Research

I consider myself a low dimensional topologist. I do think a little about some high dimensional stuff, but this is mostly in the fringe cases of smooth five and six dimensional manifolds, and mostly as tools for extracting information about my favorite dimension, 4.

• Pseudo isotopies of simply connected 4-manifolds -joint with David Gabai, David Gay, Vyacheslav Krushkal, and Mark Powell (Arxiv)

Abstract. Perron and Quinn gave independent proofs in 1986 that every topological pseudoisotopy of a simply-connected, compact topological 4-manifold is isotopic to the identity. Another result of Quinn is that every smooth pseudo-isotopy of a simply-connected, compact, smooth 4-manifold is smoothly stably isotopic to the identity. From this he deduced that π4(TOP(4)/ O(4)) = 0. A replacement criterion is used at a key juncture in Quinn’s proofs, but the justification given for it is incorrect. We provide different arguments that bypass the replacement criterion, thus completing Quinn’s proofs of both the topological and the stable smooth pseudo-isotopy theorems. We discuss the replacement criterion and state it as an open problem. 

• Unknotting 3-balls in the 5-ball (Accepted: PAMS)
  Abstract: The purpose of this note is to answer affirmatively a question posed by both Gay, and Hughes, Kim and Miller as to whether every 3--ball smoothly embedded in the 4--sphere becomes isotopic relative to the bounding 2-sphere when pushed into the 5-ball.
  
  

• Relations amongst twists along Montesinos twins in the 4-sphere - joint with David Gay (Accepted: AGT)
  Abstract: Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain twists along submanifolds analogous to Dehn twists in dimension two. The subgroup of the smooth mapping class group of the 4-sphere coming from loops of 5-dimensional handles of index 1 and 2 coincides with the subgroup generated by twists along Montesinos twins (pairs of 2-spheres intersecting transversely twice) in which one of the two 2-spheres in the twin is unknotted. In this paper we show that this subgroup is in fact trivial or cyclic of order two.