Acting Chair this semester only.
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Department of Mathematics
and Computer Science, Rutgers-Newark
Automorphisms of 3-manifolds
The most interesting automorphisms (self-homeomorphisms) of 3-manifolds occur in boundary-reducible and reducible manifolds. The paper Automorphisms of 3-dimensional handlebodies develops a theory of automorphisms of handlebodies and compression bodies analogous to the Nielsen-Thurston theory of autmorphisms of surfaces. In the theory, the analogue of the pseudo-Anosov automorphism is the generic automorphism of a handlebody or compression body, and generic automorphisms turn out, as one would expect, to be the most interesting and mysterious. Automorphisms of irreducible 3-manifolds with non-empty boundary can then be understood by combining the new methods with older methods in 3-manifold topology, involving the Jaco-Shalen-Johannson characteristic submanifold and Bonahon's characteristic compression body. Much remains to be understood about generic automorphisms, which are currently being studied further by former Rutgers graduate student Leonardo Navarro de Carvalho.
In recent work with Carvalho, the ideas used for the classification of automorphisms of handlebodies and compression bodies are extended to classify, in a certain sense, the automorphisms of any compact 3-manifold (satisfying the Thurston Geometrization Conjecture).
Automorphisms of 3-dimensional handlebodies and compression bodies, Topology 41 (2002) 363-410: Postscript file PDF file
A classification of automorphisms of
3-manifolds, with Leonardo Navarro de Carvalho. Math
Mapping class groups of compression bodies and 3-manifolds
In work related to the classification of automorphisms of 3-manifolds, I obtain a short exact sequence relating various mapping class groups (diffeotopy groups) associated to a compression body of arbitrary dimension. This is applied to obtain a short exact sequence for the mapping class group of a reducible 3-manifold M. The sequence gives the mapping class group of the disjoint union of irreducible summands of M as a quotient of the entire mapping class group of M. There are further applications.
Mapping class groups of compression bodies and
Charalampos Charitos and I have a new normal surface theory for incompressible (but not necessarily boundary incompressible) surfaces in irrecucible (but not necessarily boundary irreducible) compact 3-manifolds with boundary. It may be possible to apply this theory to obtain better invariant lamination-like objects for generic automorphisms of handlebodies and compression bodies.
Essential disks and semi-essential surfaces in 3-manifolds. Math arXiv
Contact structures and contaminations in 3-manifolds
In a project with Jacek Swiatkowski (Wroclaw University) I am studying contact structures using branched surface techniques. A natural generalization of the contact structure and the confoliation is the contamination, an object which can be carried by a branched surface. Contact structures can be represented by contaminations. The goals of this research are 1) to understand contact structures combinatorially; 2) to understand the relationships between foliations, confoliations, contact structures, laminations, and contaminations; and 3) ultimately to apply the methods to obtain information about 3-manifold topology.
A contamination carrying criterion for branched surfaces, Preprint, 2001. Math arXiv
Contact structures, sigma-confoliations, and contaminations in