Slava Krushkal

We show that an oriented surface in R⁴ containing double point singularities induces a map between the Khovanov homology groups of its boundary links in a functorial way. As part of this work, the movie moves of Carter and Saito are extended to surfaces with double points.

In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms.
This is a mathematical proposal for one of the simplest examples in a family of \(\pi_*\)(TMF)-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories.
As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in \(\pi_*\)(TMF), conjecturally generalizing the theta function of a lattice.  We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions.

Given a rational homology 3-sphere M, we introduce a triple linking form on H₁(M;Z), defined when the classical torsion linking pairing of three homology classes vanishes pairwise. If M is the boundary of a simply-connected 4-manifold N, the triple linking form can be computed in terms of the higher order intersection form on N, introduced by Matsumoto. We use these methods to formulate an embedding obstruction for rational homology spheres in S⁴, extending a 1938 theorem of Hantzsche.

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.

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.