We say that a topological n-manifold N is a cubical n-manifold if it is contained in the n-skeleton of the canonical cubulation C of R n+k (k ≥ 1). In this paper, we prove that any closed, oriented cubical 2manifold has a transverse field of 2-planes in the sense of Whitehead and therefore it is smoothable by a small ambient isotopy. * 2010 Mathematics Subject Classification. Primary: 57R10. Secondary: 57R55, 57R25.

El transporte es vital para el desarrollo social y económico de las sociedades alrededor del mundo. Por lo tanto, se tiene que considerar que el transporte masivo ha incrementado a través del tiempo con el consumo discriminado de combustibles fósiles que han servido para suplir la demanda en el transporte. Existen impactos sociales y medio ambientales que tienen que ser evaluados junto con la demanda de energía en Ecuador. La misma que en 2012 equivalió a 46 millones de BEP, donde 73% fue consumida por vehículos de carga pesada y 32% por vehículos de carga liviana. Este patrón de consumo combinado con una flota de vehículos ineficiente y un subsidio discriminado resulto en un derroche excesivo de energía. La investigación se enfoca en el análisis de la situación del transporte en Ecuador, para identificar el desperdicio de energía, el medio ambiente, y las oportunidades de eficiencia energética en este sector.

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Using the rings of Lipschitz and Hurwitz integers H(Z) and Hur(Z) in the quaternion division algebra H, we define several Kleinian discrete subgroups of P SL(2, H). We define first a Kleinian subgroup P SL(2, L) of P SL(2, H(Z)). This group is a generalization of the modular group P SL(2, Z). Next we define a discrete subgroup P SL(2, H) of P SL(2, H) which is obtained by using Hurwitz integers and in particular the subgroup of order 24 consisting of Hurwitz units. It contains as a subgroup P SL(2, L). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic half space H 1 H := {q ∈ H : (q) > 0}. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients H 1 H /P SL(2, L) and H 1 H /P SL(2, H) which are quaternionic versions of the classical modular orbifold and they are of finite volume. We give a thorough study of the Iwasawa decompositions, affine subgroups, and their descriptions by Lorentz transformations in the Lorentz-Minkowski model of hyperbolic 4-space. We give abstract finite presentations of these modular groups in terms of generators and relations via the Cayley graphs associated to the fundamental domains. We also describe a set of Selberg covers (corresponding to finite-index subgroups acting freely) which are quaternionic hyperbolic manifolds of finite volume with cusps whose sections are 3-tori. These hyperbolic arithmetic 4-manifolds are topologically the complement of linked 2-tori in the 4-sphere, in analogy with the complement in the 3-sphere of the Borromean rings and are related to the ubiquitous hyperbolic 24-cell. Finally we study the Poincaré extensions of these Kleinian groups to arithmetic Kleinian groups acting on hyperbolic 5-space and described in the quaternionic setting. In particular P SL(2, H(Z)) and P SL(2, Hur(Z)) are discrete subgroups of isometries of H 5 R and H 5 R /P SL(2, H(Z)), H 5 R /P SL(2, Hur(Z)) are examples of arithmetic 5-dimensional hyperbolic orbifolds of finite volume.