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Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and $\ell^2$-MultiplicitiesMay 14 2019We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a ... More
Smoothing finite group actions on three-manifoldsJan 30 2019We show that every continuous action of a finite group on a smooth three-manifold is a uniform limit of smooth actions.
On $p$-adic limits of topological invariantsNov 01 2018The purpose of this article is to define and study new invariants of topological spaces: the $p$-adic Betti numbers and the $p$-adic torsion. These invariants take values in the $p$-adic numbers and are constructed from a virtual pro-$p$ completion of ... More
On a homology of ternary groups with applications to knot theoryMay 28 2018We define a homology for ternary groups using both associativity and skew elements. We describe the odd-even construction which yields many examples of ternary groups. We define the ternary knot group, consider its homomorphisms into ternary groups, and ... More
On the topology of the Milnor fibration of a hyperplane arrangementJul 21 2016Mar 14 2017This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic ... More
On the topology of the Milnor fibration of a hyperplane arrangementJul 21 2016This note is mostly an expository survey, centered on the topology of complements of hyperplane arrangements, their Milnor fibrations, and their boundary structures. An important tool in this study is provided by the degree 1 resonance and characteristic ... More
A new class of homology and cohomology 3-manifoldsJun 19 2016Jun 26 2016We show that for any set of primes $\mathcal{P}$ there exists a space $M_{\mathcal{P}}$ which is a homology and cohomology 3-manifold with coefficients in $\mathbb{Z}_{p}$ for $p\in \mathcal{ P}$ and is not a homology or cohomology 3-manifold with coefficients ... More
Residually finite rationally $p$ groupsApr 07 2016Nov 29 2016In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely ... More
Residually finite rationally $p$ groupsApr 07 2016Jan 26 2019In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely ... More
Residually finite rationally $p$ groupsApr 07 2016In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or for infinitely ... More
Volume polynomials and duality algebras of multi-fansSep 10 2015We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This ... More
Involutions on sapphire Sol 3-manifolds and the Borsuk-Ulam theorem for maps into $R^n$Oct 01 2014Nov 13 2014For each sapphire Sol $3$-manifold, we classify the free involutions. For each triple $(M, \tau; R^n)$ where $M$ is a sapphire Sol $3$-manifold and $\tau$ is a free involution, we show if $(M, \tau; R^n)$ has the Borsuk-Ulam property or not. It is known ... More
Random Discrete Morse Theory and a New Library of TriangulationsMar 26 2013Nov 27 20131) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will ... More
Hyperplane arrangements and Milnor fibrationsJan 21 2013Dec 14 2013There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking ... More
The Hilbert--Smith conjecture for three-manifoldsDec 11 2011Jan 22 2013We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. ... More
On intersections of closed curves on surfacesNov 22 2011The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, ... More
Extension of the Borsuk Theorem on Non-Embeddability of SpheresJun 25 2009It is proved that the suspension of a closed n-dimensional manifold M, $n\ge1$, does not embed in a product of n+1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a far-reaching generalization the Borsuk theorem ... More
Embedding compacta into products of curvesDec 20 2007We present some results on n-dimensional compacta lying in n-dimensional products of compacta, in particular, in products of n 1-dimensional compacta. Most of our basic results are proven under the assumption that the compacta X admit essential maps into ... More
Isomorphisms in l^1-homologyDec 20 2006Aug 01 2008Taking the l^1-completion and the topological dual of the singular chain complex gives rise to l^1-homology and bounded cohomology respectively. In contrast to l^1-homology, major structural properties of bounded cohomology are well understood by the ... More