Related papers: Estimates of covering type and minimal triangulati…
The covering type of a space $X$ is defined as the minimal cardinality of a good cover of a space that is homotopy equivalent to $X$. We derive estimates for the covering type of $X$ in terms of other invariants of $X$, namely the ranks of…
The notion of covering type was recently introduced by Karoubi and Weibel to measure the complexity of a topological space by means of good coverings. When X has the homotopy type of a finite CW-complex, its covering type coincides with the…
The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X.…
In this paper we study a new notion of category weight of homology classes developing further the ideas of E. Fadell and S. Husseini. In the case of closed smooth manifolds the homological category weight is equivalent to the cohomological…
We introduce the notion of the \emph{equivariant covering type} of a space $X$ on which a finite group $G$ acts, and study its properties. The equivariant covering type measures the size of $G$-equivariant good covers of $X$ and is thus an…
We study the minimum total weight of a disk triangulation using vertices out of $\{1,\ldots,n\}$, where the boundary is the triangle $(123)$ and the $\binom{n}3$ triangles have independent weights, e.g. $\mathrm{Exp}(1)$ or…
We study lower bounds for the number of vertices in a PL-triangulation of a given manifold $M$. While most of the previous estimates are based on the dimension and the connectivity of $M$, we show that further information can be extracted…
In this paper, Lusternik-Schinrelmann and geometric category of finite spaces are considered. We define new numerical invariants of these spaces derived from the geometric category and present an algorithmic approach for its effective…
We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated…
We introduce some compact orbifolds on which there is a certain finite group action having a simple convex polytope as the orbit space. We compute the orbifold fundamental group and homology groups of these orbifolds. We calculate the…
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space…
We introduce and develop an analogous of the Auslander-Buchweitz approximation theory (see \cite{AB}) in the context of triangulated categories, by using a version of relative homology in this setting. We also prove several results…
We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative…
In this paper we study a new topological invariant $\Cat(X,\xi)$, where $X$ is a finite polyhedron and $\xi\in H^1(X;\R)$ is a real cohomology class. $\Cat(X,\xi)$ is defined using open covers of $X$ with certain geometric properties; it is…
We extend the theory of the Lusternik-Schnirelmann category to general metric spaces by means of covers by arbitrary subsets. We also generalize the definition of the strict category weight. We show that if the Bockstein homomorphism on a…
We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\times B$, whenever $A$ and $B$ are subgroups of $\pi_1(X)$ whose conjugates intersect trivially. For…
We show that the size of a minimal simplicial cover of a polytope $P$ is a lower bound for the size of a minimal triangulation of $P$, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and…
In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a…
We present estimates of number of simplices of given dimension of classical compact Lie groups. As in the previous work \cite{GMP2} the approach is a combination of an estimate of number of vertices with a use of valuation of the covering…
We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds…