Related papers: Intersections of subcomplexes in non-positively cu…

We consider two mixed curve $C,C'\subset {\Bbb C}^2$ which are defined by mixed functions of two variables $\bf z=(z_1,z_2)$. We have shown in \cite{MC}, that they have canonical orientations. If $C$ and $C'$ are smooth and intersect…

We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous…

In this note, we give a motivic characterization of the integral cohomology of dual boundary complexes of smooth quasi-projective complex algebraic varieties. As a corollary, the dual boundary complex of any stably affine space (of positive…

We give a new proof, using comparatively simple techniques, of the Sullivan conjecture: the space of pointed maps from the classifying space of the cyclic group of order $p$ to any finite-dimensional CW complex $K$ is contractible.

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the…

A description of all subsemigroups of $M_2(\mathbb{C})$ which are given by a countable intersection of constructible sets is given. Furthermore, it is shown that they are intersections of constructible semigroups.

We prove that for each positive integer $n$, the Rips complexes of the $n$-dimensional integer lattice in the $d_1$ metric (i.e., the Manhattan metric, also called the natural word metric in the Cayley graph) are contractible at scales…

We study the asymptotic dynamics of maps which are piecewise contracting on a compact space. These maps are Lipschitz continuous, with Lipschitz constant smaller than one, when restricted to any piece of a finite and dense union of disjoint…

We produce an infinite family of $2$-complexes that are intrinsically linked when embedded into four dimensions. In particular, we show that any embedding into $\mathbb{R}^4$ of the suspension of a graph containing $K_6$ as a minor contains…

A 2-complex $K$ has collapsing non-positive immersion if for every combinatorial immersion $X\to K$, where $X$ is finite, connected and does not allow collapses, either $\chi(X)\le 0$ or $X$ is point. This concept is due to Wise who also…

We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or…

Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed…

We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either by puncturing a closed surface or via orbifold coverings. As a corollary, we give new quasi-isometric embeddings…

Bifurcations in a system of coupled maps are investigated. Using symbolic dynamics it is proven that for coupled shift maps the well known space--time--mixing attractor becomes unstable at a critical coupling strength in favour of a…

Let $\pi\cln \cX\to S$ and $\pi\cln \cY\to S$ be two smooth families of projective non-uniruled manifolds over a Riemann surface $S$ (probably non-compact). Suppose these two families are pointwise isomorphic. We prove that there exists an…

If $X$ is a compact set, a {\it topological contraction} is a self-embedding $f$ such that the intersection of the successive images $f^k(X)$, $k>0$, consists of one point. In dimension 3, we prove that there are smooth topological…

What sets A \subset Z^n can be written in the form (K-K) \cap Z^n, where K is a compact subset of R^n such that K+Z^n=R^n? Such sets A are called achievable, and it is known that if A is achievable, then < A >=Z^n. This condition completely…

Fessler and Gutierrez \cite{Fe,Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve…

A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's $\mathbb{Z}/p\mathbb{Z}$ model and all possible fusions between them. Our methods…

Real Bruhat cells give an important and well studied stratification of such spaces as $GL_{n+1}$, $Flag_{n+1} = SL_{n+1}/B$, $SO_{n+1}$ and $Spin_{n+1}$. We study the intersections of a top dimensional cell with another cell (for another…