Mario Velasquez

• PhD in Mathematics
• Professor (Associate) at National University of Colombia

In this article we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum-Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the $K$-theory of the redu...

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 gro...

We provide an explicit computation of the topological $K$-theory groups $K_*(C_r^*(\mathbb{Z}^n\rtimes \mathbb{Z}/m))$ of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}Z/m$ with $m$ square-free. We want to highlight the fact that we are not impossing any conditions on the $\Z/m$-action on $\mathbb{Z}^n$. This generalizes previous c...

We provide an explicit computation of the cohomology groups (with un-twisted coefficients) of semidirect products of the form Z n ⋊ Z/m with m free of squares, by means of formulas that only depend on n, m and the action of Z/m on Z n. We want to highlight the fact that we are not impossing any conditions on the Z/m-action on Z n , and as far as we...

Given a manifold with corners X , we associate to it the corner structure simplicial complex \Sigma_X . Its reduced K-homology is isomorphic to the K-theory of the C^* -algebra \mathcal{K}_b(X) of b-compact operators on X . Moreover, the homology of \Sigma_X is isomorphic to the conormal homology of X . In this note, we construct for an arbitrary a...

Let G be a finite group and let X be a compact G-space. In this note we study the (Z+ × Z/2Z)-graded algebra FqG (X) = ⊕n ≤ 0 qn · KG∫Gn(Xn) ⊗ C, defined in terms of equivariant K-theory with respect to wreath products as a symmetric algebra, we review some properties of FqG (X) proved by Segal and Wang. We prove a Kunneth type formula for this gra...

We present a decomposition of rational twisted $G$-equivariant K-theory, $G$ a finite group, into cyclic group equivariant K-theory groups of fixed point spaces. This generalises the untwisted decomposition by Atiyah and Segal as well as the decomposition by Adem and Ruan for twists coming from group cocycles.

Given a manifold with corners $X$, we associates to it the corner structure simplicial complex $\Sigma_X$. Its reduced K-homology is isomorphic to the K-theory of the $C^*$-algebra $\mathcal{K}_b(X)$ of b-compact operators on $X$. Moreover, the homology of $\Sigma_X$ is isomorphic to the conormal homology of $X$. In this note, we constract for an a...

In this paper, we provide a framework for the study of Hecke operators acting on the Bredon (co)homology of an arithmetic discrete group. Our main interest lies in the study of Hecke operators for Bianchi groups. Using the Baum–Connes conjecture, we can transfer computations in Bredon homology to obtain a Hecke action on the [Formula: see text]-the...

In this note we present a complete computation of the topological K-theory of the reduced C*-algebra of a semidirect product of the form $\Gamma=\mathbb{Z}^n\rtimes_\rho\mathbb{Z}/2$ with no further assumptions about of the conjugacy action $\rho$. For this, we use some results for $\mathbb{Z}/2$-equivariant K-theory proved by Rosenberg and previou...

We study decompositions of G-equivariant K-theory for a large class of proper actions of Lie groups, when we have a normal subgroup acting trivially. Similar decompositions were known for the case of a compact Lie group acting on a space, but our main result applies to discrete, linear and almost connected groups. We apply this decomposition to stu...

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners $X$ of any codimension, t...

Given a connected manifold with corners of any codimension there is a very basic and computable homology theory called conormal homology defined in terms of faces and orientations of their conormal bundles, and whose cycles correspond geometrically to corner's cycles. Our main theorem is that, for any manifold with corners X of any codimension, the...

In this note we set a configuration space description of the equivariant connective K-homology groups with coefficients in a unital C*-algebra for proper actions. Over this model we define a connective assembly map and prove that in this setting is possible to recover the analytic assembly map.

In this note we prove a formula for the induced character in equivariant k-theory. More specifically, if $H$ is a subgroup of a finite group $G$ and $X$ is a $G$-CW complex, there is a homomorphism ind$_H^G$ of abelian groups generalizing the induction map of representation rings. On the other hand for the \emph{character} map $char_G$ we prove a f...

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. W...

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. W...

We compare twisted Equivariant K-theory of Sl3Z with untwisted equivariant K-Theory of its universal central extension, St3Z. We compute all twisted equivariant K-theory groups of Sl3Z and by considering work by Tezuka and Yagita, we conclude that they cannot agree with the untwisted equivariant K-theory groups of St3Z. Using universal coeffcient t...

We compare twisted equivariant K-theory of SL3Z with untwisted equivariant K-theory of a central extension St3Z. We compute all twisted equivariant K-theory groups of SL3Z, and compare them with previous work on the equivariant K-theory of BSt3Z by Tezuka and Yagita. Using a universal coefficient theorem by the authors, the computations explained h...

Using a combination of Atiyah-Segal ideas in one side and of Connes and Baum-Connes ideas in the other, we show that the Total Twisted geometric K-homology group of a Lie groupoid (includes Lie groups and discrete groups) admits a ring structure (or module structure for the odd group). This group is the left hand side of the twisted geometric Baum-...

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 comple- tion Theorem of Atiyah-Segal type for twisted equivariant K-Theory. Using a Universal coefficient Theorem, we prove a cocompletion Theor...

We use the 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 Equivariant Bredon cohomology with local coefficients in twisted representations. We furthermore give an explicit descr...

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 action...

We study the relationship between the twisted Orbifold K-theories ${^{\alpha}}K_{orb}(\textsl{X})$ and ${^{\alpha'}}K_{orb}(\textsl{Y})$ for two different twists $\alpha\in Z^3(G;S^1)$ and $\alpha'\in Z^3(G';S^1)$ of the Orbifolds $\textsl{X}=[*/G]$ and $\textsl{Y}=[*/G']$ respectively, for $G$ and $G'$ finite groups. We prove that under suitable h...

Following ideas of Graeme Segal we construct a configuration space that represents equivariant connective K-homology for group actions of finite groups and furthermore we describe explicitly the complex homology of this configuration space as a Hopf algebra. As a consequence of this work we obtain models of representing spaces for equivariant K-the...