ON INVARIANTS AND HOMOLOGY OF SPACES OF KNOTS IN ARBITRARY MANIFOLDS

VICTOR A. VASSILIEV

Abstract. Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to well-defined knot invariants are identified as 1-dimensional cohomology classes of generalized loop spaces of the manifold. Unlike the case of the 3-sphere, these obstructions can be non-trivial and provide invariants of the manifold itself.

The corresponding algebraic machinery allows us to obtain on the level of the “abstract nonsense” some of results and problems of the theory, and to extract from other the essential topological part.