HOMOLOGY CYLINDERS AND SKEIN ALGEBRAS (Geometry of discrete groups and hyperbolic spaces)

Abstract

In this paper, we introduce a construction of an invariant for a homology cylinder of a surface Σ. It is an element of the skein algebra of Σ and has two aspects. The first is a quantization of the action of homology cylinders on fundamental groups. In the second aspect, we can extend the Ohtsuki series, one for integral homology spheres, to our invariant. We use the HOMFLY-PT skein algebra in this paper. But the main theorem holds in other skein algebras.