Professor Abigail Thompson studies combinatorial methods in 3-dimensional topology. Despite the recent influence of algebraic and geometric techniques such as quantum groups, hyperbolic geometry, and algebraic varieties in the study of 3-manifolds, most of the fundamental arguments involve or can be reduced to cutting and pasting surfaces and manifolds and studying their possible combinatorial configurations. Knots and links are an especially good starting point for such reasoning, both because complements are representative examples of 3-manifolds, and because the knots and links in a 3-manifold are a fundamental part of its structure.
In joint work with Scharlemann, Professor Thompson has obtained a number of new classification and decision results and new proofs of old results using an important cut-and-paste concept called thin position. In particular, she has found a simplified argument that an algorithm of Rubinstein decides if a particular 3-manifold is homeomorphic to S^3 . She and Scharlemann also gave a much simpler proof of Waldhausen's theorem , which states that there is only one Heegaard splitting of the 3-sphere of each genus. This work has the following two consequences, among others. First, it relates primitive descriptions of 3-manifolds, such as triangulations and Heegaard splittings, to their deep instrinsic structure, and it may lead to generalizations such as an algorithm to decide if any two 3-manifolds are homeomorphic. Second, it is progress towards the Poincaré conjecture. For example, if one is presented with a 3-manifold and happens to know that it is simply-connected, Rubinstein's algorithm is a way to conclusively determine whether or not it is a counterample to Poincaré.
Professor Thompson also showed that all knots which are band-connected sums have Property P, meaning that Dehn surgery on such a knot cannot produce a counterexample to the conjecture . Since band-connected sums do not appear to be fundamentally simpler than other types of knots, this result is good evidence that all knots have Property P, which in turn is an important special case of the Poincaré conjecture.
- Property P for the band-connect sum of two knots, Topology 26 (1987),
- Link genus and the Conway moves (with M. Scharlemann), Comment. Math. Helv.
64 (1989), 527-535.
- Detecting unknotted graphs in 3-space (with M. Scharlemann), J.
Differential Geom. 34 (1991), 539-560.
- Thin position and Heegaard splittings of the 3-sphere (with M.
Scharlemann), J. Differential Geom. 39 (1994), 343-357.
- Thin position and Rubinstein's solution to the recognition problem for S3, Math. Res. Lett. 1 (1994), 613-630.
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