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.
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48 results for “proper homotopy equivalence”
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.
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 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.
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.
Extends geometric group theory techniques to arbitrary proper metric ARs.
problem Generalizing geometric group actions to non-freely acting groups with torsion.
method Extends techniques from geometric group theory to arbitrary proper metric ARs, eliminating freeness requirements.
result New theorems on proper homotopy equivalence and Z-structures for geometric actions.
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.
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.
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.
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.
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.
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.
We prove that if is a lattice in a classical simple Lie group , then the symmetric space of is -equivariantly homotopy equivalent to a proper cocompact -CW complex of dimension the virtual cohomological dimension of .
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.
The paper classifies maps from vector bundles to Euclidean spaces.
problem Classifying homotopy classes of maps from vector bundles to Euclidean spaces.
method Using homotopy theory and vector bundles.
result Computed homotopy classes of proper maps and stability range.
The study reveals non-homotopy equivalent subspaces of curves with curvature constraints.
problem Understanding the homotopy type of subspaces of curves with curvature constraints.
method Used a version of the h-principle to prove results.
result Explicit construction of exotic generators for some homotopy and cohomology groups.
Classifies -injective maps between non-compact surfaces.
problem Characterizing maps with injective fundamental groups.
method Proper homotopy classification of maps.
result All -injective proper maps are classified.
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.
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.
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.
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.
Motivated by string topology and the arc operad, we introduce the notion of quasi-operads and consider four (quasi)-operads which are different varieties of the operad of cacti. These are cacti without local zeros (or spines) and cacti proper as well as both varieties with fixed constant size one of the constituting lo…
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.
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…
Proves stable properties of proper maps between manifolds.
problem Stability of homotopy classes of proper maps and Pontryagin-Thom construction.
method Explicit construction and proof of bijection in a stable range.
result Stabilization of homotopy classes of proper maps and Pontryagin-Thom type bijection.
New examples of manifolds with similar homotopy but different simple homotopy types.
problem Characterizing groups for which high-dimensional manifolds can be homotopy equivalent but not simple homotopy equivalent.
method Construction of doubles of thickenings and use of a formula for Whitehead torsion.
result Examples of high-dimensional manifolds exist for any finitely presented group with a nontrivial Whitehead group involution.
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 is said to be properly 3-realizable if there exists a compact 2-polyhedron with whose universal cover $\til…
One proves that there exists an obstruction to an open simply connected -manifold of dimension being geometrically simply connected. In particular there exist uncountably many simply connected -manifolds which are not w.g.s.c. One proves that for an -manifold proper homotopy equivalent to a…
New examples of manifolds that are homotopy but not simple homotopy equivalent.
problem Characterizing simple homotopy types of even dimensional manifolds.
method Using algebraic K-theory, surgery obstruction map, and homotopy automorphisms.
result Construction of infinite families of manifolds that are homotopy equivalent but not simple homotopy equivalent.
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
Generalizes manifold results for Lie groups, proving equivariant homotopy type.
problem Hilbert-Smith conjecture for topological -manifolds.
method Generalization of previous results, verification of n-classifying spaces.
result Any Palais-proper topological -manifold has the equivariant homotopy type of a countable proper -CW complex.
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 .
The paper shows that certain hypersurfaces of spheres in nonpositive curvature space forms are both topologically and homotopically rigid.
problem The rigidity of hypersurfaces in nonpositive curvature space forms.
method Analyzing the space of closed hypersurfaces with principal curvatures in a specified interval and showing their weak homotopy equivalence to the group of diffeomorphisms of the sphere.
result Closed hypersurfaces with principal curvatures in a specified interval are weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of the sphere.
Solves a problem related to Nielsen realization for certain groups.
problem Whether a cocompact proper topological manifold is equivariantly homotopy equivalent to a classifying space.
method Assumes a zero-dimensional singular set and uses properties of hyperbolic groups and aspherical manifolds.
result Solves the problem for specific groups containing a normal torsion-free subgroup.
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
Study delocalized eta invariants for signature operators on proper manifolds.
problem Define and analyze delocalized eta invariants for signature operators on proper manifolds.
method Develop detailed heat-kernel analysis and apply to proper manifolds with boundary.
result Prove index formulas relating delocalized eta invariants to Atiyah-Patodi-Singer indices.
New infinite family of 4-manifolds with same stable properties but not homotopy equivalent.
problem Finding infinite homotopy stable classes of 4-manifolds with boundary.
method Construction of an infinite family of topological 4-manifolds with specific properties.
result Infinite family of 4-manifolds that are stably homeomorphic but not homotopy equivalent.
New 3D shape not homotopy equivalent to any hyperbolic shape.
problem Finding 3D shapes not homotopy equivalent to hyperbolic ones.
method Constructed a locally hyperbolic 3-manifold with specific fundamental group properties.
result The constructed manifold is not homotopy equivalent to any hyperbolic manifold.
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.
Spaces of metrics with invertible Dirac operator are homotopy equivalent for cobordant manifolds.
problem Proving homotopy equivalence of spaces of metrics with invertible Dirac operator.
method Using cobordism theory and properties of Dirac operators.
result Spaces of metrics with invertible Dirac operator are homotopy equivalent for cobordant manifolds.
Homotopy equivalences of 3-manifolds have a bounded power.
problem Understanding the behavior of self-homotopy equivalences of 3-manifolds.
method Proving the existence of a constant for every self-homotopy equivalence of a 3-manifold such that is homotopic to a homeomorphism for some integer .
result There exists a constant depending only on the manifold such that for every self-homotopy equivalence of , there is an integer with for which is homotopic to a homeomorphism.
Proves loop coproduct invariance under simple homotopy equivalences.
problem Invariance of loop coproduct under simple homotopy equivalences.
method Transformation formula involving Whitehead torsion.
result Loop coproduct is invariant under simple homotopy equivalences.
Variant of string topology coproduct invariance formula found.
problem Failure of string topology coproduct invariance under homotopy equivalences.
method Obstruction class built from higher homotopy data and fake diagonal.
result Vanishing of obstruction class measures smallness of homotopy equivalences.
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of -morphisms twisted by gauge equivalent elements.
result Globalized formalities with different covariant derivatives are homotopic.
We compute the homotopy type of the space of proper d-dimensional submanifolds of 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…
Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a…
The paper finds manifold structures on complex spaces.
problem Constructing manifold structures on highly connected Poincaré complexes.
method Constructing examples and determining homotopy types.
result Examples of highly connected Poincaré complexes are found to be homotopy equivalent to manifolds but not smooth.
By defining combinatorial moves, we can define an equivalence relation on Gauss words called homotopy. In this paper we define a homotopy invariant of Gauss words. We use this to show that there exist Gauss words that are not homotopically equivalent to the empty Gauss word, disproving a conjecture by Turaev. In fact, …