arXiv research

Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks…

New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.

Classifies $π_1$-injective maps between non-compact surfaces.

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

Groups of homotopy equivalences of graphs help realize compact subgroups.

Complex equivalence classes found in graph homotopy.

We prove that an open 3-manifold proper homotopy equivalent to a geometrically simply connected polyhedron is simply connected at infinity, generalizing a theorem of V.Poenaru.

The study explores ends in coarse homotopy of proper geodesic spaces.

Study on descent properties of complex affine surfaces under proper morphisms.

A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…

How different is the universal cover of a given finite 2-complex from a 3-manifold (from the proper homotopy viewpoint)? Regarding this question, we recall that a finitely presented group $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ whose universal cover $\til…

Generalizes Pontryagin's construction for proper maps in stable dimensions.

Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb R}^n$ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the h…

Strong rigidity proven for non-compact surfaces.

The study shows that certain cubical presentations lead to aspherical spaces.

In this paper we study the set of projective maps between compact proper convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When the target manifold is strictly convex, our results imply t…

Study of area minimizing surfaces in homotopy classes of maps.

Study shows weak homotopy equivalences for complete minimal surfaces.

We extend several techniques and theorems from geometric group theory so that they apply to geometric actions on arbitrary proper metric ARs (absolute retracts). A second way that we generalize earlier results is by eliminating freeness requirements often placed on the group actions. In doing so, we allow for groups wi…

We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold $W$, the homotopy classes of proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$ stabilise as $n\to\infty$, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for …

Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.

Every nonflat conformal minimal surface is homotopic to a proper one.

The thesis defines and proves invariants for manifolds of bounded geometry.

Corrects a 1998 proof about free factors of free groups.

Solves a problem related to classifying spaces for proper actions and Nielsen Realization.

We prove that if $Γ$ is a lattice in a classical simple Lie group $G$, then the symmetric space of $G$ is $Γ$-equivariantly homotopy equivalent to a proper cocompact $Γ$-CW complex of dimension the virtual cohomological dimension of $Γ$.

For Riem(M) the space of Riemannian metrics over a compact 3-manifold without boundary $M$, we study topological properties of the dense open subspace Riem'(M) of metrics which possess no Killing vectors. Given the stratification of Riem(M), we work under the condition that, in a sense defined in the text, the connecte…

Let G be a Lie group with finitely many connected components and let K be a maximal compact subgroup. We assume that G satisfies the rapid decay (RD) property and that G/K has non-positive sectional curvature. As an example, we can take G to be a connected semisimple Lie group. Let M be a G-proper manifold with compact…

Let $G$ be a matrix group. Topological $G$-manifolds with Palais-proper action have the $G$-homotopy type of countable $G$-CW complexes (3.2). This generalizes E Elfving's dissertation theorem for locally linear $G$-manifolds (1996). Also we improve the Bredon--Floyd theorem from compact groups $G$ (1960).

The Hilbert-Smith conjecture states, for any connected topological manifold $M$, any locally compact subgroup of $\mathrm{Homeo}(M)$ is a Lie group. We generalize basic results of Segal-Kosniowski-tomDieck (2.6), James-Segal (2.12), G Bredon (3.7), Jaworowski-Antonyan et al. (5.5), and E Elfving (7.3). The last is our …

The paper improves collapsing Alexandrov spaces results using good coverings.

Study delocalized eta invariants for signature operators on proper manifolds.

In this paper, we show that the class of all properly 3-realizable groups is closed under amalgamated free products (and HNN-extensions) over finite groups. We recall that $G$ is said to be properly 3-realizable if there exists a compact 2-polyhedron $K$ with $π_1(K) \cong G$ and whose universal cover $\tilde{K}$ has t…

Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.

The paper generalizes the Borsuk-Ulam theorem to surfaces and cyclic actions.

Unified theory of orbifolds and cohomology.

Synthetic theory defines orbifolds as microlinear types with finite identifications.

We study when a smooth variety $X$, embedded diagonally in its Cartesian square, is the zero scheme of a section of a vector bundle of rank $\dim(X)$ on $X\times X$. We call this the diagonal property (D). It was known that it holds for all flag manifolds ${\rm SL}_n/P$. We consider mainly the cases of proper smooth va…

Solves a problem related to Nielsen realization for certain groups.

Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there …

New coarse LS-category introduced for groups and spaces.

While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to understand the homotopy type of its subspaces consisting of the curves whose geodesic curvatures are constrained to a prescribed p…

This paper extends some results of Hatcher and Quinn beyond the metastable range. We give a bordism theoretic obstruction to deforming a map between manifolds simultaneously off of a collection of pairwise disjoint submanifolds under the assumption that it can be deformed off of any proper subcollection in a homotopy c…

One proves that there exists an obstruction to an open simply connected $n$-manifold of dimension $n\geq 5$ being geometrically simply connected. In particular there exist uncountably many simply connected $n$-manifolds which are not w.g.s.c. One proves that for $n\neq 4$ an $n$-manifold proper homotopy equivalent to a…

The paper generalizes Sperner's lemma to higher dimensions and calculates a new invariant.

In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups $G$ and $H$ are "proper $2$-equivalent" if there exist (equivalently, for all)…