25,880 research outputs found
Euler Characteristic in Odd Dimensions
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
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 -connected graphs
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 -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -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
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Euler complexes and geometry of modular varieties
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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