Related papers: An upper bound on the LS-category in presence of t…

We prove that $$ \cat X\le cd(\pi_1(X))+\bigg\lceil\frac{\dim X-1}{2}\bigg\rceil$$ for every CW complex $X$ where $cd(\pi_1(X))$ denotes the cohomological dimension of the fundamental group of $X$.

Dranishnikov~\cite{D2} proved that \[{\rm cat} X\leq {\rm cd}(\pi_1(X))+\Bigl\lceil\frac{{\rm hd} (X)-1}{2}\Bigr\rceil.\] where ${\rm cd}(\pi)$ denotes the cohomological dimension of a group $\pi$ and ${\rm hd}(X)$ denotes the homotopy…

The following inequality \cat X\le \cat Y+\lceil\frac{hd(X)-r}{r+1}\rceil holds for every locally trivial fibration between $ANE$ spaces $f:X\to Y$ which admits a section and has the $r$-connected fiber where $hd(X)$ is the homotopical…

Let $X$ be a finite simply connected CW complex of dimension $n$. The loop space homology $H\_*(\Omega X;\mathbb Q)$ is the universal enveloping algebra of a graded Lie algebra $L\_X$ isomorphic with $ pi\_{*-1} (X)\otimes \mathbb Q$. Let…

In the derived category of the category of modules over a commutative Noetherian ring $R$, we define, for an ideal $\fa$ of $R$, two different types of cohomological dimensions of a complex $X$ in a certain subcategory of the derived…

The $\pi_2$-diffeomorphism finiteness result (\cite{FR1,2}, \cite{PT}) asserts that the diffeomorphic types of compact $n$-manifolds $M$ with vanishing first and second homotopy groups can be bounded above in terms of $n$, and upper bounds…

Sormani and Wei proved in 2004 that a compact geodesic space has a categorical universal cover if and only if its covering/critical spectrum is finite. We add to this several equivalent conditions pertaining to the geometry and topology of…

We prove that for every compactum X with dim_Z X <= n >= 2 there is a cell-like resolution r: Z --> X from a compactum Z onto X such that dim Z <= n and for every integer k and every abelian group G such that dim_G X <= k >= 2 we have dim_G…

In this paper, we study necessary and sufficient conditions for the existence of categorical universal coverings using open covers of a given space $X$. As some applications, first we present a generalized version of the Shelah Theorem…

This paper is about geometric and topological properties of a proper CAT(0) space $X$ which is cocompact - i.e. which has a compact generating domain with respect to the full isometry group. It is shown that geodesic segments in $X$ can…

We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its…

We give a homological characterization of $n$-manifolds whose universal covering $\Wi M$ has Gromov's macroscopic dimension $\dim_{mc}\Wi M<n$. As the result we distinguish $\dim_{mc}$ from the macroscopic dimension $\dim_{MC}$ defined by…

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually…

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 give an upper bound on the dimension of the bounded derived category of an abelian category. We show that if $\CX$ is a sufficiently nice subcategory of an abelian category, then derived dimension of $\CA$ is at most $\CX$-dim$\CA$,…

If $X$ is a topological group, then its fundamental groupoid $\pi_1X$ is a group-groupoid which is a group object in the category of groupoids. Further if $X$ is a path connected topological group which has a simply connected cover, then…

This paper is a synthesis and extension of three earlier papers on $PD_4$-complexes $X$ with fundamental group $\pi$ such that $c.d.\pi=2$ and $\pi$ has one end. Our goal is to show that the homotopy types of such complexes are determined…

We prove that every finite connected simplicial complex has the homology of the classifying space for some $\mathrm{CAT}(0)$ cubical duality group. More specifically, for any finite simplicial complex $X$, we construct a locally…

Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.

We prove an upper bound of higher topological complexity $TC_n(X)$ using higher $\mathcal{D}$-topological complexity $TC_n^{\mathcal{D}}(X)$ of a space $X$. An intermediate invariant $\widetilde{TC}_n(X)$ is used in the proof. We interpret…