Related papers: Inequivalent Cantor Sets in $R^{3}$ Whose Compleme…

To each isolated critical point of a smooth function on a 3-manifold we put in correspondence a tree (graph without cycles). We will prove that functions are topologically equivalent in the neighborhoods of critical points if and only if…

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic…

For $G$ and $H_1,\dots, H_n$ finite groups, does there exist a $3$-manifold group with $G$ as a quotient but no $H_i$ as a quotient? We answer all such questions in terms of the group cohomology of finite groups. We prove non-existence with…

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using…

The Isbell, compact-open and point-open topologies on the set $C(X,\mathbb{R})$ of continuous real-valued maps can be represented as the dual topologies with respect to some collections $\alpha(X)$ of compact families of open subsets of a…

A continuum $K$ is a common model for the family ${\mathcal K}$ of continua if every member of ${\mathcal K}$ is a continuous image of $K$. We show that none of the following classes of spaces has a common model: 1) the class of strongly…

A homotopy equivalence between a hyperbolic 3-manifold and a closed irreducible 3-manifold is homotopic to a homeomorphsim provided the hyperbolic manifold satisfies a purely geometric condition. There are no known examples of hyperbolic…

We study noncompact, complete, finite volume, Riemannian 4-manifolds $M$ with sectional curvature $-1<K<0$. We prove that $\pi_1 M$ cannot be a 3-manifold group. A classical theorem of Gromov says that $M$ is homeomorphic to the interior of…

We use nonsmooth critical point theory and the theory of geodesics with obstacle to show a multiplicity result about orthogonal geodesic chords in a Riemannian manifold (with boundary) which is homeomorphic to an $N$-disk. This applies to…

We show that for any connected smooth manifold $M$ of dimension different from $3$ the restriction of the compact-open topology to the diffeomorphism group of $M$ is minimal, i.e. the group does not admit a strictly coarser Hausdorff group…

We construct infinitely many examples of pairs of isospectral but non-isometric $1$-cusped hyperbolic $3$-manifolds. These examples have infinite discrete spectrum and the same Eisenstein series. Our constructions are based on an…

We show that every $PD_3$-complex $P$ bounds a $PD_4$-pair $(Z,P)$. If $P$ is orientable we may assume that $\pi_1(Z)=1$. We show also that if $P$ has a manifold 1-skeleton then it is homotopy equivalent to a closed 3-manifold, and that if…

We consider closed acylindrical surfaces in 3-manifolds and in knot and link complements, and show that the genus of these surfaces is bounded linearly by the number of tetrahedra in the triangulation of the manifold and by the number of…

We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply…

We consider the self-similar structure of the class of generalized Cantor sets $$\Gamma_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_n\beta^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<\beta<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of…

We show that if $X$ is an indecomposable $PD_3$-complex and $\pi_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge…

Let G be a simple complex algebraic group. By using a notion of a G-category we define invariants of tangles with flat G-connections in their complements. We also show that quantized universal enveloping algebras at roots of unity provide…

Two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or a finite index subgroup (or viceversa). Virtually isomorphic groups are…

K. Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_\omega.$ The problem of non-triviality of this third homotopy group has been open. We show that it is an infinitely generated group.…

For a generic anti-canonical hypersurface in each smooth toric Fano 4-fold with rank 2 Picard group, we prove there exist three isolated rational curves in it. Moreover, for all these 4-folds except one, the contractions of generic…