# Alejandro Adem

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## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - Mar 2019)

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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##### Master's Student Supervision (2010-2017)

Arrow’s Impossibility Theorem is a classical result in social choice theory (a branch of economic theory), which states that any system of rules for combining (“aggregating”) individual preference relations into a single representative relation results in a “dictatorship” where the combined preference only reflects the wishes of a single individual (provided that the aggregation rule satisfies two basic criteria). Since the 1980s, this result has been reformulated and understood using algebraic topology. The topological approach offers some geometric intuition as to why Arrow’s theorem holds, and can also be used to find alternative hypotheses which may escape the dictatorship outcome. A thorough examination of such topological models constitutes the main body of this thesis.Recently, social choice theory has been generalized (resulting in a field called “judgment aggregation”), and results analogous to Arrow's theorem have been established. The second part of this thesis introduces this field of study, and studies how some of the techniques from topological social choice theory can be extended to understand dictatorship outcomes in the theory of judgment aggregation. Although the analysis is restricted to a rather simple case, it nonetheless highlights the potential for a more general topological model of judgment aggregation, and exposes the main challenges that must be overcome in constructing such a theory.

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