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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…
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…
This paper investigates sufficient and necessary conditions for the existence of a homotopy equivalence between two finite simplicial complexes from an algorithmic point of view. As a result, the conditions are formulated in terms of the…
Given a compact metric space $X$, we associate to it an inverse sequence of finite $T_0$ topological spaces. The inverse limit of this inverse sequence contains a homeomorphic copy of $X$ that is a strong deformation retract. We provide a…
The two pillars of Algebraic topology - Homology and homotopy theory rely on the availability of basic building blocks called cells. Cells take the form of simplexes, and have properties such as faces, sub-cells, convexity and…
Given CW complexes X and Y, let map(X,Y) denote the space of continuous functions from X to Y with the compact open topology. The space map(X,Y) need not have the homotopy type of a CW complex. Here the results of an extensive investigation…
A phantom map is a potentially nontrivial map which induces the zero map on every homology theory and on homotopy groups. Zabrodsky has shown that in the presence of particular finiteness conditions on spaces $X$ and $Y$ every map $X\to Y$…
Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the…
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…
Let M be an arbitrary Riemannian homogeneous space, and let Omega be a space of tilings of M, with finite local complexity (relative to some symmetry group Gamma) and closed in the natural topology. Then Omega is the inverse limit of a…
We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at…
In this paper, we study a construction of homotopy invariants of open or closed covers, where the homotopy class is defined relative to a pair $(V,r)$, with $V$ a finite set of points in $\mathbb{R}^d$ and $r$ a point in the interior of…
We present two models for the space of knots which have endpoints at fixed boundary points in a manifold with boundary, one model defined as an inverse limit of spaces of maps between configuration spaces and another which is cosimplicial.…
We study the inverse problem for persistent homology: For a fixed simplicial complex $K$, we analyse the fiber of the continuous map $\mathrm{PH}$ on the space of filters that assigns to a filter $f: K \to \mathbb R$ the total barcode of…
Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty $) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1, $ a…
In the present paper we study bundles equipped with extra homotopy conditions, in particular so-called simplicial $n$-bundles. It is shown that (under some condition) the classifying space of 1-bundles is the double coset space of some…
We prove that every finite connected simplicial complex has the homology of the classifying space for some $\mathrm{CAT}(0)$ cubical duality group. More specifically, for any finite simplicial complex $X$, we construct a locally…
Finite-sheeted covering mappings onto compact connected groups are studied. It is shown that a finite-sheeted covering mapping from a connected Hausdorff topological space onto a compact connected abelian group G must be a homeomorphism…
We construct anisotropic conductivities with the same Dirichlet-to-Neumann map as a homogeneous isotropic conductivity. These conductivities are singular close to a surface inside the body.
A well-known theorem of Paulsen says that if $\mathcal{A}$ is a unital operator algebra and $\phi:\mathcal{A}\to B(\mathcal{H})$ is a unital completely bounded homomorphism, then $\phi$ is similar to a completely contractive map $\phi'$.…