1,171 research outputs found
On invariants and homology of spaces of knots in arbitrary manifolds
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
Homology of spaces of non-resultant polynomial systems in R^2 and C^2
The resultant veriety in the space of systems of homogeneous polynomials of
given degrees consists of such systems having non-trivial solutions. We
calculate the integer cohomology groups of all spaces of non-resultant systems
of polynomials , and also the rational cohomology groups of similar
systems in
Topology of two-connected graphs and homology of spaces of knots
We propose a new method of computing cohomology groups of spaces of knots in
, , based on the topology of configuration spaces and
two-connected graphs, and calculate all such classes of order As a
byproduct we define the higher indices, which invariants of knots in
define at arbitrary singular knots. More generally, for any finite-order
cohomology class of the space of knots we define its principal symbol, which
lies in a cohomology group of a certain finite-dimensional configuration space
and characterizes our class modulo the classes of smaller filtration
- β¦