## Bio

Professor of Mathematics, University of Virginia.

Research interests: Low-dimensional geometry and topology

## Bio

Professor of Mathematics, University of Virginia.

Research interests: Low-dimensional geometry and topology

- Freedman MH, Krushkal V, Lidman T. Spineless 5-manifolds and the deformation conjecture. arXiv:2401.03498. 2024.
We construct a compact PL 5-manifold M (with boundary) which is homotopy equivalent to the wedge of eleven 2-spheres, V

_{11}S^{2},

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.PDF109.02 KB - Khovanov M, Krushkal V, Nicholson J. On the universal pairing for 2-complexes. arXiv:2312.07429. 2023.
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.

PDF557.73 KB - Gabai D, Gay D, Hartman D, Krushkal V, Powell M. Pseudo-isotopies of simply connected 4-manifolds. arXiv:2311.11196. 2023.
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 π

_{4}(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.PDF2.11 MB - Akhmechet R, Johnson P, Krushkal V. Lattice cohomology and q-series invariants of 3-manifolds. J. Reine Angew. Math. 796 (2023), 269-299. 2023.
An invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin

^{c}-structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU(2) quantum invariants of 3-manifolds at roots of unity.

In particular, our work gives rise to a 2-variable refinement of the \(\widehat Z\)-invariant.PDF786.04 KB - Arone G, Krushkal V. Embedding obstructions in Rd from the Goodwillie-Weiss calculus and Whitney disks. Asian J. Math. 27 (2023), no. 2, 135-186. 2023.
Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings of $K$ into a Euclidean space $\R^d$. For $2$-dimensional complexes in $\R^4$, a geometric analogue is also introduced, based on intersections of Whitney disks and more generally on the intersection theory of Whitney towers developed by Schneiderman and Teichner. The focus in this paper is on the first obstruction beyond the classical embedding obstruction of van Kampen. In this case we show the two approaches lead to essentially the same obstruction. We also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are shown to be realized in a family of examples. Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometric and cohomological theories.

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