V.~A.~Vassiliev

On invariants and homology of spaces of knots in
arbitrary manifolds

The construction of finite-order knot invariants
in $\R^3$, based on resolutions of discriminant sets, can be carried
over immediately to the case of knots in an arbitrary
three-dimensional manifold $M$ (maybe nonorientable) and, moreover,
to the calculation of cohomology groups of spaces of knots in
arbitrary manifolds of dimension $\ge3$.

Obstructions to the integrability of admissible weight systems to
well-defined knot invariants in $M$ are identified as $1$-dimensional
cohomology classes of certain generalized loop spaces of $M$. Unlike
the case $M=\R^3$, these obstructions can be nontrivial and provide
invariants of the manifold $M$ itself.

The corresponding algebraic machinery allows us to obtain on the
level of ``abstract nonsense'' some of the results and problems of
theory, and to extract from others the essential topological (in
particular, low-dimensional) part.