Slava Krushkal

We prove a localization theorem for exotic diffeomorphisms, showing that every diffeomorphism of a compact simply-connected 4-manifold that is isotopic to the identity after stabilizing with one copy of S²×S², is smoothly isotopic to a diffeomorphism that is supported on a contractible submanifold. For those that require more than one copy of S²×S², we prove that the diffeomorphism can be isotoped to one that is supported in a submanifold homotopy equivalent to a wedge of 2-spheres, with null-homotopic inclusion map. We investigate the implications of these results by applying them to known exotic diffeomorphisms.

We extend a construction of Jones to associate (n, n)-tangles with elements of Thompson's group F and prove that it is asymptotically faithful as \(n \to\infty\). Using this construction we show that the oriented Thompson group \(\vec F\) admits a lax group action on a category of Khovanov's chain complexes.

We construct a compact PL 5-manifold M (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, V₁₁ S²,
which is ``spineless'', meaning M is not the regular neighborhood of any 2-complex PL embedded in M.
We formulate a related question about the existence of exotic smooth structures on 4-manifolds which is of interest in relation to the deformation conjecture for 2-complexes, also known as the generalized Andrews-Curtis conjecture. 

The universal pairing for manifolds was defined and shown to lack positivity in dimension 4 by Freedman et al. We prove an analogous result for 2-complexes, and also show that the universal pairing does not detect the difference between simple homotopy equivalence and 3-deformations. The question of whether these two equivalence relations are different for 2-complexes is the subject of the Andrews-Curtis conjecture. We also discuss the universal pairing for higher-dimensional complexes and show that it is not positive.

Perron and Quinn gave independent proofs in 1986 that every topological pseudo-isotopy 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.