Study new bounds on TC of spaces with subgroup inclusions.
problem Lower bounds on TC of spaces with subgroup inclusions.
method Generalizes TC results from aspherical spaces to spaces with subgroup inclusions.
result Establishes new lower bounds on sequential TCs of aspherical spaces.
3D spaces without boundaries are found that aren't manifolds.
problem Finding compact aspherical Alexandrov spaces without boundaries.
method Constructing specific examples of 3D spaces.
result Examples of 3D spaces without boundaries that are not topological manifolds.
The "polyhedral product functor" produces a space from a simplicial complex L and a collection of pairs of spaces, {(A(i),B(i))}, where i ranges over the vertex set of L. We give necessary and sufficient conditions for the resulting space to be aspherical. There are two similar constructions, each of which starts with …
The study of PD3-pairs extends results for aspherical 3-manifolds.
problem Understanding PD3-pairs with aspherical ambient spaces. method Attaching 1-handles to PD3-pairs with aspherical ambient space and π1-injective boundary. result There are only finitely many PD3-pairs with a specific group property. 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.
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional closed aspherical manifold M such that the fundamental group π1(M) coarsely co…
Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.
problem Understanding actions of finite groups on aspherical manifolds.
method Analyzing the outer automorphism group and homeomorphism group of the fundamental group.
result Proves the homeomorphism group is Jordan and bounds the discrete degree of symmetry.
We show that any open aspherical manifold of dimension n>3 is tangentially homotopy equivalent to an n-manifold whose universal cover is not homeomorphic to the Euclidean space.
Study finite group actions on exotic aspherical space forms.
problem Classify finite group actions on M#Σ where M is a closed aspherical space form and Σ is an exotic n-sphere. method Combines geometric and topological rigidity results with smoothing theory and spectral sequence computations.
result Classification of free actions of finite groups on M#Σ when M is 7-dimensional. Every compact aspherical Riemannian manifold admits a canonical series of orbibundle structures with infrasolv fibers which is called its infrasolv tower. The tower arises from the solvable radicals of isometry group actions on the universal covers. Its length and the geometry of its base measure the degree of continuo…
Proves conjecture on graph configuration spaces' complexity.
problem Topological complexity of graph configuration spaces.
method Lower bound derived from insights into aspherical spaces.
result Proves Farber's conjecture on stable topological complexity.
This paper studies limits of aspherical manifolds with specific curvature conditions.
problem Understanding the Gromov-Hausdorff limits of aspherical manifolds with given curvature constraints.
method Analyzing sequences of compact manifolds with Ricci curvature or sectional curvature conditions, and using diffeomorphism or homeomorphism properties.
result If the manifolds are diffeomorphic or homeomorphic to nilmanifolds, their limits are also diffeomorphic or homeomorphic to nilmanifolds.
Let G/H be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold Γ\G/H is by definition a quotient of G/H by a discrete uniform subgroup Γ≤G. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…
Study of circle configurations in the plane, proving aspherical space and computing fundamental groups.
problem Understanding the space of configurations of circles in the plane.
method Proved the space is aspherical and computed fundamental groups of its components.
result Fundamental groups are iterated semidirect products of braid groups, with structure dictated by a finite rooted tree.
Paper confirms Whitehead's conjecture for aspherical 2-complexes.
problem Whitehead's conjecture about aspherical 2-complexes.
method Argument on ribbon sphere-links, generalized for aspherical 2-complexes.
result Whitehead's conjecture confirmed for aspherical 2-complexes.
The paper proves spaces associated to certain cubical presentations are aspherical.
problem Proving the asphericity of cubical presentations.
method Using coned-off spaces and properties of fundamental groups.
result The fundamental groups of cubical presentations have aspherical associated spaces.
The paper proves asphericity of configuration spaces and covers them with entire functions.
problem Proving asphericity of configuration spaces and classifying covering spaces.
method Using algebraic topology and covering spaces theory.
result Locally finite infinite configuration spaces are aspherical and coverings can be realized by entire functions.
Aspherical configuration Lie groupoids complement is proven for a class of orbifolds.
problem Proving aspherical property of configuration Lie groupoids complement for orbifolds.
method Generalization to Lie groupoids, focusing on orbifolds.
result Complement is aspherical for a class of aspherical 2-dimensional orbifolds.
We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fund…
For n>6, we show that if G is a torsion-free hyperbolic group whose visual boundary is an (n-2)-dimensional Sierpinski space, then G=π_1(W) for some aspherical n-manifold W with nonempty boundary. Concerning the converse, we construct, for each n>3, examples of aspherical manifolds with boundary, whose fundamental grou…
The paper shows non-aspherical path components in G2-moduli spaces.
problem Characterizing non-aspherical path components in G2-moduli spaces.
method Using generalised Kummer construction and resolving singularities with Eguchi-Hanson spaces, the authors establish a fibration over each path component with Eilenberg Mac Lane spaces as fibres.
result The comparison map is a fibration over each path component with Eilenberg Mac Lane spaces as fibres, indicating non-trivial families of G2-manifolds.
This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …
Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.
problem Obstructing the existence of positive scalar curvature in higher dimensions.
method Introduced a relative aspherical condition and a new geometric quantity called spherical width.
result Proved results on how 3-manifolds obstruct the existence of positive scalar curvature.
Relative notions of combinatorial asphericity have been used to prove that injective labeled oriented trees (which encode spines of ribbon 2-knots) are aspherical. This article presents an overview and comparison of the different notions of relative combinatorial asphericity. It also contains new results concerning cha…
We investigate periodic diffeomorphisms of non-compact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with non-vanishing Euler characteristic. In the i…
The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.
problem Proving the Singer conjecture and refining Gromov's inequality for aspherical complex surfaces.
method Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group, and refinement of Gromov's inequality for non-residually finite surfaces.
result Proof of the Singer conjecture for aspherical complex surfaces with residually finite fundamental group.
Constructs 4-manifolds with positive Euler characteristic proving a conjecture.
problem Finding aspherical 4-manifolds with positive Euler characteristic.
method Explicit construction of 4-manifolds with specified properties.
result Proves a conjecture by constructing a 4-manifold with Euler characteristic 1.
Study on simplicial volume and Euler characteristic of aspherical manifolds.
problem Whether vanishing simplicial volume implies vanishing Euler characteristic.
method Various strategies for both affirmative and negative answers, context with other problems, and comparative analysis of additivity properties.
result Found counterexamples among aspherical spaces that are homology equivalent to manifolds but not manifolds themselves.
Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.
problem Characterizing Sasakian manifolds with nilpotent fundamental groups.
method Proved diffeomorphism to Heisenberg nilmanifolds.
result Compact aspherical Sasakian manifolds with nilpotent fundamental groups are Heisenberg nilmanifolds.
This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed.
We develop a new purely combinatorial approach to N. Steenrod's problem on realisation of cycles. We prove that every n-dimensional homology class of every topological space can be realised with some multiplicity by an image of a finite-fold covering over the manifold M^n, where M^n is the isospectral manifold of real …
The paper extends a vanishing theorem for hypersurfaces in aspherical manifolds.
problem The vanishing of rational homology for hypersurfaces in aspherical manifolds.
method Generalization of Gromov's reduction from aspherical conjecture to filling radius conjecture.
result Continuous maps from certain 4-manifolds to aspherical 5-manifolds induce zero maps in H4(⋅,Q). A {\em word labeled oriented graph} (WLOG) is an oriented graph G on vertices X={x1,…,xk}, where each oriented edge is labeled by a word in X±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …
Non-asphericity of strata of genus-one differentials
problem Strata of genus-one differentials
method Non-asphericity
result Infinitely many counterexamples to conjectures
Study shows some complex shapes don't fit a certain property.
problem Some aspherical manifolds lack the Bounded Index Property (BIP).
method Analyzes specific aspherical manifolds to prove they don't have BIP.
result Proves existence of aspherical manifolds without BIP.
Tautological classes, or generalised Miller-Morita-Mumford classes, are basic characteristic classes of smooth fibre bundles, and have recently been used to describe the rational cohomology of classifying spaces of diffeomorphism groups for several types of manifolds. We show that rationally tautological classes depend…
Study aspherical manifolds with boundaries, proving homological criteria.
problem Classify compact aspherical manifolds with boundaries.
method Develop homological criteria, use Poincaré pairs, and surgery theory.
result Prove conditions for realizing manifolds as boundaries of aspherical manifolds.
Study symplectically aspherical Kähler manifolds with unique properties.
problem Existence and properties of symplectically aspherical Kähler manifolds.
method Detailed study and analysis of geometric and topological features.
result Existence of symplectically aspherical Kähler manifolds with large fundamental groups.
We establish isosystolic inequalities for a class of manifolds which includes the aspherical manifolds. In particular, we relate the systolic volume of aspherical manifolds first to their minimal entropy, then to the algebraic entropy of their fundamental groups.
Researchers propose a new approach to the Hopf problem for aspherical varieties.
problem The Hopf problem for aspherical smooth projective varieties.
method An approach recently proposed by Liu, Maxim, and Wang in [LMW21].
result An intriguing conjecture regarding the geography of aspherical surfaces of general type.
A labeled oriented tree is called injective if each generator occurs at most once as an edge label. We show that injective labeled oriented trees are aspherical. The proof relies on a new relative asphericity test based on a lemma of Stallings.
Surveying topological complexity of graph configurations, unifying traditional and modern approaches.
problem Understanding the topological complexity of configuration spaces of graphs.
method Exploring traditional cohomology methods and modern asphericity/fundamental group approaches.
result Unified understanding of topological complexity through both traditional and modern methods.
Study simplicial volume in fiber bundles with connected groups.
problem Understanding simplicial volume in fiber bundles.
method Investigated fiber bundles with connected structure groups.
result Simplicial volume of total space matches trivial bundle under certain conditions.
Diagrammatic reducibility DR and its generalization vertex asphericity VA are combinatorial tools developed for detecting asphericity of a 2-complex. Here we present tests for a relative version of VA that apply to pairs of 2-complexes (L,K), where K is a subcomplex of L. We show that a relative weight test holds…
A 2-manifold's group structure is deduced from orbit configuration spaces.
problem Understanding the fundamental groups of orbit configuration spaces.
method Relating the four-term exact sequence of orbifold pure braid groups to the fundamental groups of the orbit configuration spaces.
result Fundamental groups of orbit configuration spaces form a four-term exact sequence.
No 5D aspherical manifolds can have uniformly positive scalar curvature.
problem Proving the non-existence of metrics with positive scalar curvature on certain 5D manifolds.
method Uniform acyclicity and toric symmetrization of stable μ-bubbles.
result Compact aspherical 5-manifolds cannot have metrics with uniformly positive scalar curvature.
The paper describes the structure of injective LOT-complexes and proves they are aspherical.
problem The unresolved asphericity question for labeled oriented trees encoding spines of ribbon discs.
method Complete description of the link of a reduced injective LOT complex, proving asphericity.
result Reduced injective LOT complexes are aspherical, with specific conditions for non-boundary sub-LOTs.
Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.
problem Understanding the fundamental groups of aspherical manifolds under certain curvature conditions.
method Analyzing the collapsing behavior of manifolds with bounded Ricci curvature and diameter.
result The fundamental groups of such manifolds have non-trivial finitely generated abelian normal subgroups.