University of Oxford

Mathematical Institute News

Whitehead Prize

A Whitehead prize is awarded to Marc Lackenby of St. Catherine's College and the University of Oxford for his contributions to three dimensional topology and to combinatorial group theory.

He has proved two unexpected results about Dehn surgery, which is a much used method to construct a three-dimensional manifold M₂ from another one M₁ based on a knot K M₁ and a twisting coefficient p/q. The first is a uniqueness result: If one performs a surgery that is 'far' from the trivial one on a knot K M₁ which is a null-homotopic and H₂ (M₁) nontrivial, then the homeomorphism class of M₂ determines M₁, K and p/q uniquely. The second result is that there is a constant c depending only on M₁ such that if M₂ is 'exceptional' then |q| < c.

With Daryl Cooper he also proved a remarkable finiteness result that for a given M₂ there are only finitely many hyperbolic knots K S³ such that M₂ can be obtained by a p/q surgery if q > 22.

He has found other remarkable results about hyperbolic three dimensional manifolds. One is a simple algorithm enhancing Thurston's famous result giving the existence of hyperbolic structures on a large class of three dimensional manifolds. The algorithm allows one to calculate (up to explicit bounds) the volume of the (hyperbolic) complement of a class of knots. Another of his theorems is related to the famous 2p theorem of Gromov and Thurston that a Dehn filling of a cusped hyperbolic manifold M³ along a curve of length more than 2p always gives rise to a negatively curved manifold. Using new methods Lackenby has shown that if 2p is replaced by 6 then the fundamental group of the resulting manifold is Gromov hyperbolic. A consequence is that at most 12 manifolds obtained by surgery on a hyperbolic knot can have non-negatively curved fundamental group. This is close to the best possible general result since the figure eight knot has ten exceptional surgeries.

His recent work on the Heegaard genus of coverings has opened up new relations with other areas of mathematics. By using comparatively elementary methods, he has found novel connections between the isoperimetric value of a Cayley graph of a finite group and the Betti numbers of a 2-complex associated with the presentation of the group. There are exciting possible consequences of this work in combinatorial group theory.