25,880 research outputs found

    Euler Characteristic in Odd Dimensions

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    It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes

    A C*-algebra of geometric operators on self-similar CW-complexes. Novikov-Shubin and L^2-Betti numbers

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    A class of CW-complexes, called self-similar complexes, is introduced, together with C*-algebras A_j of operators, endowed with a finite trace, acting on square-summable cellular j-chains. Since the Laplacian Delta_j belongs to A_j, L^2-Betti numbers and Novikov-Shubin numbers are defined for such complexes in terms of the trace. In particular a relation involving the Euler-Poincare' characteristic is proved. L^2-Betti and Novikov-Shubin numbers are computed for some self-similar complexes arising from self-similar fractals.Comment: 30 pages, 7 figure

    Complexes of not ii-connected graphs

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    Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev. In this paper we study the complexes of not ii-connected kk-hypergraphs on nn vertices. We show that the complex of not 22-connected graphs has the homotopy type of a wedge of (n−2)!(n-2)! spheres of dimension 2n−52n-5. This answers one of the questions raised by Vassiliev in connection with knot invariants. For this case the SnS_n-action on the homology of the complex is also determined. For complexes of not 22-connected kk-hypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n−2)(n-2)-connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n−3)(n-3)-connected graphs we provide a formula for the generating function of the Euler characteristic

    Euler complexes and geometry of modular varieties

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    There is a mysterious connection between the multiple polylogarithms at N-th roots of unity and modular varieties. In this paper we "explain" it in the simplest case of the double logarithm. We introduce an Euler complex data on modular curves. It includes a length two complex on every modular curve. Their second cohomology groups recover the Beilinson-Kato Euler system in K_2 of modular curves. We show that the above connection in the double logarithm case is provided by the specialization at a cusp of the Euler complex data on the modular curve Y_1(N). Furthermore, specializing the Euler complexes at CM points we find new examples of the connection with geometry of modular varieties, this time hyperbolic 3-folds.Comment: Dedicated to Joseph Bernstein for his 60th birthday. The final version. Some corrections were made. To appear in GAFA, special volume dedicated to J. Bernstei
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