Васильев Виктор Анатольевич

On invariants and homology of spaces of knots in arbitrary manifolds.

1998 · ARTICLE · en

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.

Singularity theory. I. Dynamical systems. VI

1998 · BOOK · en

On decision trees for orthants

1997 · ARTICLE · en

M. Rabin's principle asserts that the depth of any algebraic decision tree, recognizing a closed orthant in scRn, is no less than n. Using the techniques of Newton polyhedra, we give the shortest possible proof of this fact, extending it to arbitrary collections of open or closed orthants, and apply it to trees distinguishing real polynomials having at least l real roots.

Holonomic links and Smale principles for multisingularities

1997 · ARTICLE · en

A loop S^1 → ℝ^n is holonomic if it is the (n - 1)-jet extension of a function S^1 → ℝ^1. We prove that for n = 3 any tame link in ℝ^n is isotopy equivalent to a holonomic one; for n > 3 the space of holonomic links is holotopy equivalent to the space of all differentiable links.

Ramification in integral geometry and monodromy of complex links

1997 · ARTICLE · en

Введение в топологию

1997 · BOOK · ru

Топология дополнений к дискриминантам

1997 · BOOK · ru

Топологическая сложность и вещественность

1996 · ARTICLE · ru

О пространствах полиномиальных узлов

1996 · ARTICLE · ru

Вычислены группы гомологий неособых полиномиальных вложений R1→Rn степени ⩽4. Развита общая алгебраическая техника подобных вычислений для пространств полиномиальных узлов произвольной степени.

Stratified Picard-Lefschetz theory

1995 · ARTICLE · en

The monodromy action in the homology of level sets of Morse functions on stratified singular analytic varieties is studied. The local variation operators in both the standard and the intersection homology groups defined by the loops around the critical values of such functions are reduced to similar operators in the homology groups of the transversal slices of the corresponding strata.