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.
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.
We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.
New method controls surface extrinsic diameter for positive scalar curvature metrics.
problem Preventing complete metrics with positive scalar curvature on surfaces within manifolds.
method Interior control for extrinsic diameter of surfaces with positive scalar curvature.
result Closed aspherical manifolds cannot have complete metrics with positive scalar curvature when subsets are removed.
Study pro-p completions of orientable PD_n groups, proving best results in three cases.
problem Understanding pro-p completions of orientable PD_n groups. method Examined four cases of orientable PD_3-groups and some PD_n groups (n≤5), providing examples and proving best results in three cases.
result Best results in three out of four cases of orientable PD_3-groups.
New concept of quasi-ribbon surface-links simplifies complex surface-links.
problem Complexity in surface-links of trivial components.
method Introducing quasi-ribbon surface-links as a generalization of ribbon surface-links.
result Every F-link of trivial components on a surface F with at most one aspheric component is a quasi-ribbon surface-link.
Formula for winding numbers on non-null-homotopic curves on surfaces.
problem Determining winding numbers for non-null-homotopic curves on surfaces.
method Generalized a Whitney-type formula for winding numbers of non-null-homotopic curves on aspherical surfaces.
result Formula for winding numbers on non-null-homotopic curves on aspherical surfaces.
Constructs odd Euler characteristic 4-manifolds.
problem Finding 4-manifolds with odd Euler characteristics.
method Explicit construction of aspherical 4-manifolds with odd Euler characteristics.
result Explicit examples of aspherical 4-manifolds with odd Euler characteristics greater than 12.
The Borel Conjecture predicts that closed aspherical manifolds are topological rigid. We want to investigate when a non-aspherical oriented connected closed manifold M is topological rigid in the following sense. If f: N --> M is an orientation preserving homotopy equivalence with a closed oriented manifold as target, …
Proves strong version of Hopf problem for hyperbolic circle bundles.
problem Strong version of Hopf problem for circle bundles over aspherical manifolds with hyperbolic fundamental groups.
method Analyzes circle bundles over aspherical manifolds with hyperbolic fundamental groups.
result Every self-map of non-zero degree is either homotopic to a homeomorphism or a non-trivial covering and the bundle is virtually trivial.
We define winding numbers of regular closed curves on surfaces with a nice euclidean or hyperbolic geometry. We prove that two regular closed curves are regularly homotopic if and only if they are freely homotopic and have the same winding number.
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.
We study random 2-dimensional complexes in the Linial - Meshulam model and find torsion in their fundamental groups at various regimes. We find a simple algorithmically testable criterion for a subcomplex of a random 2-complex to be aspherical; this implies that any aspherical subcomplex of a random 2-complex satisfies…
Proves connection between ℓ2-Betti numbers and BNSR invariants.
problem Relationship between ℓ2-Betti numbers and BNSR invariants. method Analyzes ℓ2-Betti numbers and BNSR invariants of groups. result If the nth ℓ2-Betti number is non-zero, then the nth BNSR invariant over Q is empty. Symplectic fillings of prequantization bundles are shown to be disk bundles under certain conditions.
problem Characterizing symplectic fillings of prequantization bundles with finite capacities.
method Analysis of symplectic capacities and diffeomorphisms.
result Symplectic fillings of prequantization bundles are diffeomorphic to disk bundles under finite capacity conditions.
The study compares and characterizes different notions of relative combinatorial asphericity.
problem Proving the asphericity of injective labeled oriented trees encoding spines of ribbon 2-knots.
method Overview and comparison of different notions of relative combinatorial asphericity, new characterizations, and tests.
result New tests that imply relative combinatorial asphericity and examples illustrating the concepts.
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.
The paper shows how certain Kähler groups are uniquely determined by their profinite completions.
problem Understanding the uniqueness of Kähler groups within residually finite groups.
method Holomorphic fibrations and profinite completions of fundamental groups.
result Aspherical smooth projective varieties are determined by their algebraic fundamental groups.
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 …
We provide estimates on the Bartnik mass of constant mean curvature (CMC) surfaces which are diffeomorphic to spheres and have positive mean curvature. We prove that the Bartnik mass is bounded from above by the Hawking mass and a new notion we call the asphericity mass. The asphericity mass is defined by applying Hami…
New tests detect asphericity in complex pairs, simplifying previous proofs.
problem Detecting asphericity in complex pairs (L,K) where K is a subcomplex of L. method Developed relative weight tests for injective labeled oriented trees.
result Injective labeled oriented trees are aspherical, strengthening previous results.
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.
New surfaces in 3D manifolds are found that cannot be smoothly deformed into each other.
problem Finding surfaces in 3D manifolds that cannot be smoothly deformed into each other.
method Constructing infinite families of homotopic surfaces in closed genus-g surfaces, showing they are not smoothly image-concordant. result Closed surfaces with common framed dual sphere can be π1-injective but not smoothly image-concordant. Study shows asphericity isn't inherited in certain arrangements.
problem Asphericity in hyperplane arrangements is not always inherited.
method Examples of K(π,1)-arrangements with restricted asphericity. result Asphericity is not hereditary among hyperplane arrangements.
The paper classifies fixed subgroups in a specific group product.
problem Classifying fixed subgroups in a specific group product.
method Complete classification through direct product analysis.
result Infinitely many fixed subgroups if and only if k ≥ 2.
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 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.
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…
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. 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.
The study creates symplectic examples with non-trivial homotopy groups.
problem Constructing symplectic manifolds with specific homotopy groups.
method Examples of symplectic manifolds with non-trivial second homotopy groups in various dimensions.
result Examples of symplectic manifolds with non-trivial homotopy groups in dimensions 4 and greater than 6.
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.
Locally homogeneous aspherical Sasaki manifolds are quasi-regular S1-Seifert bundles.
problem Characterizing locally homogeneous aspherical Sasaki manifolds.
method Analyzing homogeneous Sasaki manifolds and their quotients by discrete subgroups.
result Compact locally homogeneous aspherical Sasaki manifolds are quasi-regular S1-Seifert bundles.
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.
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 …
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 …
Study aspherical manifolds without Anosov diffeomorphisms.
problem Identify conditions under which certain manifolds do not support Anosov diffeomorphisms.
method Weakened conditions on Gogolev and Lafont, use group theoretic and topological results.
result Product of certain manifolds does not support Anosov diffeomorphisms.
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.
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.
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.
New examples of manifolds with large symmetry found.
problem Finding manifolds with significant continuous symmetry.
method Analyzing the solvable radicals of isometry group actions on universal covers.
result Examples of aspherical manifolds with large symmetry that do not support locally homogeneous metrics.
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.
Study counts special paths on complex shapes.
problem Counting special paths on complex shapes.
method Proved a lower bound using gradient flow and homology.
result Lower bound on the number of special paths.