Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
Characterizes Stein surfaces with finite homotopy rank-sum.
problem Finite homotopy rank-sum in Stein spaces.
method Rational homotopy theory, classification of Stein surfaces.
result Affine Stein surfaces with finite fundamental group are either simply connected or of order 2.
Characterizes compact complex surfaces with finite homotopy rank-sum.
problem Compact complex surfaces with finite homotopy rank-sum.
method Characterization and proof of Steinness of universal cover.
result Smooth compact complex Kaehler surfaces with finite homotopy rank-sum.
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.
The paper shows plentiful non-homotopy finite Poincaré duality spaces.
problem The existence of non-homotopy finite Poincaré duality spaces.
method Constructing a finitely dominated Poincaré space with a non-trivial 2-divisible element in the reduced Grothendieck group.
result The existence of finitely dominated Poincaré spaces that are not homotopy finite.
Homotopy classification for certain 4-manifolds with dihedral fundamental groups.
problem Classifying the homotopy types of specific 4-manifolds with dihedral fundamental groups.
method Using quadratic 2-type and combining with results from Hambleton-Kreck and Bauer.
result Homotopy types of finite oriented Poincaré 4-complexes are determined by their quadratic 2-type when fundamental group is dihedral.
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. Added examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
problem Constructing examples of S^1-manifolds with finite 2nd homotopy group and non-zero A-genus.
method Explicit equivariant surgeries to construct examples.
result Construction of new examples with finite 2nd homotopy group and non-zero A-genus.
Generalizes π2-diffeomorphism finiteness to non-zero first homotopy groups.
problem Bounding diffeomorphic types of compact manifolds with vanishing first and second homotopy groups.
method Generalizing the π2-diffeomorphism finiteness theorem to include non-zero first homotopy groups. result Diffeomorphic types of compact manifolds with non-zero first homotopy groups can be bounded.
Some properties of [L]-homotopy group for finite complex L are investigated. It is proved that for complex L whose extension type lying between Sn and Sn+1 n-th [L]-homotopy group of Sn is isomorphic to Z.
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. 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.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
We prove the homotopy invariance of L^2 torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L^2 cohomology of the covering space vanishes, the homotopy invariance was establi…
We describe the second homotopy group of any CW-complex K by analyzing the universal cover of a locally finite model of K using the notion of G-coloring of a partially ordered set. As applications we prove a generalization of the Hurewicz theorem, which relates the homotopy and homology of non-necessarily simply-…
Let X be a finite CW-complex of dimension q. If its fundamental group π1(X) is polycyclic of Hirsch number h>q we show that at least one of the homotopy groups πi(X) is not finitely generated. If h=q or h=q-1 the same conclusion holds unless X is an Eilenberg-McLane space K(π1(X),1).
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.
We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…
Study shows automorphism groups of certain hyperbolic manifolds are infinitely generated.
problem Understanding the homotopy groups of automorphism groups of hyperbolic manifolds.
method Analyzing the homotopy groups of specific topological groups and applying these results to automorphism groups of hyperbolic manifolds.
result The (n-4)-th homotopy group of the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (n >= 4) is infinitely generated.
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…
Researchers show a complex structure is not a counterexample to a topological problem.
problem Wall's D2 problem about finite CW-complexes.
method Introduced and analyzed new presentations of quaternion groups to prove homotopy types.
result The complex structure is not a counterexample to Wall's D2 problem.
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.
Let G be a finite group. The unit sphere in a finite-dimensional orthogonal G-representation motivates the definition of homotopy representations, due to tom Dieck. We introduce an algebraic analogue, and establish its basic properties including the Borel-Smith conditions and realization by finite G-CW-complexes.
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.
We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
problem Proving a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.
method Homological approach to show finitely many nonzero homology groups, each finitely generated.
result BDiff(M, rel ∂) has finitely many nonzero homology groups, each finitely generated, for connected sums of irreducible 3-manifolds with nontrivial and non-spherical boundaries.
The paper shows geometric realisation over specific groups and knots.
problem Geometric realisation of modules over aspherical groups and knots.
method Using extensions of scalars of relation modules and constructing specific 2-complexes.
result Exotic presentations of groups and stably free non-free modules over Baumslag-Solitar groups.
The paper classifies 4-manifolds based on their fundamental groups and orientation characters.
problem Classifying 4-manifolds with specific fundamental groups and orientation characters.
method Developed a criterion for groups and homomorphisms to classify 4-manifolds up to homotopy equivalence.
result Closed 4-manifolds with specified fundamental groups and orientation characters are classified by their quadratic 2-types.
Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
The paper classifies 4-manifolds with given boundaries.
problem Extending homotopy equivalences of boundaries to 4-manifolds.
method Surgery theory and classification of groups.
result Conditions for extending homotopy equivalences to homeomorphisms.
In this note, we fill in a gap in the literature by proving that the Teichmueller modular groups (mapping class groups) are not Poincare duality groups and the complexes of curves of surfaces have infinite homotopy type (i.e. are not homotopy equivalent to a finite CW-complex).
Periodic geodesics on Hilbert half-Lie groups exist whenever the fundamental group is nontrivial.
problem Existence of periodic geodesics on Hilbert half-Lie groups
method Using completeness results and Lyusternik-Fet type theorem
result Periodic geodesics exist whenever the fundamental group is nontrivial
A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…
Study shows exotic Dehn twists on certain 3-sphere fillings.
problem Extending group actions from boundaries to interiors of 4-manifolds.
method Analyzing Dehn twists on Seifert homology spheres and their fillings.
result Dehn twists on certain fillings are infinite order exotic.
Study homotopy groups of open books and their pages, pages, and bindings.
problem Homotopy groups of open books and their components.
method Homotopy theoretic conditions on monodromy, integral and rational loop space decompositions.
result Integral and rational loop space decompositions for open books under specific conditions.
If a finite group G is isomorphic to a subgroup of SO(3), then G has the D2-property. Let X be a finite complex satisfying Wall's D2-conditions. If π1(X)=G is finite, and χ(X)≥1−Def(G), then X∨S2 is simple homotopy equivalent to a finite 2-complex, whose simple homotopy type depends only on …
Stable approach solves equivariant Hopf theorem for G-manifolds.
problem Describe homotopy classes of G-equivariant maps into a G-sphere.
method Equivariant stable homotopy theory with semi-free G-universe.
result Degrees of maps are characterized by congruences.
Study homotopy types of 4-manifolds, finding decompositions and conditions for desuspension.
problem Determine homotopy types of double suspensions of 4-manifolds with 2-torsion.
method Use Postnikov square and analyze homology groups to find decompositions and conditions for desuspension.
result Homotopy decompositions of double suspensions as wedge sums of specific complexes.
Study Swan modules and homotopy types, resolving Wall and Dyer questions.
problem Homotopy classification of CW-complexes and Swan modules.
method Analysis of Swan modules and their stably free properties.
result Existence of non-free stably free Swan modules and implications for homotopy equivalence.
The paper defines and proves the existence of train track maps on graphs of groups.
problem Understanding homotopy equivalences in graphs of groups.
method Developed the theory of train track maps on graphs of groups, defining maps and homotopy equivalences.
result Any homotopy equivalence of a graph of groups may be represented by a relative train track map under certain conditions.
Novikov theorem extended to rational Pontryagin classes for cyclic group C4.
problem Classifying stable Cp-smoothings of high-dimensional manifolds. method Computing equivariant homotopy groups and applying to C4. result Novikov's theorem extended to rational Pontryagin classes for C4. This paper categorifies Quinn's TQFTs and computes them for specific omega-groupoids.
problem Constructing and computing finite total homotopy TQFTs.
method Direct homotopy theoretical construction, categorification of Quinn's TQFTs, explicit computation for omega-groupoids.
result Categorification and explicit computation of Quinn's TQFTs for omega-groupoids.
We give an analytical proof of the Poincare-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.
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.
Scalable spaces are simply connected manifolds with nice cohomology properties.
problem Understanding the limitations of formality in higher homotopy groups.
method Analyzing the embedding of cohomology algebras into differential forms.
result Spaces that are formal but not scalable provide counterexamples to Gromov's conjecture.
This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles X, from a perspective of homotopy theory: Specifically, for any prime ℓ which does not divide the type of X, the ℓ-torsion of this invariants is equal to a sum of the colouring po…
We develop the foundations of the deformation theory of compact complete affine space forms and affine crystallographic groups. Using methods from the theory of linear algebraic groups we show that these deformation spaces inherit an algebraic structure from the space of crystallographic homomorphisms. We also study th…
Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negative…
Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.
problem Understanding the topological properties of random branched covers of groups.
method Constructing a random model for branched covers and showing asymptotic homotopy equivalence to geometrically small cancellation complexes.
result The fundamental group of a random branched cover is Gromov hyperbolic and has small cohomological dimension.