arXiv subjects
Publications and source records attributed to Noe Barcenas.
We examine positive and negative results for the Gromov-Lawson-Rosenberg Conjecture within the class of crystallographic groups. We give necessary conditions within the class of split extensions of free abelian by cyclic groups to satisfy the unstable Gromov-Lawson-Rosenberg Conjecture. We also give necessary conditions within the same class of groups, producing an infinite number of counterexamples for the conjecture.
We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by computing the connective real k-homology of the classifying space with the Adams spectral sequence and two types of detection theorems for the kernel of the alpha invariant: one based on eta-invariants, closely following work of Botvinnik-Gilkey-Stolz, and a second one based on homological methods. Along the way, we determine differentials of the Adams spectral sequence for classifying spaces involved in the computation, and we study the cap structure of the Adams spectral sequence for sub-hopf algebras of the Steenrod algebra relevant to the computation of connective real and complex k-homology.
We present an overview of computational methods for Bredon cohomology with a special focus on infinite groups
We show that the twisted K-theory of the classifying space of a p-local finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.
We compare different algebraic structures in twisted equivariant K-Theory for proper actions of discrete groups. After the construction of a module structure over untwisted equivariant K-Theory, we prove a completion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theorem for Twisted Borel K-Homology for discrete Groups.
We use a spectral sequence developed by Graeme Segal in order to understand the twisted G-equivariant K-theory for proper and discrete actions. We show that the second page of this spectral sequence is isomorphic to a version of Bredon cohomology with local coefficients in twisted representations. We furthermore explain some phenomena concerning the third differential of the spectral sequence, and we recover known results when the twisting comes from finite order elements in discrete torsion.
We define twisted equivariant K-homology groups using geometric cycles. We compare them with approaches using Kasparov KK-Theory and (twisted) group C*-algebras.
We extend a work of Bartsch, Clapp and Puppe on the Mountain pass theorems. We consider functionals invariant with respect to infinite discrete groups satisfying a maximality condition on the finite subgroups.
We develop an Eilenberg-Moore spectral sequence to compute Bredon cohomology of spaces with an action of a group given as a pullback. Using several other spectral sequences, and positive results on the Baum-Connes Conjecture, we are able to compute Equivariant K-theory and K-Homology of the reduced group C*-algebra of a 6-dimensional crystallographic group $Γ$ introduced by Vafa and Witten. We also use positive results on the Farrell-Jones Conjecture to give a vanishing result for the negative algebraic K-theory of the integral group ring of $Γ$.
We compare twisted Equivariant K-theory of Sl3Z with untwisted equivariant K-Theory of its universal central extension, St3Z. Using universal coefficient theorems by the authors, the computations explained here give the domain of Baum-Connes assembly maps landing on the topological K-theory of twisted group C*-algebras related to Sl3Z, for which a version of Poincaré Duality studied previously by Echterhoff, Emerson and Kim is verified.
We define equivariant projective unitary stable bundles as the appropriate twists when defining K-theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah-Segal.
Replaces Previous version. Includes comments on poincare duality for twisted equivariant in the context of proper and discrete actions and the Baum-Connes Conjecture. We use a spectral sequence proposed by C. Dwyer and previous work by Sanchez-Garcia and Soule to compute Twisted Equivariant K-theory groups of the classifying space for proper actions of Sl3(Z). After proving a Universal coefficient theorem in Bredon Cohomology with specific coefficients, we compute the twisted equivariant K-homology and state a relation to the Baum-Connes Conjecture with coefficients.
In this article we extend the classical definitions of equivariant cohomotopy theory to the setting of proper actions of Lie groups. We combine methods originally developed in the analysis of nonlinear differential equations, mainly in connection with Leray-Schauder theory, and on the other hand from developments of equivariant $K$-Theory by N.C. Phillips. We prove the correspondence with a previous construction of W. Lück by constructing an index. As an illustration of these methods, we introduce a Burnside ring defined in analytical terms. With this definition, we extend a weak version of the Segal Conjecture to a certain family of Lie groups and comment the relation to an invariant in Gauge Theory, due to Bauer and Furuta.