Related papers: A Short Note on Mapping Cylinders

We introduce a new map between configuration spaces of points in a background manifold - the replication map - and prove that it is a homology isomorphism in a range with certain coefficients. This is particularly of interest when the…

In this article we investigate rigidity properties of integrable area-preserving twist maps of the cylinder. More specifically, we prove that if a deformation of the standard integrable map preserves rotational invariant circles (i.e.,…

We prove that a locally compact space with an upper curvature bound is a topological manifold if and only if all of its spaces of directions are homotopy equivalent and not contractible. We discuss applications to homology manifolds, limits…

Let X be a Stein manifold, A a closed complex subvariety of X, and f a continuous map from X to a complex manifold Y whose restriction to A is holomorphic. After a homotopic deformation of the Stein structure outside a neighborhood of A in…

We consider the general problem of constructing the structure of a smooth manifold on a given space of loops in a smooth finite dimensional manifold. By generalising the standard construction for smooth loops, we derive a list of conditions…

It is a well-known result of C.T.C. Wall's that one may decompose a simply connected 6-manifold as a connected sum of two simpler manifolds. Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincar\'e…

We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various…

In this paper we prove the homotopy lifting property for symmetric products $SP_{m}(X)$ and $F_{m}(X)$, with $X$ a Hausdorff topological space. Furthermore, we introduce a new tool, the theory of topological puzzles, to get a useful…

A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial…

We define and develop a homotopy invariant notion for the topological complexity of a map $f:X \to Y$, denoted TC($f$), that interacts with TC($X$) and TC($Y$) in the same way cat($f$) interacts with cat($X$) and cat($Y$). Furthermore,…

Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the sense that $f\circ h = f$. We will show that if $h$ leaves invariant each regular…

The aim of this paper is to show how the homotopy type of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of finite approximations of the corresponding space. This recovering allows us to define…

We prove two theorems about homotopies of curves on 2-dimensional Riemannian manifolds. We show that, for any epsilon > 0, if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through…

The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm-one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any…

In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a…

Given pointed cellular spaces $X$ and $Y$, $X$ compact, and an integer $r\ge0$, we define a relation $\overset r\approx$ on $[X,Y]$ and argue for the conjecture that it always coincides with the $r$-similarity $\overset r\sim$.

We develop a theory of $\times$-homotopy, fundamental groupoids and covering spaces that apply to non-simple graphs, generalizing existing results for simple graphs. We prove that $\times$-homotopies from finite graphs can be decomposed…

For a fixed integer $d\geq 1$, we show that two quasitoric manifolds over a product of $d$-simplices are homotopy equivalent after appropriate localization, provided that their integral cohomology rings are isomorphic.

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the…

We describe a way to compute mapping spaces of cyclic operads through modules. As an application we compute the homotopy automorphism space of the cyclic Batalin-Vilkovisky (Hopf co-)operad.