arXiv subjects

Victor A. Vassiliev

Publications and source records attributed to Victor A. Vassiliev.

24 records · Page 2Linked to original sources

arXiv2014

On combinatorial formulas for cohomology of spaces of knots

We develop homological techniques for finding explicit combinatorial expressions of finite-type cohomology classes of spaces of knots in $R^n, n \ge 3,$ generalizing Polyak--Viro formulas for invariants (i.e. 0-dimensional cohomology classes) of knots in $R^3$. As the first applications we give such formulas for the (reduced mod 2) {\em generalized Teiblum--Turchin cocycle} of order 3 (which is the simplest cohomology class of {\em long knots} $R^1 \hookrightarrow R^n$ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of {\em compact knots} $S^1 \hookrightarrow R^n$. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in $R^3.$

math.GT↗

arXiv2014

Topology of plane arrangements and their complements

This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality, twisted homology groups, monodromy theory and multidimensional hypergeometric functions. The emphasis on the most geometrical explanation is done; applications and analogies in the differential topology are outlined.

math.GT↗

arXiv2014

Spaces of Hermitian operators with simple spectra and their finite-order cohomology

The topology of spaces of Hermitian operators in $C^n$ with non-simple spectra was studied by V.Arnold in a relation with the theory of adiabatic connections and the quantum Hall effect. The natural filtration of these spaces by the sets of operators with fixed numbers of eigenvalues defines the spectral sequence, providing interesting combinatorial and homological information on this stratification. We construct a different spectral sequence, also counting the homology groups of these spaces and based on the universal techniques of {\em topological order complexes} and resolutions of algebraic varieties, generalizing the combinatorial inclusion-exclusion formula and similar to the construction of finite degree knot invariants. This spectral sequence degenerates at the term $E_1$, is (conjecturally) multiplicative, and as $n$ grows then it converges to a stable spectral sequence counting the cohomology of the space of infinite Hermitian operators without multiple eigenvalues, all whose terms $E^{p,q}_r$ are finitely generated. It allows us to define the finite degree cohomology classes of this space, and to apply the well-known facts and methods of the topological theory of flag manifolds to the problems of geometrical combinatorics, especially concerning the continuous partially ordered sets of subspaces and flags.

math.AT↗

arXiv2014

On $r$-neighborly submanifolds in $R^N$

A submanifold $M \subset R^N$ is $r$-neighborly if for any $r$ points in $M$ there is a hyperplane, supporting $M$ and touching it at exactly these $r$ points. We prove that the minimal dimension $Δ(k,r)$ of the Euclidean space, containing a stably $r$-neighborly submanifold, is asymptotically not smaller than $2kr-k$.

math.GT↗

arXiv1997

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 $\R^n$, $n \ge 3$, based on the topology of configuration spaces and two-connected graphs, and calculate all such classes of order $\le 3.$ As a byproduct we define the higher indices, which invariants of knots in $\R^3$ 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.

math.GT↗

arXiv1997

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.

math.GT↗