Euler Characteristics of Categories and Homotopy Colimits

In a previous article, we introduced notions of finiteness obstruction, Euler characteristic, and L^2-Euler characteristic for wide classes of categories. In this sequel, we prove the compatibility of those notions with homotopy colimits of I-indexed categories where I is any small category admitting a finite I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit Formula include formulas for products, homotopy pushouts, homotopy orbits, and transport groupoids. We also apply our formulas to Haefliger complexes of groups, which extend Bass--Serre graphs of groups to higher dimensions. In particular, we obtain necessary conditions for developability of a finite complex of groups from an action of a finite group on a finite category without loops.Comment: 44 pages. This final version will appear in Documenta Mathematica. Remark 8.23 has been improved, discussion of Grothendieck construction has been slightly expanded at the beginning of Section 3, and a few other minor improvements have been incoporate

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and M\"obius inversion for wide classes of categories. The finiteness obstruction of a category Gamma of type (FP) is a class in the projective class group K_0(RGamma); the functorial Euler characteristic and functorial L^2-Euler characteristic are respectively its RGamma-rank and L^2-rank. We also extend the second author's K-theoretic M\"obius inversion from finite categories to quasi-finite categories. Our main example is the proper orbit category, for which these invariants are established notions in the geometry and topology of classifying spaces for proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler characteristic are special cases of the L^2-Euler characteristic. Some of Leinster's results on M\"obius-Rota inversion are special cases of the K-theoretic M\"obius inversion.Comment: Final version, accepted for publication in the Advances in Mathematics. Notational change: what was called chi(Gamma) in version 1 is now called chi(BGamma), and chi(Gamma) now signifies the sum of the components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25 summarizes when all Euler characteristics are equal. Minor typos have been corrected. 88 page

The Ore condition, affiliated operators, and the lamplighter group

Let G be the wreath product of Z and Z/2, the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1-x is invertible, with x a generator of Z considered as subset of G. Then R is not flat over kG. If k is the field of complex numbers, this applies in particular to the algebra UG of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.Comment: LaTex2e, 7 pages. Added a new proof of the main result (due to Warren Dicks) which is shorter, easier and more elementary, and at the same time yields a slightly more general result. Additionally: misprints removed. to appear in Proceedings of "Higher dimensional manifold theory", Conference at ICTP Trieste 200

Finite group extensions and the Baum-Connes conjecture

In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups, or certain fundamental groups of complements of links in S^3.Comment: AMS-Latex, logical structure clarified, final version, to appear in Geometry and Topolog

Auswirkungen der Eurokrise auf Sachsen und allgemeine Implikationen der Krise

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Berufserfolg Bamberger Soziologen: Ergebnisse der ersten Bamberger Absolventenstudie

Die Autoren berichten über die Ergebnisse einer Studie über den Berufsverlauf von Bamberger HochschulabsolventInnen, die im Zeitraum 1997/98 erstmals vom Lehrstuhl für Soziologie I durchgeführt wurde. Dargestellt wird zum einen die Berufseinmündung und die aktuelle Beschäftigungssituation der insgesamt 116 Soziologie-AbsolventInnen, die an der schriftlichen Befragung teilnahmen. Zum anderen werden die Einflussfaktoren auf die Berufseinmündung und den Berufserfolg beschrieben, die sich bei den sozio-demographischen Variablen, den studieninternen und -externen Qualifikationen, bei der Wahl der Studienschwerpunkte und beim Erreichen individuell gesetzter Studienziele erkennen lassen. Für die verschiedenen Studienschwerpunkte innerhalb der Soziologie bzw. für die verschiedenen anvisierten Berufsfelder lassen sich ferner unterschiedliche Karrierechancen ausmachen. Von den für Bamberg charakteristischen Berufsfeldern scheinen die PR-Branche und die wissenschaftliche Karriere die günstigsten Vorzeichen zu setzen. (ICI

Finiteness obstructions and Euler characteristics of categories

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Characterizing degradation gradients through land cover change analysis in rural Eastern Cape, South Africa

CITATION: Munch, Z., et al. 2017. Characterizing degradation gradients through land cover change analysis in rural Eastern Cape, South Africa. Geosciences, 7(1):7, doi:10.3390/geosciences7010007.The original publication is available at http://www.mdpi.comLand cover change analysis was performed for three catchments in the rural Eastern Cape, South Africa, for two time steps (2000 and 2014), to characterize landscape conversion trajectories for sustained landscape health. Land cover maps were derived: (1) from existing data (2000); and (2) through object-based image analysis (2014) of Landsat 8 imagery. Land cover change analysis was facilitated using land cover labels developed to identify landscape change trajectories. Land cover labels assigned to each intersection of the land cover maps at the two time steps provide a thematic representation of the spatial distribution of change. While land use patterns are characterized by high persistence (77%), the expansion of urban areas and agriculture has occurred predominantly at the expense of grassland. The persistence and intensification of natural or invaded wooded areas were identified as a degradation gradient within the landscape, which amounted to almost 10% of the study area. The challenge remains to determine significant signals in the landscape that are not artefacts of error in the underlying input data or scale of analysis. Systematic change analysis and accurate uncertainty reporting can potentially address these issues to produce authentic output for further modelling.http://www.mdpi.com/2076-3263/7/1/7Publisher's versio

Euler characteristics of categories and homotopy colimits

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