arXiv research

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 …

Constructs 4-manifolds with positive Euler characteristic proving a conjecture.

Nilpotent Sasakian manifolds are Heisenberg nilmanifolds.

Aspherical configuration Lie groupoids complement is proven for a class of orbifolds.

The paper extends a vanishing theorem for hypersurfaces in aspherical manifolds.

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 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 $π_1(M)$ coarsely co…

Study finite group actions on aspherical manifolds, proving rigidity and symmetry bounds.

3D spaces without boundaries are found that aren't manifolds.

Study aspherical manifolds with boundaries, proving homological criteria.

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.

Study symplectically aspherical Kähler manifolds with unique properties.

Study shows some complex shapes don't fit a certain property.

The study of $PD_3$-pairs extends results for aspherical 3-manifolds.

Aspherical manifolds with bounded curvature have non-trivial abelian subgroups in their fundamental groups.

This paper studies limits of aspherical manifolds with specific curvature conditions.

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…

Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.

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…

Let $G/H$ be a contractible homogeneous Sasaki manifold. A compact locally homogeneous aspherical Sasaki manifold $Γ\big\backslash G/H$ is by definition a quotient of $G/H$ by a discrete uniform subgroup $Γ\leq G$. We show that a compact locally homogeneous aspherical Sasaki manifold is always quasi-regular, that is, $…

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.

The study constructs geometrically decomposable aspherical 4-manifolds with non-zero signature and explores their properties.

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.

Proves non-existence of certain metrics on specific manifolds.

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.

We prove a lower bound on the number of maximally broken trajectories of the negative gradient flow of a Morse-Smale function on a closed aspherical manifold in terms of integral (torsion) homology.

We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomor…

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 $(M,ω)$ satisfying the condition $[ω]|_{π_2M}=0$. 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.

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.

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…

Study on simplicial volume and Euler characteristic of aspherical manifolds.

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.

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.

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 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 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…

New 4D shapes found that defy smoothness rules.

The paper confirms a conjecture about knots in aspherical 3-manifolds.

A 5-manifold fibers over a circle with nonpositive curvature.