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.
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.
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.
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.
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.
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.
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.
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 AM for every self-homotopy equivalence f of a 3-manifold M such that fk is homotopic to a homeomorphism for some integer k. result There exists a constant AM depending only on the manifold M such that for every self-homotopy equivalence f of M, there is an integer k with 1≤k≤AM for which fk 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.
Given a bundle of chain complexes, the algebra of functions on its shifted cotangent bundle has a natural structure of a shifted Poisson algebra. We show that if two such bundles are homotopy equivalent, the corresponding Poisson algebras are homotopy equivalent. We apply this result to L∞-algebroids to show th…
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of L∞-morphisms twisted by gauge equivalent elements. result Globalized formalities with different covariant derivatives are homotopic.
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.
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.
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, …
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…
A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to th…
New invariant detects non-homotopy equivalent 4-manifolds.
problem Detecting non-homotopy equivalent 4-manifolds.
method Extending Kreck and Schafer's doubling construction to 2-complexes with finite fundamental group.
result Existence of k closed smooth 4-manifolds that are stably diffeomorphic but not homotopy equivalent. Constructs Lepage equivalents for arbitrary-order Lagrangians.
problem Creating Lepage equivalents for complex Lagrangians.
method Uses variational bicomplex and symmetric linear connections to construct Lepage equivalents satisfying the closure property.
result Shows how to extend global Lepage equivalents to ones satisfying the closure property.
The paper studies obstructions to homotopy invariance of loop coproducts.
problem Characterizing obstructions to homotopy invariance of loop coproducts.
method Using a construction of Geoghegan and Nicas, the paper defines the Reidemeister trace and realizes the Goresky-Hingston coproduct as a map of spectra.
result The failure of a map to entwine spectral coproducts can be characterized by Chas-Sullivan multiplication with the Reidemeister trace.
Survey on finite group actions on CW-complexes homotopy to spheres.
problem Understanding finite group actions on CW-complexes homotopy equivalent to spheres.
method Survey of extensive literature on finite G-CW-complexes homotopy equivalent to spheres. result Finite G-CW-complexes homotopy equivalent to spheres have finite group actions. Paper defines weak (1, 3) homotopy for knot projections and classifies trivial knots.
problem Classifying knot projections under weak (1, 3) homotopy.
method Defines weak (1, 3) homotopy, introduces a map to knot isotopy classes, and determines trivial knots.
result Determines which knot projections are trivial under weak (1, 3) homotopy.
Equivalence relations can be defined on Gauss phrases using combinatorial moves. In this paper we consider two closely related equivalence relations on Gauss phrases, homotopy and open homotopy. In particular, in each case, we define a new invariant and determine the values that it can attain.
We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen …
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
The paper characterizes G-ANR spaces and their properties.
problem Characterizing G-ANR spaces and their properties for compact groups. method Proving conditions for a metrizable G-space to be a G-ANR. result Conditions for a metrizable G-space to be a G-ANR are provided. Homotopy equivalent boundaries of cube complexes are studied.
problem The equivalence of different boundaries of cube complexes.
method Using a partial order on a quotient of the Roller boundary, we obtain the simplicial Roller boundary and show homotopy equivalence among the Tits, simplicial, and simplicial Roller boundaries.
result The Tits, simplicial, and simplicial Roller boundaries are homotopy equivalent.
Study the embedding space of a Hopf link in 3D and 3-manifolds.
problem Homotopy equivalence of embedding spaces of Hopf links.
method Prove homotopy equivalence using subspace inclusions and manifold properties.
result Embedding spaces are homotopy equivalent to specific 3-manifolds.
This paper refines homotopy theory for cubical sets and uniform spaces.
problem Classical homotopy theory limitations in cubical sets and uniform spaces.
method Develops a uniform-theoretic refinement for cubical sets and uniform spaces, lifting to a full and faithful embedding.
result Lifts classical homotopy categories to new uniform homotopy categories, generalizing cohomology theories.
Two string links are equivalent up to 2n-moves and link-homotopy if and only if their all Milnor link-homotopy invariants are congruent modulo n. Moreover, the set of the equivalence classes forms a finite group generated by elements of order n. The classification induces that if two string links are equivalent u…
The paper defines a new equivalence relation for knot projections and finds an infinite number of distinct classes.
problem Classifying knot projections based on weak homotopy equivalence.
method Defining weak (1, 2, 3) homotopy and using it to find an invariant.
result There are an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections.
New homotopy theory reveals the structure of stable curves.
problem Understanding the structure of the moduli stack of stable curves.
method Using stratified homotopy theory, the category of stable curves captures the stratified homotopy type of the moduli stack.
result The category of stable curves classifies constructible sheaves via an exodromy equivalence.
Proves Khovanov homology has no torsion for bipartite circle graphs.
problem Proving properties of Khovanov homology for bipartite circle graphs.
method Proved homotopy equivalence of independence complexes to wedges of spheres.
result Extreme Khovanov homology has no torsion.
The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.
problem Spaces of unbounded Fredholm operators and their properties.
method Analyzing the spaces and proving homotopy equivalences.
result Natural maps between four spaces of unbounded Fredholm operators are homotopy equivalences.
We prove that for cobordant closed spin manifolds of dimension n≥3 the associated spaces of metrics with invertible Dirac operator are homotopy equivalent. This is the spinorial counterpart of a similar result on positive scalar curvature of Chernysh/Walsh and generalizes the surgery result of Ammann-Dahl-Humbert…
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.
If M and N are equivariantly homotopy equivalent G-manifolds, then the fixed sets M^G and N^G are also homotopy equivalent. The replacement problem asks the converse question: If F is homotopy equivalent to the fixed set M^G, is F = N^G for a G-manifold equivariantly homotopy equivalent to M? We prove that for locally …
Classifies smooth manifolds homotopy equivalent to sphere products
problem Classifying smooth manifolds homotopy equivalent to sphere products
method Using normal-invariant map and explicit families of manifolds
result Classifies smooth manifolds up to almost diffeomorphism
Study rigidity of self-maps and classify manifolds homotopy equivalent to Stiefel manifolds.
problem Rigidity of self-maps and classification of manifolds homotopy equivalent to Stiefel manifolds.
method Finding explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences.
result Classification of manifolds tangentially homotopy equivalent to Vn,2imesSk up to almost diffeomorphism. 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.
We show that the Prüfer surface, which is a separable non-metrizable 2-manifold, has not the homotopy type of a CW-complex. This will follow easily from J. H. C. Whitehead's result: if one has a good approximation of an arbitrary space by a CW-complex, which fails to be a homotopy equivalence, then the given space is n…
We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaintrob's Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two rep…
The paper defines new homotopy relations on knot projections and classifies certain knot types.
problem Defining and classifying knot homotopy relations.
method Introducing cross chord numbers and using them to define strong and weak (1, 3) homotopies.
result Complete classification of knot projections with trivializing number two.
The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-di…
This is the second of a series of papers which are devoted to a comprehensive theory of maps between orbifolds. In this paper, we develop a basic machinery for studying homotopy classes of such maps. It contains two parts: (1) the construction of a set of algebraic invariants -- the homotopy groups, and (2) an analog o…
Homotopy types of spaces of metrics with positive scalar curvature are shown to be equivalent to spheres.
problem Understanding the homotopy types of spaces of metrics with positive scalar curvature.
method Analyzing the spaces of metrics on manifolds with positive scalar curvature and showing homotopy equivalence to spheres.
result Spaces of metrics with positive scalar curvature on certain manifolds are homotopy equivalent to spheres.
We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.
The aim of this paper is to give an s-cobordism classification of topological 4-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of 4-manifolds. Using this braid together with the m…