Building on previous work, Ulrich Oertel will study the topology of 3-manifolds, using essential laminations and essential maps of surfaces. The goals include generalizing results about Haken manifolds to manifolds containing essential laminations or admitting essential maps of surfaces. In addition, with Jacek Swiatkowski, Oertel will study contact structures, confoliations, and contaminations in 3-manifolds. This may also lead to results about the topology of 3-manifolds admitting appropriately restricted (essential or tight) versions of these (partial) plane fields on the 3-manifolds. Another goal is to establish further connections between contact structures and foliations and laminations in 3-manifolds. Finally, Oertel will continue to study automorphisms of 3-manifolds, refining and generalizing existing work that deals with the important case of automorphisms of handlebodies and compression bodies. The goal here, already partially achieved, is to classify automorphisms of 3-manifolds up to isotopy, as in the Nielsen-Thurston classification of automorphisms of surfaces. A closed 3-manifold is a space that is locally like ordinary 3-dimensional space. Thus, understanding 3-manifolds amounts to understanding all possible 3-dimensional universes. Especially since humans live in a 3-dimensional space, one wishes to determine properties of 3-manifolds, and ultimately, also to classify them. From the 1950's to the 1970's, progress on 3-manifolds was based to a large degree on methods using certain embedded surfaces, called incompressible surfaces. From the 3-manifolds containing incompressible surfaces, one can, for example, extract algebraic information sufficient to distinguish any two of them. The investigator will use objects in 3-manifolds similar to incompressible surfaces, namely essential laminations and essential maps of surfaces, to explore the unknown territory further. There are other structures on 3-manifolds, called contact structures, that have their origins in geometry rather than topology. Work of Eliashberg and Thurston has shown that contact structures are related, via confoliations, to certain laminations called foliations. With Swiatkowski, the investigator is finding further connections among an entire range of related objects: contact structures, confoliations, foliations, laminations, and contaminations, the last being a newly invented object. It is typical in mathematics to study a space by studying suitable maps from the space to itself. For 3-manifolds, one can consider maps called automorphisms or self-diffeomorphisms. In continuing research, Oertel is classifying automorphisms of 3-manifolds (up to isotopy). The work is revealing a rich and interesting theory of the automorphisms of some of the simplest 3-manifolds. ***
