New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.
problem Characterizing nonrigidity of open contractible manifolds.
method Construction of infinitely many pairwise nonhomeomorphic smooth open contractible manifolds.
result Existence of infinitely many pairwise nonhomeomorphic smooth open contractible manifolds with same proper homotopy type.
The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.
problem Understanding when surfaces are homotopy equivalent to graphs.
method Analyzes second-countable orientable surfaces with noncompact boundary.
result Defines a necessary and sufficient condition for surfaces to be homotopy equivalent to graphs.
We compute the homotopy type of the space of proper d-dimensional submanifolds of Rn 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…
Study on descent properties of complex affine surfaces under proper morphisms.
problem Understanding descent behavior of homotopy-theoretic properties of smooth affine surfaces.
method Examined Eilenberg-MacLane property and introduced finite homotopy rank-sum property. Proved descent under proper morphisms for surfaces of log Kodaira dimension ≤0.
result Finite homotopy rank-sum property descends under proper morphisms for smooth affine surfaces of log Kodaira dimension ≤0.
Classifies non-linear Fredholm maps linking to stable homotopy groups of spheres.
problem Classifying non-linear proper Fredholm maps between Hilbert spaces.
method Using stable homotopy groups of spheres to classify maps up to proper homotopy.
result Determines the non-trivial kernel of the map from stable homotopy groups to non-linear proper Fredholm maps.
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 π1(K)≅G whose universal cover $\til…
In this paper we classify the homotopy classes of proper maps E→Rk, where E is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps Rn→Rk. We find a stability range of such maps. We conclude with some remarks…
Classifies π1-injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All π1-injective proper maps are classified. Strong rigidity proven for non-compact surfaces.
problem Proving rigidity of non-compact surfaces.
method Generalized result showing proper homotopy to homeomorphism.
result All non-compact orientable surfaces, except plane and punctured plane, are topologically rigid.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
Complex equivalence classes found in graph homotopy.
problem Complexity of proper homotopy equivalence in graphs.
method Demonstrated Borel completeness and comeager equivalence classes.
result Complex equivalence classes exist in infinite graphs.
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.
problem Understanding ends in coarse homotopy of proper geodesic spaces.
method Recontextualizing ends as a functor and proving properties of coarse path components.
result Existence of a natural surjection from coarse path components to ends, not always an injection.
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×Rn→Rk+n stabilise as n→∞, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for …
Study of area minimizing surfaces in homotopy classes of maps.
problem Existence and regularity of area minimizing surfaces in metric spaces.
method Introducing relative 1-homotopy type for Sobolev maps, using local quadratic isoperimetric inequality, and analog for closed surfaces.
result Existence and local Hölder regularity of area minimizing surfaces in proper geodesic metric spaces.
Synthetic theory defines orbifolds as microlinear types with finite identifications.
problem Defining orbifolds in traditional set-level foundations with internal symmetries.
method Synthetic differential cohesive homotopy type theory, microlinearity, finite identifications.
result Proper étale groupoids are orbifolds in synthetic theory.
Researchers found multiple ways to end-sum 4-manifolds, contradicting a previous conjecture.
problem Nonuniqueness of end-sums in 4-manifolds.
method Explicit examples and detailed discussion of end-cohomology algebra.
result Uncountably many distinct proper homotopy types from end-sums.
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…
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).
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…
Generalizes Pontryagin's construction for proper maps in stable dimensions.
problem Mapping submanifolds to homotopy classes of proper maps.
method Introduces a new bijection between cobordism sets and homotopy classes for proper maps.
result Provides a bijection for cobordism sets of submanifolds embedded in WimesRn. 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…
The Hilbert-Smith conjecture states, for any connected topological manifold M, any locally compact subgroup of 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 …
Study of endperiodic maps on infinite graphs, proving homotopy and eigenvalue properties.
problem Understanding endperiodic maps on infinite graphs with finitely many ends.
method Adapting relative train track maps and combinatorial techniques to infinite type setting.
result Any generalized endperiodic map is homotopic to a relative train track map.
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 π1(K)≅G and whose universal cover K~ has t…
Researchers prove the automorphism group of a sphere complex matches the mapping group of a graph.
problem Understanding the automorphisms of sphere complexes associated with graphs.
method Analyzing the group of proper homotopy equivalences and constructing an exhaustion of the sphere complex.
result The automorphism group of the sphere complex is isomorphic to the mapping group of the graph.
A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, the first author proved that the restriction to Brunnian links of any Goussarov-Vassiliev finite type invariant of (n+1)-component links of degree<2n is trivial. The purpose of this paper is to study the first nont…
The study shows that certain cubical presentations lead to aspherical spaces.
problem Understanding the asphericity of cubical presentations in 2D.
method Analyzing the second homotopy group of coned-off spaces associated with cubical presentations.
result The coned-off space is aspherical under specific conditions.
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…
While the topology of the space of all smooth immersed curves on the 2-sphere S2 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…
Study shows weak homotopy equivalences for complete minimal surfaces.
problem Understanding complete minimal surfaces and their properties.
method Analyzes algebraic null immersions and conformal minimal immersions.
result Inclusion and differential mappings are weak homotopy equivalences.
Study the topology of stable vector fields and Lyapunov functions on R^n.
problem Topology of stable vector fields and Lyapunov functions on R^n.
method Differential topology, Lyapunov theory, and results on diffeomorphism groups of discs.
result Path-connected and simply connected spaces of stable vector fields for n≠4,5 and weakly contractible for n≤3.
In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inve…
Coarse homotopy theory connects Euclidean cones to shape theory of compact spaces.
problem Establishing connections between coarse homotopy theory and shape theory.
method Using pointed shape invariants and inverse mapping telescopes.
result Proving two compact spaces are strong shape equivalent if their Euclidean cones are coarsely homotopy equivalent.
The paper generalizes the Borsuk-Ulam theorem to surfaces and cyclic actions.
problem Generalizing the Borsuk-Ulam theorem to surfaces and cyclic actions.
method Algebraic criterion involving braid groups and homology groups.
result Determines the Borsuk-Ulam property for maps from surfaces to R^2.
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×X. We call this the diagonal property (D). It was known that it holds for all flag manifolds SLn/P. We consider mainly the cases of proper smooth va…
For each link L in S^3 and every quantum grading j, we construct a stable homotopy type X^j_o(L) whose cohomology recovers Ozsvath-Rasmussen-Szabo's odd Khovanov homology, H_i(X^j_o(L)) = Kh^{i,j}_o(L), following a construction of Lawson-Lipshitz-Sarkar of the even Khovanov stable homotopy type. Furthermore, the odd Kh…
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
The paper generalizes Sperner's lemma to higher dimensions and calculates a new invariant.
problem Generalizing Sperner's lemma to higher dimensions and quantifying its outcomes.
method Using triangulations of (m+1)-discs and simplicial mappings, the authors define a new invariant and prove a theorem about fully colored simplices. result The number of fully colored simplices is not less than the new invariant μ([f]).
Study shows equivariant Khovanov homotopy types are equivalent.
problem Understanding equivariant structures in Khovanov homotopy types.
method Investigates group actions on homotopy coherent diagrams to prove equivalence.
result Equivariant Khovanov homotopy types are equivariantly stably homotopy equivalent.
Defines homotopy type for links in thickened surfaces.
problem Homotopical Khovanov homology of links in higher genus surfaces.
method Stable homotopy type for links in thickened torus and higher genus surfaces.
result Definition of Khovanov-Lipshark-Sarkar homotopy type for links in thickened surfaces.
Every nonflat conformal minimal surface is homotopic to a proper one.
problem Proving homotopy of nonflat conformal minimal surfaces to proper ones.
method Analyzing immersions and fluxes of Riemann surfaces into \(\mathbb{R}^n\) and \(\mathbb{C}^n\).
result Every nonflat conformal minimal immersion is homotopic to a proper one.
The paper introduces a new equivalence relation for finitely presented groups based on their asymptotic topology.
problem Classifying finitely presented groups based on their asymptotic topology.
method Introducing a new equivalence relation (proper 2-equivalence) and studying its properties.
result Finitely presented groups with specific properties are classified up to proper 2-equivalence.
The thesis defines and proves invariants for manifolds of bounded geometry.
problem Lipschitz-homotopy invariants for manifolds of bounded geometry.
method Definition of a controvariant functor and invariance of the Roe index and ρ-class.
result Lipschitz-homotopy invariants are defined and proven for manifolds of bounded geometry.
Corrects a 1998 proof about free factors of free groups.
problem Verifying the proof of free factors in free groups.
method Analyzes the geometric realization of free factors.
result Geometric realization is homotopy equivalent to a wedge of spheres.
4-manifolds with specific groups have unique homotopy types.
problem Classifying 4-manifolds with finite abelian 2-generator fundamental groups.
method Showed homotopy type is determined by quadratic 2-type.
result Homotopy type of 4-manifolds is determined by their quadratic 2-type.
Solves a problem related to classifying spaces for proper actions and Nielsen Realization.
problem Whether a cocompact proper topological Γ-manifold is equivariantly homotopy equivalent to the classifying space for proper actions.
method Using Poincaré models and assuming a zero-dimensional singular set, the problem is solved in the Poincaré category.
result New results about Brown's problem are obtained under certain conditions on the underlying group.
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.