arXiv research

Study simplicial volume in fiber bundles with connected groups.

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

New example shows noncontractible fiberings for a 3-manifold.

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…

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…

In 2007 Agol showed that if N is an aspherical compact 3-manifold with empty or toroidal boundary such that its fundamental group is virtually RFRS, then $N$ is virtually fibered. We give a largely self-contained proof of Agol's theorem using complexities of sutured manifolds.

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

New trick builds hyperbolic manifolds from compact ones, proving some don't virtually fiber.

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive sc…

The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this ca…

The paper proves non-existence of positive scalar curvature on certain fiber bundles.

Maps between circle bundles are studied, proving fiber-preserving and finiteness results for mapping degrees.

We prove a basic inequality for the d-invariants of a splice of knots in homology spheres. As a result, we are able to prove a new relation on the rank of reduced Floer homology under maps between Seifert fibered homology spheres, improving results of the first and second authors. As a corollary, a degree one map betwe…

Let M be a closed orientable Seifert fibered 3-manifold with a hyperbolic base 2-orbifold, or equivalently, admitting a geometry modeled on H^2 \times R or the universal cover of SL(2,R). Our main result is that the connected component of the identity map in the diffeomorphism group Diff(M) is either contractible or ho…

Paper confirms Whitehead's conjecture for aspherical 2-complexes.

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

An interesting question in symplectic topology, which was posed by C. H. Taubes, concerns the topology of closed (i.e. compact and without boundary) connected oriented three dimensional manifolds whose product with a circle admits a symplectic structure. The only known examples of such manifolds are those which fiber o…

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bund…

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.

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…

The paper proves the Singer conjecture for aspherical complex surfaces and refines Gromov's inequality.

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

Study new bounds on TC of spaces with subgroup inclusions.

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…

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

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

Nilpotent Sasakian manifolds 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.

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

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

A {\em word labeled oriented graph} (WLOG) is an oriented graph $\cal G$ on vertices $X=\{ x_1,\ldots ,x_k\}$, where each oriented edge is labeled by a word in $X^{\pm1}$. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree are of …

Defines Floer homology with DG coefficients for symplectic 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 shows some complex shapes don't fit a certain property.

Study aspherical manifolds with boundaries, proving homological criteria.

Study symplectically aspherical Kähler manifolds with unique properties.

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.

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.

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…

The paper describes the structure of injective LOT-complexes and proves they are aspherical.

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

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".

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, $…

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

Using small cancellation for rotating families of groups, we construct new examples of aspherical polyhedra.

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…