Vyacheslav Krushkal

Subexponential groups in 4{manifold topology

We present a new, more elementary proof of the Freedman{Teichner result that the geometric classication techniques (surgery, s{cobordism, and pseudoisotopy) hold for topological 4{manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of coarse geometry.

Alexander Duality, Gropes and Link Homotopy

We prove a geometric renement of Alexander duality for certain 2{complexes, the so-called gropes, embedded into 4{space. This renement can be roughly formulated as saying that 4{dimensional Alexander duality preserves the disjoint Dwyer ltration. In addition, we give new proofs and extended versions of two lemmas of Freedman and Lin which are of central importance in the A-B{slice problem ,t he main open problem in the classication theory of topological 4{manifolds. Our methods are group theoretical, rather than using Massey products and Milnor {invariants as in the original proofs. AMS Classication numbers Primary: 55M05, 57M25 Secondary: 57M05, 57N13, 57N70

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 . 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 . C2/ for any triangulation of the sphere to .. C1// 2 for the same graph, where denotes 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 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 QD C1 arises from a Jones‐Wenzl projector in the Temperley‐Lieb algebra. We generalize this identity to each value QD 2C 2 cos.2 j=.nC 1// for j < n positive integers. When jD 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. 57M15; 05C15, 57R56, 81R05

Surgery on closed 4-manifolds with free fundamental group

Even though the disk embedding theorem is not available in dimension 4 for free fundamental groups, some surgery problems may be shown to have topological solutions. We prove that surgery problems may be solved if one considers closed 4-manifolds and the intersection pairing is extended from the integers, and prove a related splitting result.

Exponential separation in 4{manifolds

We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4{manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating 1 {null surfaces in 4{manifolds. In our applications to link-homotopy, grope splitting serves as a geometric substitute for the Milnor group.

Graphs, links, and duality on surfaces

We introduce a polynomial invariant of graphs on surfaces,

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

P_G

SO.3/ homology of graphs and links

, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for

Slicing the Hopf link

P_G

Geometric complexity of embeddings in {\mathbb{R}^{d}}

, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs,