Related papers: Smooth approximations and their applications to ho…

We formulate a conjecture that arithmetic locally symmetric manifolds have simple homotopy type, and prove it for the non-compact case. More precisely, we show that, for any symmetric space S of non-compact type without Euclidean de Rham…

The paper is concerned with defining a topology on the set of ideals of codimension d of the algebra C^\infty(M,R) with M being a compact smooth manifold. Its main property is that it is compact Hausdorff and it contains as a subspace the…

In this paper we prove weak L^{1,p} (and thus C^{\alpha}) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We…

For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank }a\leq k$. Motivated by applications to $C^{*}$-algebra theory, we investigate the…

In this paper we introduce a general notion of weak extension property for embeddings induced by a group actions. As an example, for the group H(M, m) of measure-preserving homeomorphisms of a noncompact manifold M, we deduce weak type…

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property…

Given two real algebraic varieties X and Y, we denote by R(X,Y) the set of all regular maps from X to Y. The set R(X,Y) is regarded as a topological subspace of the space C(X,Y) of all continuous maps from X to Y endowed with the…

In the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\mathbb…

We study the space of Riemannian metrics with positive scalar curvature on a compact manifold with boundary. These metrics extend a fixed boundary metric and take a product structure on a collar neighbourhood of the boundary. We show that…

The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$ for some subgroup $\operatorname{SK}_1(A)$, where $A$ is a commutative ring and $A^\times$ denotes the set of units in $A$. Consider an o-minimal expansion…

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…

The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set $E\subset\mathbb{R}^n$, how can we tell if there exists a $d$-dimensional, $C^m$-smooth manifold $\mathcal{M}\supset E$? We…

Given based cellular spaces X and Y, X compact, we define a sequence of increasingly fine equivalences on the based-homotopy set [X,Y].

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…

In this paper the weak topology on a normed space is studied from the viewpoint of infinite-dimensional topology. Besides the weak topology on a normed space $X$ (coinciding with the topology of uniform convergence on finite subsets of the…

We show that if $Y$ is a compact topological manifold and $X$ is a locally flat submanifold, then the complement $Y - X$ is homotopy equivalent to a finite CW complex. This is a direct proof, and does not rely on much of the theory of…

We first show that, for a fixed locally compact manifold $N,$ the space $L^2(S^1,N)$ has not the homotopy type odf the classical loop space $C^\infty(S^1,N),$ by two theorems: - the inclusion $C^\infty(S^1,N) \subset L^2(S^1,N)$ is null…

Let $f,g:M \rightarrow N$ be two maps between simply-connected smooth manifolds $M$ and $N$, such that $M$ is compact and $N$ is of finite $\mathbb{R}$-type. The goal of this paper is to use integration of certain differential forms to…

This paper explores various differentiable structures on the product manifold $M \times \mathbb{S}^k$, where $M$ is either a 4-dimensional closed, oriented, smooth manifold or a simply connected 5-dimensional closed, smooth manifold. We…

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of…