EQUIVARIANT STABLE HOMOTOPY THEORY FOR PROPER ACTIONS OF DISCRETE GROUPS

Following ideas of Graeme Segal [Segal(1973)], [Segal(1968)], Christian Schlichtkrull [Schlichtkrull(2007)] and Kazuhisa Shimakawa [Shimakawa(1989)] we construct equivariant stable homotopy groups for proper equivariant CW complexes with an action of a discrete group

The ideal-valued index of fibrations with total space a G2G_{2} flag manifold

Using the cohomology of the $G_2$-flag manifolds $G_2/U(2)_{\pm}$, and their structure as a fiber bundle over the homogeneous space $G_2/SO(4)$, we compute their Borel cohomology and the $\mathbb{Z}_2$ Fadell-Husseini index of such fiber bundles, for the $\mathbb{Z}_2$ action given by complex conjugation. Considering the orthogonal complement of the tautological bundle $\gamma$ over $\widetilde{G}_{3}( \mathbb{R}^{7})$, we compute the $\mathbb{Z}_2$ Fadell-Husseini index of the pullback bundle of $s\gamma^{\perp}$ along the composition of the embedding between $G_2/SO(4)$ and $\widetilde{G}_{3}( \mathbb{R}^{7})$, and the fiber bundle $G_2/U(2)_{\pm} \to G_2/SO(4)$. Here $s\gamma^{\perp}$ means the associated sphere bundle of the orthogonal bundle $\gamma^{\perp}$. Furthermore, we derive a general formula for the $n$-fold product bundle $(s\gamma^{\perp})^n$ for which we make the same computations.Comment: 20 pages, 6 figure

Rigidity of actions on metric spaces close to three dimensional manifolds

In this article we propose a metric variation on the C^0-version of the Zimmer program for three manifolds. After a reexamination of the isometry groups of geometric three-manifolds, we consider homomorphisms defined on higher rank lattices to them and establish a dichotomy betweeen finite image or infinite volume of the quotient. Along the way, we enumerate classification results for actions of finite groups on three manifolds where available, and we give an answer to a metric variation on topological versions of the Zimmer program for aspherical three manifolds, as asked by Weinberger and Ye, which are based on the dichotomy stablished in this work and known topological rigidity phenomena for three manifolds. Using results by John Pardon and Galaz-Garc\'ia-Guijarro, the dichotomy for homomorphisms of higher rank lattices to isometry groups of three manifolds implies that a C 0 -isometric version of the Zimmer program is also true for singular geodesic spaces closely related to three dimensional manifolds, namely three dimensional geometric orbifolds and Alexandrov spaces. A topological version of the Zimmer Program is seen to hold in dimension 3 for Alexandrov spaces using Pardon's ideas

Brown representability and spaces over a category

We prove a Brown Representability Theorem in the context of spaces over a category. We discuss two applications to the representability of equivariant cohomology theories, with emphasis on Bredon cohomology with local coefficients.Probamos un teorema de representabilidad de Brown en el contexto de espacios sobre una categoría. Discutimos dos aplicaciones a la representabilidad de teorías de cohomología, con énfasis en cohomología de Bredon con coeficientes locales

Proper actions, nonlinearity and homotopy theory

In dieser Arbeit wird die Erweiterung der Methoden der äquivarianten stabilen Homotopietheorie zu breiteren Kontexten untersucht. Die klassische Theorie voraussetzt Kompaktheit oder sogar Endlichkeit an der wirkenden Gruppe. Äquivariante Homotopie und Kohomotopie werden durch Spektren und analytische Methoden für eigentliche G-CW Komplexe konstruiert. Die Übereinstimmung mit der klassischen Definition, sowie zu einer von Lück 2005 veröffentlichten Konstruktion mittels Vektorraumbündeln wird bewiesen. Die Segal Vermutung wird in zwei Versionen verallgemeinert (für familien endlicher Untergruppen in diskreten Gruppen, bzw. für halb-einfache Liegruppen deren maximale kompakte Untergruppe keine Fundamentaldarstellung quaternionischen Typs aufweist). Eine bivariante , äquivariante homotopietheorie für C*-Algebren wird auch definiert

Subclinical endometritis and pregnancy rate in dairy cows in Mexico

En el presente estudio se determinó la prevalencia de endometritis subclínica y su efecto en la tasa de gestación en vacas lecheras en manejo intensivo. Entre los días 35 y 45 posparto se seleccionaron 237 vacas sin patologías uterinas (vacas con útero involucionado y sin secreciones vaginales anormales); a todas ellas se les tomó una muestra citológica de endometrio mediante la técnica de Cytobrush. Los frotis se procesaron con la tinción de Diff-Quick. Se contaron un total de 200 células y se determinó la proporción de células polimorfonucleares (PMN). Las vacas con ≥ 6 % de PMN se consideraron con endometritis subclínica. Cuarenta y tres por ciento (103/237) de las vacas presentó endometritis subclínica. Las vacas con puerperio patológico tuvieron mayor probabilidad de padecer endometritis subclínica que las vacas con puerperio normal (60 vs 36 %, respectivamente; razón de probabilidad= 2.69). Las vacas normales tuvieron mayor tasa de gestación en el primer servicio que la vacas con endometritis subclínica (31.4 vs 20.0 %; respectivamente; razón de probabilidad= 2.02) y la proporción de vacas gestantes en el día 120 posparto fue mayor en las vacas normales que en las vacas con endometritis subclínica (51.6 vs 37.9 %, respectivamente; razón de probabilidad= 1.98). Se concluye que la prevalencia de endometritis subclínica en vacas lecheras entre los días 35 y 45 posparto es de 43 %, lo cual disminuyó en un 11.4 % la tasa de gestación en el primer servicio y en un 13.7 % la tasa de gestación acumulada en el día 120 posparto.The present study determined the prevalence of subclinical endometritis and its effect on pregnancy rate in intensively managed dairy cows. Between d 35 and 45 postpartum, 237 cows with no uterine pathologies judged to have completed uterine involution and no abnormal vaginal secretions were selected. Endometrial cytology samples were taken from each cow using the Cytobrush technique, and smears were stained with Diff-Quick. A total of 200 sample cells were counted and the proportion of polymorphonuclear leukocytes (PMN) was estimated. Cows with ≥6 % PMN were considered to have subclinical endometritis; 43 % (103/237) of cows sampled met this criterion. Cows with pathological puerperium had a higher probability of undergoing subclinical endometritis than cows with normal puerperium (60 vs 36 %, respectively; odds ratio= 2.69). Cows not presenting subclinical endometritis had a higher pregnancy rate at first service than cows with this condition (31.4 vs 20.0 %, respectively; odds ratio= 2.02), and the percentage of pregnant cows at d 120 postpartum was higher in cows not experiencing subclinical endometritis than in cows with this condition (51.6 vs 37.9 %, respectively; odds ratio= 1.98). It concludes that 43 % of cows suffered subclinical endometritis between d 35 and 45 postpartum, which decreased pregnancy rate at first service, as well as cumulative pregnancy rate at d 120 postpartum.Consejo Nacional de Investigaciones Científicas y TécnicasFacultad de Ciencias Veterinaria