Žiga Virk

On semilocally simply connected spaces

Abstract The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces. While the condition “semilocally simply connected” plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition “homotopically Hausdorff” instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.

Multiple perturbations of a singular eigenvalue problem

Abstract We study the perturbation by a critical term and a ( p − 1 ) -superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.

A combinatorial approach to coarse geometry

Using ideas from shape theory we embed the coarse category of metric spaces into the category of direct sequences of simplicial complexes with bonding maps being simplicial. Two direct sequences of simplicial complexes are equivalent if one of them can be transformed to the other by contiguous factorizations of bonding maps and by taking infinite subsequences. That embedding can be realized by either Rips complexes or analogs of Roe’s antiČech approximations of spaces. In that model coarse n-connectedness of K = {K1 → K2 → . . .} means that for each k there is m > k such that the bonding map from Kk to Km induces trivial homomorphisms of all homotopy groups up to and including n. The asymptotic dimension being at most n means that for each k there is m > k such that the bonding map from Kk to Km factors (up to contiguity) through an n-dimensional complex. Property A of G.Yu is equivalent to the condition that for each k and for each ǫ > 0 there is m > k such that the bonding map from |Kk| to |Km| has a contiguous approximation g : |Kk| → |Km| which sends simplices of |Kk| to sets of diameter at most ǫ. Date: June 8, 2009. 2000 Mathematics Subject Classification. Primary 54F45; Secondary 55M10.

Homotopical smallness and closeness

Abstract The aim of this paper is to introduce the concepts of homotopical smallness and closeness. These are the properties of homotopical classes of maps that are related to recent developments in homotopy theory and to the construction of universal covering spaces for non-semi-locally simply connected spaces, in particular to the properties of being homotopically Hausdorff and homotopically path Hausdorff. The definitions of notions in question and their role in homotopy theory are supplemented by examples, extensional classifications, universal constructions and known applications.

Compact maps and quasi-finite complexes

Abstract The simplest condition characterizing quasi-finite CW complexes K is the implication X τ h K ⇒ β ( X ) τ K for all paracompact spaces X . Here are the main results of the paper: Theorem 0.1 If { K s } s ∈ S is a family of pointed quasi-finite complexes, then their wedge ⋁ s ∈ S K s is quasi-finite. Theorem 0.2 If K 1 and K 2 are quasi-finite countable CW complexes, then their join K 1 * K 2 is quasi-finite. Theorem 0.3 For every quasi-finite CW complex K there is a family { K s } s ∈ S of countable CW complexes such that ⋁ s ∈ S K s is quasi-finite and is equivalent, over the class of paracompact spaces, to K. Theorem 0.4 Two quasi-finite CW complexes K and L are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of X τ F , where F is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.

Inducing Maps Between Gromov Boundaries

It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper, we introduce the class of visual functions f that do induce continuous maps

Approximations of 1-dimensional intrinsic persistence of geodesic spaces and their stability

{\tilde{f}}

Small loop spaces

f~ between Gromov boundaries. Its subclass, the class of radial functions, induces Hölder maps between Gromov boundaries. Conversely, every Hölder map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large-scale properties of f and small-scale properties of