Related papers: A singular variety associated to the smallest degr…
For given polynomial map $F:\C^2\to\C^2$ with nonvanishing jacobian we associate a variety whose homology or intersection homology describes the geometry of singularities at infinity of this map.
In the paper "Geometry of polynomial mapping at infinity via intersection homology" the second and third authors associated to a given polynomial mapping $F : \C^2 \to \C^2$ with nonvanishing jacobian a variety whose homology or…
In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce `higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously…
We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…
We study the arithmetical ranks and the cohomological dimensions of an infinite class of Cohen-Macaulay varieties of minimal degree. Among these we find, on the one hand, infinitely many set-theoretic complete intersections, on the other…
Bounds on the minimum degree and on the number of vertices at- taining it have been much studied for finite edge-/vertex-minimally k- connected/k-edge-connected graphs. We give an overview of the results known for finite graphs, and show…
The purpose of this paper is to introduce a version of singular homology based on smooth mappings of manifolds with corners. Although variants of such a theory exists in the literature, we felt that certain points were not adequately…
We describe the relation of $r$-similarity and finite-order invariants on the homotopy set $[S^1,Y]=\pi_1(Y)$.
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…
Intersection homology is defined for simplicial, singular and PL chains and it is well known that the three versions are isomorphic for a full filtered simplicial complex. In the literature, the isomorphism, between the singular and the…
We characterize the smallest finite spaces with the same homotopy groups of the spheres. Similarly, we describe the minimal finite models of any finite graph. We also develop new combinatorial techniques based on finite spaces to study…
We study properties of contiguity distance between simplicial maps. In particular, we show that simplicial versions of $LS$-category and topological complexity are particular cases of this more general notion.
We study the degree of an $L$-Lipschitz map between Riemannian manifolds, proving new upper bounds and constructing new examples. For instance, if $X_k$ is the connected sum of $k$ copies of $\mathbb CP^2$ for $k \ge 4$, then we prove that…
Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the…
The aim of this note is to describe a geometric relation between simple plane curve singularities classified by simply laced Cartan matrices and cluster varieties of finite type also classified by the simply laced Cartan matrices. We…
We compute numerically the homology of several graph complexes in low loop orders, extending previous results.
We introduce a geometry on the cone of positive closed currents of bidegree (p,p) and apply it to define the intersection of such currents. We also construct and study the Green currents and the equilibrium measure for horizontal-like…
We consider a possibility of the existence of intersection homology morphism, which would be associated to a map of analytic varieties. We assume that the map is an inclusion of codimension one. Then the existence of a morphism follows from…
A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the…
The definition of the intersection number of a map with a closed manifold can be extended to the case of a closed stratified set such that the difference between dimensions of its two biggest strata is greater than $1$. The set Sigma of…