50 research outputs found
On the Connectivity of Cobordisms and Half-Projective TQFT's
We consider a generalization of the axioms of a TQFT, so called
half-projective TQFT's, with an anomaly, , in the composition law.
is a coboundary on the cobordism categories with non-negative, integer
values. The element of the ring over which the TQFT is defined does not
have to be invertible. In particular, it may be 0. This modification makes it
possible to extend quantum-invariants, which vanish on , to
non-trivial TQFT's. (A TQFT in the sense of Atiyah with this property has to be
trivial all together). Under a few natural assumptions the notion of a
half-projective TQFT is shown to be the only possible generalization. Based on
separate work with Lyubashenko on connected TQFT's, we construct a large class
of half-projective TQFT's with . Their invariants vanish on , and they coincide with the Hennings invariant for non-semisimple Hopf
algebras. Several toplogical tools that are relevant for vanishing properties
of such TQFT's are developed. They are concerned with connectivity properties
of cobordisms, as for example maximal non-separating surfaces. We introduce in
particular the notions of ``interior'' homotopy and homology groups, and of
coordinate graphs, which are functions on cobordisms with values in the
morphisms of a graph category. For applications we will prove that
half-projective TQFT's with vanish on cobordisms with infinite interior
homology, and we argue that the order of divergence of the TQFT on a cobordism
in the ``classical limit'' can be estimated by the rank of its maximal free
interior group.Comment: 55 pages, Late
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
On asymptotically hereditarily aspherical groups
We undertake a systematic study of asymptotically hereditarily aspherical
(AHA) groups - the class of groups introduced by Tadeusz Januszkiewicz and the
second author as a tool for exhibiting exotic properties of systolic groups. We
provide many new examples of AHA groups, also in high dimensions. We relate AHA
property with the topology at infinity of a group, and deduce in this way some
new properties of (weakly) systolic groups. We also exhibit an interesting
property of boundary at infinity for few classes of AHA groups.Comment: 36 pages, minor modifications to v
Chordal bipartite, strongly chordal, and strongly chordal bipartite graphs
AbstractRobert E. Jamison characterized chordal graphs by the edge set of every k-cycle being the symmetric difference of k−2 triangles. Strongly chordal (and chordal bipartite) graphs can be similarly characterized in terms of the distribution of triangles (respectively, quadrilaterals). These results motivate a definition of ‘strongly chordal bipartite graphs’, forming a class intermediate between bipartite interval graphs and chordal bipartite graphs
Characterizing 2-Trees Relative to Chordal and Series-Parallel Graphs
The 2-connected 2-tree graphs are defined as being constructible from a single 3-cycle by recursively appending new degree-2 vertices so as to form 3-cycles that have unique edges in common with the existing graph. Such 2-trees can be characterized both as the edge-minimal chordal graphs and also as the edge-maximal series-parallel graphs. These are also precisely the 2-connected graphs that are simultaneously chordal and series-parallel, where these latter two better-known types of graphs have themselves been both characterized and applied in numerous ways that are unmotivated by their interaction with 2-trees and with each other.
Toward providing such motivation, the present paper examines several simple, very closely-related characterizations of chordal graphs and 2-trees and, after that, of series-parallel graphs and 2-trees. This leads to showing a way in which chordal graphs and series-parallel graphs are special---indeed, extremal---in this regard