68,116 research outputs found
Algebraic Torsion in Contact Manifolds
We extract a nonnegative integer-valued invariant, which we call the "order
of algebraic torsion", from the Symplectic Field Theory of a closed contact
manifold, and show that its finiteness gives obstructions to the existence of
symplectic fillings and exact symplectic cobordisms. A contact manifold has
algebraic torsion of order zero if and only if it is algebraically overtwisted
(i.e. has trivial contact homology), and any contact 3-manifold with positive
Giroux torsion has algebraic torsion of order one (though the converse is not
true). We also construct examples for each nonnegative k of contact 3-manifolds
that have algebraic torsion of order k but not k - 1, and derive consequences
for contact surgeries on such manifolds. The appendix by Michael Hutchings
gives an alternative proof of our cobordism obstructions in dimension three
using a refinement of the contact invariant in Embedded Contact Homology.Comment: 53 pages, 4 figures, with an appendix by Michael Hutchings; v.3 is a
final update to agree with the published paper, and also corrects a minor
error that appeared in the published version of the appendi
Contractors for flows
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and
connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a
contractor for the graph parameter counting the number of B-flows of a graph,
where B is a subset of a finite Abelian group closed under inverses. We prove
our main result using the duality between flows and tensions and finite Fourier
analysis. We exhibit several examples of contractors for B-flows, which are of
interest in relation to the family of B-flow conjectures formulated by Tutte,
Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur
- …