arXiv subjects

Dmitry N. Kozlov

Publications and source records attributed to Dmitry N. Kozlov.

39 records · Page 3Linked to original sources

arXiv2001

Topology of spaces of hyperbolic polynomials and combinatorics of resonances

In this paper we study the topology of the strata, indexed by number partitions $λ$, in the natural stratification of the space of monic hyperbolic polynomials of degree $n$. We prove stabilization theorems for removing an independent block or an independent relation in $λ$. We also prove contractibility of the one-point compactifications of the strata indexed by a large class of number partitions, including $λ=(k^m,1^r)$, for $m\geq 2$. Furthermore, we study the maps between the homology groups of the strata, induced by imposing additional relations (resonances) on the number partition $λ$, or by merging some of the blocks of $λ$.

math.CO↗

arXiv2001

Rational homology of spaces of complex monic polynomials with multiple roots

We study rational homology groups of one-point compactifications of spaces of complex monic polynomials with multiple roots. These spaces are indexed by number partitions. A standard reformulation in terms of quotients of orbit arrangements reduces the problem to studying certain triangulated spaces $X_{λ,μ}$. We present a combinatorial description of the cell structure of $X_{λ,μ}$ using the language of marked forests. As applications we obtain a new proof of a theorem of Arnold and a counterexample to a conjecture of Sundaram and Welker, along with a few other smaller results.

math.CO↗

arXiv2001

A comparison of Vassiliev and Ziegler-Zivaljevic models for homotopy types of subspace arrangements

In this paper we represent the Vassiliev model for the homotopy type of the one-point compactification of subspace arrangements as a homotopy colimit of an appropriate diagram over the nerve complex of the intersection semilattice of the arrangement. Furthermore, using a generalization of simplicial collapses to diagrams of topological spaces over simplicial complexes, we construct an explicit deformation retraction from the Vassiliev model to the Ziegler-Zivaljevic model.

math.CO↗