On invariants and homology of spaces of knots in arbitrary manifolds

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.