2,084 research outputs found
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
Bucolic Complexes
We introduce and investigate bucolic complexes, a common generalization of
systolic complexes and of CAT(0) cubical complexes. They are defined as simply
connected prism complexes satisfying some local combinatorial conditions. We
study various approaches to bucolic complexes: from graph-theoretic and
topological perspective, as well as from the point of view of geometric group
theory. In particular, we characterize bucolic complexes by some properties of
their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several
known results are generalized. We also show that locally-finite bucolic
complexes are contractible, and satisfy some nonpositive-curvature-like
properties.Comment: 45 pages, 4 figure
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