Courses:

      Math Seminar 21:640:492:Q2; Tuesday, Thursday 11:30-12:50,  Hill Hall, Room 202
 
Course web page on Blackboard

Office Hours:  Tuesday , Thursday 2:30-3:30 pm, and by appointment;

Office Phone - 973-353-3909;           Email - oertel@rutgers.edu

Department of Mathematics and Computer Science, Rutgers-Newark

Current Research 

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 compact 3-manifolds,  with Leonardo Navarro de Carvalho.   Math arXiv
 

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 3-manifolds. Math arXiv
 

Normal surfaces

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 3-manifolds, Preprint, 2002.      Math arXiv
 

Some Papers

• On the existence of infinitely many essential surfaces of bounded genus, Pac. J. Math.  202, No. 2, 2002, 449-458. Postscript file
• Spaces which are not negatively curved, joint with Lee Mosher. Communications in Analysis and Geometry, 6 , No1, (1998), 67-140.
• Two-dimensional measured laminations of positive Euler characteristic, joint with Lee Mosher.  Oxford Q. J. Math. 52 (2001), no. 2, 195--216.
• Invariant laminations for diffeomorphisms of handlebodies, Oxford Quarterly Journal of Mathematics, 51, No. 4, (2000)
• 485-507.
• Boundaries of pi₁-injective surfaces, Topology and its Applications 78 (1997), 215-234.
• Affine foliations and covering hyperbolic structures, joint with Athanase Papadopoulos, Manuscripta Mathematica 104 (2001) 383-406.
• Affine laminations and their stretch factors, Pacific Journal of Mathematics 182, No. 2, (1998), 303-327.
• Full laminations in 3-manifolds, joint with Allen Hatcher, Math. Proc. Camb. Phil. Soc. 119 (1996), 73-82.
• Affine lamination spaces for surfaces, joint with Allen Hatcher, Pac. J. of Math. 154 (1992), No.1, 87-101.
• Boundary slopes for Montesinos knots, joint with Allen Hatcher, Topology 28 (1989), 453-480.
• Essential laminations in 3-manifolds, joint with David Gabai, Annals of Mathematics 130 (1989), 41-73.
• Sums of incompressible surfaces, Proceedings AMS 102 (1988) No.3, 711-719.
• Measured laminations in 3-manifolds, Transactions AMS. 305 (1988) No. 2, 531-573.
• Homology branched surfaces: Thurston's norm on H₂(M ³), LMS Lecture Note Series 112, Low-dimensional Topology and Kleinian Groups, Editor D.B.A. Epstein, 1986.
• An algorithm to decide if a 3-manifold is a Haken manifold, joint with William Jaco, Topology 23 (1984), 195-209 .
• Incompressible branched surfaces, Invent. Math. 76 (1984), 385-410.
• Closed incompressible surfaces in complements of star links, Pac. J. of Math. 111 (1984), 209-230.
• Incompressible surfaces via branched surfaces, joint with William Floyd, Topology 23 (1984), 117-125.