Slava Krushkal

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.

We extend the skein lasagna theory of Morrison–Walker–Wedrich to 4-manifolds with corners and formulate gluing formulas for 4-manifolds with boundary and, more generally, with corners. As an application, we
develop a categorical framework for a presentation of the skein lasagna module of trisected closed 4-manifolds.
Further, we extend the theory to dimension two by introducing bicategories for closed oriented surfaces and proving a gluing formula for the categories associated with 3-manifolds with boundary.

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.