arXiv subjects

Vyacheslav Krushkal

Publications and source records attributed to Vyacheslav Krushkal.

44 records · Page 3Linked to original sources

arXiv2009

Robust 4-manifolds and robust embeddings

A link in the 3-sphere is homotopically trivial, according to Milnor, if its components bound disjoint maps of disks in the 4-ball. This paper concerns the question of what spaces give rise to the same class of homotopically trivial links when used in place of disks in an analogous definition. We show that there are 4-manifolds for which this property depends on their embedding in the 4-ball. This work is motivated by the A-B slice problem, a reformulation of the 4-dimensional topological surgery conjecture. As a corollary this provides a new, secondary, obstruction in the A-B slice problem for a certain class of decompositions of D^4.

math.GT↗

arXiv2008

Tutte chromatic identities from the Temperley-Lieb algebra

This paper introduces a conceptual framework, in the context of quantum topology and the algebras underlying it, for analyzing relations obeyed by the chromatic polynomial χ(Q) of planar graphs. Using it we give new proofs and substantially extend a number of classical results concerning the combinatorics of the chromatic polynomial. In particular, we show that Tutte's golden identity is a consequence of level-rank duality for SO(N) topological quantum field theories and Birman-Murakami-Wenzl algebras. This identity is a remarkable feature of the chromatic polynomial relating χ({ϕ+2}) for any triangulation of the sphere to (χ({ϕ+1}))^2 for the same graph, where ϕdenotes the golden ratio. The new viewpoint presented here explains that Tutte's identity is special to these values of the parameter Q. A natural context for analyzing such properties of the chromatic polynomial is provided by the {\em chromatic algebra}, whose Markov trace is the chromatic polynomial of an associated graph. We use it to show that another identity of Tutte's for the chromatic polynomial at Q=ϕ+1 arises from a Jones-Wenzl projector in the Temperley-Lieb algebra. We generalize this identity to each value Q= 2+2\cos(2πj/(n+1)) for j< n positive integers. When j=1, these Q are the Beraha numbers, where the existence of such identities was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations.

math.CO↗

arXiv2008

A counterexample to the strong version of Freedman's conjecture

A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean rings are not even weakly A-B slice, where one drops the equivariant aspect of the problem, has been the main focus in search for an obstruction to surgery. We show that the Borromean rings, and more generally all links with trivial linking numbers, are in fact weakly A-B slice. This result shows the lack of a non-abelian extension of Alexander duality in dimension 4, and of an analogue of Milnor's theory of link homotopy for general decompositions of the 4-ball.

math.GT↗

arXiv2008

Link groups and the A-B slice problem

The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.-S. Lin. This unifies and extends the previously known partial obstructions in the A-B slice program. As a consequence, link groups satisfy Alexander duality when restricted to the class of model decompositions, but not for general submanifolds of the 4-ball.

math.GT↗

arXiv2005

Surfaces in 4-manifolds and the surgery conjecture

We give a survey of geometric approaches to the topological 4-dimensional surgery and 5-dimensional s-cobordism conjectures, with a focus on the study of surfaces in 4-manifolds. The geometric lemma underlying these conjectures is a statement about smooth immersions of disks and of certain 2-complexes, capped gropes, in a 4-manifold. We also mention a reformulation in terms of the A,B-slice problem, and the relation of this question to recent developments in the study of the classical knot concordance group.

math.GT↗

arXiv2004

On the asymptotics of quantum SU(2) representations of mapping class groups

We investigate the rigidity and asymptotic properties of quantum SU(2) representations of mapping class groups. In the spherical braid group case the trivial representation is not isolated in the family of quantum SU(2) representations. In particular, they may be used to give an explicit check that spherical braid groups and hyperelliptic mapping class groups do not have Kazhdan's property (T). On the other hand, the representations of the mapping class group of the torus do not have almost invariant vectors, in fact they converge to the metaplectic representation of SL(2,Z) on L^2(R). As a consequence we obtain a curious analytic fact about the Fourier transform on the real line which may not have been previously observed.

math.QA↗

arXiv2003

Dwyer's filtration and topology of 4-manifolds

Topological 4-dimensional surgery is conjectured to fail, in general, for free fundamental groups. M. Freedman and P. Teichner have shown that surgery problems with an arbitrary fundamental group have a solution, provided they satisfy a certain condition on Dwyer's filtration on second homology. We give a new geometric proof of this result, and analyze its relation to the canonical surgery problems.

math.GT↗

arXiv2002

On disk embedding up to s-cobordism

The disk embedding lemma is a technique underlying the topological classification results in 4-manifold topology for good fundamental groups. The purpose of this paper is to develop new tools for disk embedding that work up to s-cobordism, without restrictions on the fundamental group. As an application we show that a surgery problem gives rise to a collection of capped gropes that fit in the framework of control theory.

math.GT↗