arXiv subjects

Wolfgang Lück

Publications and source records attributed to Wolfgang Lück.

21 records · Page 2Linked to original sources

arXiv2011

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.

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arXiv2010

Finiteness obstructions and Euler characteristics of categories

We introduce notions of finiteness obstruction, Euler characteristic, L^2-Euler characteristic, and Möbius 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 Möbius 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 Möbius-Rota inversion are special cases of the K-theoretic Möbius inversion.

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arXiv2009

Obstructions to Fibering a Manifold

Given a map f: M \to M of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S^1, these torsion obstructions are identified with the ones due to Farrell. We have changed the exposition according to the comments of the referee and corrected some typos. The paper will appear in Geometriae Dedicata.

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