). We also provide a lower boundon the number of components for the case G = Spin(n) for suffciently largen. In the second part, we describe the space Hom (Г,SU(2)), where Г is agroup arising from a central extension of the form0 → ℤr → Г → ℤk → 0.The description of this space is good enough that, using some known results,it allows us to compute its cohomology groups.

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In the first part of the thesis we discuss the rank conjecture of Benson and Carlson. In particular, we prove that if G is a finite p-group of rank 3 and with p odd, or if G is a central extension of abelian p-groups, then there is a free finite G-CW-complex homotopy equivalent to the product of rk(G) spheres; where rk(G) is the rank of G.We also treat an extension of the rank conjecture to groups of finite virtual cohomological dimension. In this context, for p a fixed odd prime, we show that there is an infinite group L satisfying the two following properties: every finite subgroup G
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In this thesis we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the third power of modulo two group via the pairing of projective representations where the three cocycles are chosen using the inverse transgression map. We find the rank of the fusion algebra as well as the relation between its basis elements. We also give some applications to topological gauge theories. We next show that the twisted fusion algebra of the third power of modulo p group is isomorphic to the non-twisted fusion algebra of the extraspecial p-group of order p³ and exponent p.The final point of my thesis is to explicitly compute the cohomology groups of toroidal orbifolds which are the quotient spaces obtained by the action of modulo p group on the k-dimensional torus. We compute the particular case where the action is induced by the n-th power of augmentation ideal.

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Equivariant K-theory for actions of groupoids is defined and shown to bea cohomology theory on the category of finite equivariant CW-complexes.Under some conditions, these theories are representable. We use this fact todefine twisted equivariant K-theory for actions of groupoids. A classificationof possible twistings is given. We also prove a completion theorem for twistedand untwisted equivariant K-theory. Finally, some applications to proper actions of Lie groups are discussed.

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