Homological algebra of quantum invariants in dimension four

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DMS-0104139Mikhail G. KhovanovThe project aims to construct quantum invariants of 4-dimensional objects. It is based on the author's recent discovery of a doubly-graded cohomology theory of links in the 3-sphere. The Euler characteristic of the cohomology groups is equal to the Jones polynomial. We would like to extend this theory to link cobordisms. The invariant of a cobordism will be a homomorphism between cohomology groups assigned to the boundaries of the cobordism. Furthermore, the theory should extend to tangles and tangle cobordisms. To a tangle we'll associate a functor between triangulated categories associated to the boundaries of the tangle, and to a tangle cobordism a natural transformation between functors. These triangulated categories will be related to highest weightcategories of modules over simple Lie algebras, as well as categories of modules over certain Frobenius algebras, such as cyclotomic Hecke algebras. In addition, we will look for cohomology theories lifts of other quantum invariants of links and 3-manifolds, including the Alexander and HOMFLY polynomials and Witten-Reshetikhin-Turaev invariants. An n-dimensional manifold is an object that locally looks like an n-dimensional space. A circle can be approximated by a tangent line in the neighbourhood of a point, and is a one-dimensional manifold (n-manifold, for short). The global structure distinguishes the circle from the line, though. Surfaces provide examples of two-manifolds. It turns out that one and two-manifolds are easy to classify, while in higher dimensions classification is hard. It is a theorem that in dimensions greater than three there can be no satisfactory classification, and topologists seem to be fairly close to finding one for three-manifolds. Given a pair of manifolds, it is a tough question to decide whether or not they are isomorphic. One approach is to extract some tangible invariant out of a manifold, such as a number, or a polynomial, and then compare these numbers. Most of the times the numbers are different and tell us that the manifolds are different, too. Dimension three is special in that there is a wealth of such invariants. These invariants, moreover, link three-manifold topology with deep algebraic structures. There are indications that the invariants can be lifted to the next dimension, to invariants of four-manifolds, and my goal is to find them and compare to analytical invariants of 4-manifolds that arise from solutions of certain partial differential equations.