Trisections of 4-manifolds

Abstract

The study of n-dimensional manifolds has seen great advances in the last half century. In dimensions greater than four, surgery theory has reduced classification to homotopy theory except when the fundamental group is nontrivial, where serious algebraic issues remain. In dimension 3, the proof by Perelman of Thurston’s Geometrization Conjecture (1) allows an algorithmic classification of 3-manifolds. The work of Freedman (2) classifies topological 4-manifolds if the fundamental group is not too large. Also, gauge theory in the hands of Donaldson (3) has provided invariants leading to proofs that some topological 4-manifolds have no smooth structure, that many compact 4-manifolds have countably many smooth structures, and that many noncompact 4-manifolds, in particular 4D Euclidean space R 4 , have uncountably many. However, the gauge theory invariants run into trouble with small 4-manifolds, such as those with the same homology groups as the 4D sphere, S 4 . In particular, the smooth 4D Poincare Conjecture, the last remaining case of that hallowed conjecture, is still open. (In higher dimensions, the smooth Poincare Conjecture is sometimes true in the following sense. In dimensions 3, 5, 6, 12, and 61, a homotopy sphere is diffeomorphic to the standard one, and in all other known cases, there are increasingly many exotic smooth structures on the topological sphere; however, it is possible that there may be more high-dimensional cases with no exotic spheres.) The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic. What is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions. The s-cobordism theorem states that, if M 0 m and M 1 m are the two boundary components of an m + 1 -dimensional manifold W and if … [↵][1]1Email: kirby{at}math.berkeley.edu. [1]: #xref-corresp-1-1