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Lawrence Conlon

Publications and source records attributed to Lawrence Conlon.

6 recordsLinked to original sources
arXiv2016

Open Saturated Sets Without Holonomy

Open, connected, saturated sets W without holonomy in codimension one foliations play key roles as fundamental building blocks. Here, for the case of foliated 3-manifolds, we produce a finite system of closed, convex, non-overlapping polyhedral cones in the first cohomology of W with real coefficients such that the isotopy classes of possible foliations of W without holonomy, either dense leaved in W or proper, correspond one-to-one to the rays in the interiors of these cones. This generalizes our classification of depth one foliations to foliations of finite depth and more general foliations.

math.GT↗
arXiv2014

Hyperbolic Geometry and Homotopic Homeomorphisms of Surfaces

The Epstein-Baer theory of curve isotopies is basic to the remarkable theorem that homotopic homeomorphisms of surfaces are isotopic. The groundbreaking work of R. Baer was carried out on closed, orientable surfaces and extended by D. B. A. Epstein to arbitrary surfaces, compact or not, with or without boundary and orientable or not. We give a new method of deducing the theorem about homotopic homeomorphisms from the results about homotopic curves via the hyperbolic geometry of surfaces. This works on all but 13 surfaces where ad hoc proofs are needed.

math.GT↗
arXiv2010

Endperiodic Automorphisms of Surfaces and Foliations

We extend the unpublished work of M. Handel and R. Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel-Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic lamaniations, show the geodesic laminations satisfy the axioms, and prove that paeudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the "transfer theorem" for foliations of 3-manifolds., namely that, if two depth one foliations are transverse to a common one-dimensional foliation whose monodromy on the noncompact leaves of the first foliation exhibits the nice dynamics of Handel-Miller theory, then the transverse one-dimensional foliation also induces monodromy on the noncompact leaves of the second foliation exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.

math.GT↗
arXiv2006

The sutured Thurston norm

For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In many examples, the faces of the unit ball of the sutured norm are related to cones of depth one foliations of M but examples indicate this is not a general relationship.

math.GT↗
arXiv1999

Foliation Cones

David Gabai showed that disk decomposable knot and link complements carry taut foliations of depth one. In an arbitrary sutured 3-manifold M, such foliations F, if they exist at all, are determined up to isotopy by an associated ray [F] issuing from the origin in H^1(M;R) and meeting points of the integer lattice H^1(M;Z). Here we show that there is a finite family of nonoverlapping, convex, polyhedral cones in H^1(M;R) such that the rays meeting integer lattice points in the interiors of these cones are exactly the rays [F]. In the irreducible case, each of these cones corresponds to a pseudo-Anosov flow and can be computed by a Markov matrix associated to the flow. Examples show that, in disk decomposable cases, these are effectively computable. Our result extends to depth one a well known theorem of Thurston for fibered 3-manifolds. The depth one theory applies to higher depth as well.

math.GT↗