Related papers: A counterexample to the extension space conjecture…

J.-P. Roudneff conjectured in 1991 that every arrangement of $n \ge 2d+1\ge 5$ pseudohyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ complete cells (i.e., cells bounded by each…

In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and…

We establish a loop space decomposition for certain $CW$-complexes with a single top cell in the presence of a spherical pair, thereby generalizing several known decompositions of Poincar\'{e} duality complexes in which a loop of a product…

The category of differential graded operads is a cofibrantly generated model category and as such inherits simplicial mapping spaces. The vertices of an operad mapping space are just operad morphisms. The 1-simplices represent homotopies…

There are two related poset structures, the higher Stasheff-Tamari orders, on the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In this paper it is shown that both of them have the homotopy type of a sphere of…

We show that any codimension one hyperbolic attractor of a diffeomorphism of a (d+1)-dimensional closed manifold is shape equivalent to a (d+1)-dimensional torus with a finite number of points removed, or, in the non-orientable case, to a…

In the present paper we prove the integrability (in the sense of existence of formal symmetry of infinite rank) for a class of block-triangular inhomogeneous extensions of (1+1)-dimensional integrable evolution systems. An important…

Let b(M) denote the maximal number of disjoint bases in a matroid M. It is shown that if M is a matroid of rank d+1, then for any continuous map f from the matroidal complex M into the d-dimensional Euclidean space there exist t \geq…

Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been…

We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an…

Extending the results of reconstruction of compact metric spaces by inverse limits, we show that if $(X, d), (Y, d)$ are compact metric spaces, then the mapping space $Y^X$ is homotopy equivalent to the inverse limit of an inverse system of…

Some widely known compact extended formulations have the property that each vertex of the corresponding extension polytope is projected onto a vertex of the target polytope. In this paper, we prove that for heptagons with vertices in…

In this article we disprove the conjectures asserting the positivity of the coefficients of the Ehrhart polynomial of matroid polytopes by De Loera, Haws and K\"oppe (2007) and of generalized permutohedra by Castillo and Liu (2015). We…

Let $X$ be a complex manifold containing a hypersurface $D$ and let $D^s$ denote the singular locus. We study the problem of extending a flat connection with logarithmic poles along $D$ from the complement $X \setminus D^s$ to all of $X$.…

This paper identifies the homotopy theories of topological stacks and orbispaces with unstable global homotopy theory. At the same time, we provide a new perspective by interpreting it as the homotopy theory of `spaces with an action of the…

We establish a parametric extension $h$-principle for overtwisted contact structures on manifolds of all dimensions, which is the direct generalization of the $3$-dimensional result from \cite{Eli89}. It implies, in particular, that any…

We prove affirmatively the conjecture raised by J. Mostovoy; the space of short ropes is weakly homotopy equivalent to the classifying space of the topological monoid (or category) of long knots in $\mathbb{R}^3$. We make use of techniques…

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary, we obtain the classification of closed, $n$-dimensional,…

Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space} of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially equivalent to~$P$, modulo affine transformations. We report on work by the first…

Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and whose curvatures are constrained to lie in a…