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 …
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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.
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.
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 .
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 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 such that the fundamental group coarsely co…
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.
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.
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.
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.
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.
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.
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.
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.
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.
Aspherical -pairsmath.GT
The study of -pairs extends results for aspherical 3-manifolds.
problem Understanding -pairs with aspherical ambient spaces.
method Attaching 1-handles to -pairs with aspherical ambient space and -injective boundary.
result There are only finitely many -pairs with a specific group property.
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.
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.
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.
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical which is a purely homotopy theoretical condition implies…
Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvaturemath.DG
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.
We define a new class of irreducible groups, called groups not infinite-index presentable by products or not IIPP. We prove that certain aspherical manifolds with fundamental groups not IIPP do not admit maps of non-zero degree from direct products. This extends previous results of Kotschick and Loeh, providing new cla…
We show that the aspherical manifolds produced via the relative strict hyperbolization of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automor…
The paper explores automorphism groups of parabolic structures on aspherical manifolds.
problem Characterizing the automorphism groups of parabolic structures on aspherical manifolds.
method Analyzing properties of closed aspherical parabolic ${\sfG}$-manifolds and their automorphism groups.
result Certain parabolic ${\sfG}$-structures impose strong restrictions on the topology of compact aspherical manifolds.
The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.
problem Characterizing geometrically decomposable aspherical 4-manifolds with non-zero signature.
method Constructing examples and proving inequalities for geometrically decomposable aspherical 4-manifolds.
result All geometrically decomposable aspherical 4-manifolds with non-zero signature satisfy the inequality \( \chi \geq 3|σ| \).
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.
We show any set of degrees can be realized by manifolds.
problem Realizing any set of mapping degrees between manifolds.
method Using aspherical manifolds as building blocks.
result Any finite set containing zero is a mapping degree set between manifolds.
Proves non-existence of certain metrics on specific manifolds.
problem Existence of metrics with nonnegative scalar curvature.
method Connected sum technique and non-compact domination method.
result Connected sum of aspherical manifolds with non-compact manifolds does not admit metrics with nonnegative scalar curvature.
In this paper, Problem 4.17 on R. Kirby's problem list is solved by constructing infinitely many aspherical 4-manifolds that are homology 4-spheres
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.
Cobordism and signatures of manifolds with similar fundamental groups.
problem Understanding cobordism and signatures of manifolds with specific fundamental groups.
method Analyzing smooth closed connected aspherical manifolds with good fundamental groups.
result Cobordism and signatures of manifolds are preserved under certain conditions on their fundamental groups.
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, …
We consider closed symplectically aspherical manifolds, i.e. closed symplectic manifolds satisfying the condition . Rudyak and Oprea [RO] remarked that such manifolds have nice and controllable homotopy properties. Now it is clear that these properties are mostly determined by the fact that the s…
Classifies 4-manifolds with elementary amenable groups and their boundaries.
problem Characterizing compact aspherical 4-manifolds with elementary amenable fundamental groups.
method Classification based on fundamental group properties and Farrell-Jones Conjecture.
result Such manifolds are either polycyclic or solvable Baumslag-Solitar groups.
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…
We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold.
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.
We show that in all dimensions >7 there are closed aspherical manifolds whose fundamental groups have nontrivial center but do not possess any topological circle actions. This disproves a conjectured converse (proposed by Conner and Raymond) to a classical theorem of Borel.
Nonpositive towers property in 3-manifolds spines.
problem Properties of 3-manifold spines.
method Analysis of 2-dimensional spines in aspherical 3-manifolds and 3-ball.
result Nonpositive towers property in 2-dimensional spines of aspherical 3-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 …
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.
On a problem of Hopf for circle bundles over aspherical manifolds with hyperbolic fundamental groupsmath.GT
We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known…
We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …
We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a f…
2-knot manifolds have Seifert fibered base orbifolds.
problem Characterizing Seifert fibered 2-knot manifolds.
method Analyzing aspherical 2-orbifolds with orbifold fundamental groups of weight 1.
result Aspherical 2-orbifolds with orbifold fundamental groups of weight 1 are Seifert fibered bases of 2-knot manifolds.
New 4D shapes found that defy smoothness rules.
problem Smoothness rules in 4D shapes don't always hold.
method Applied reflection group trick to exotic 4-manifolds.
result Found counterexamples to smooth Borel conjecture.