arXiv subjects
Publications and source records attributed to Edward Becerra.
In this paper it is shown how to construct a finite topological space $X$ for a given finitely presentable group $G$ such that $π_1(X)\cong G$. Our construction is not optimal in the sense that the cardinality of the space $X$ might not be the smallest possible. Our main result applies to a large class of interesting groups, including all finite groups.
In this paper we study a natural decomposition of $G$-equivariant $K$-theory of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup $A$ acting trivially. Our decomposition could be understood as a generalization of the theory known as Mackey machine under suitable hypotheses, since it decomposes $G$-equivariant K-theory in terms of twisted equivariant K-theory groups respect to some subgroups of $G/A$. 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 also apply this decomposition to study equivariant $K$-theory of spaces with only one isotropy type. We provide a rich class of examples in order to expose the strength and generality of our results. We also study the decomposition for equivariant connective $K$-homology for actions of compact Lie groups using a suitable configuration space model, based on previous papers published by the third author.
In this paper we put together some tools from differential topology and analysis in order to study second order semi-linear partial differential equations on a Riemannian manifold $M$. We look for solutions that are constants along orbits of a given group action. Using some results obtained by Helgason in [J DIFFER GEOM,6(3), 411-419] we are able to write a (reduced) second order semi-linear problem on a submanifold $Σ$. This submanifold is, in certain sense, transversal to the orbits of the group actions and its existence is assumed. We describe precise conditions on the Riemannian Manifold $M$ and the submanifold $Σ$ in order to be able to write the reduced equation on $Σ$. These conditions are satisfied by several particular cases including some examples treated separately in the literature such as the sphere, surfaces of revolution and others. Our framework also includes the setup of polar actions or exponential coordinates. Using this procedure, we are left with a second order semi-linear equation posed on a submanifold. In particular, if the submanifold $Σ$ is one-dimensional, we can use suitable tools from analysis to obtain existence and properties of solutions.
We study the relationship between the twisted Orbifold K-theories ${^α}K_{orb}(\textsl{X})$ and ${^{α'}}K_{orb}(\textsl{Y})$ for two different twists $α\in Z^3(G;S^1)$ and $α'\in Z^3(G';S^1)$ of the Orbifolds $\textsl{X}=[*/G]$ and $\textsl{Y}=[*/G']$ respectively, for $G$ and $G'$ finite groups. We prove that under suitable hypothesis over the twisting $α'$ and the group $G'$ we obtain an isomorphism between these twisted K-theories.
In this paper we present a model to calculate the stringy product on twisted orbifold K-theory of Adem-Ruan-Zhang for abelian complex orbifolds. In the first part we consider the non-twisted case on an orbifold presented as the quotient of a manifold acted by a compact abelian Lie group. We give an explicit description of the obstruction bundle, we explain the relation with the product defined by Jarvis-Kaufmann-Kimura and, via a Chern character map, with the Chen-Ruan cohomology, and we explicitely calculate the stringy product for a weighted projective orbifold. In the second part we consider orbifolds presented as the quotient of a manifold acted by a finite abelian group and twistings coming from the group cohomology. We show a decomposition formula for twisted orbifold K-theory that is suited to calculate the stringy product and we use this formula to calculate two examples when the group is $(\integer/2)^3$.