arXiv research

Classifies involutions on spherical 3-manifolds.

The paper finds hyperbolic small knots in many 3-manifolds.

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

Study $Θ$-invariants for spherical 3-manifolds via $\mathbb{Z}π$-homology equivalences.

Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version…

Classifies 3-manifolds with uniformly positive scalar curvature.

The paper studies pseudo-isotopies of spherical 3-manifolds and computes ranks of certain groups.

Describes the space of spherical triangles on a smooth 3-manifold.

3-manifolds are CR uniformized on spheres, proving a conjecture.

Geometrically, spherical 3-manifolds emerge from flat SU(2)-bundles over hyperbolic surfaces.

The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by…

Uniformizes CR structure on a specific 3-manifold.

We consider the discrete representations of 3-manifold groups into $PU(2,1)$ that appear in the Falbel-Koseleff-Rouillier census, such that the peripheral subgroups have cyclic unipotent holonomy. We show that two of these representations have conjugate images, even though they represent different 3-manifold groups. Th…

From the homotopy groups of two cubic spherical 3-manifolds we construct the isomorphic groups of deck transformations acting on the 3-sphere. These groups become the cyclic group of order eight and the quaternion group respectively. By reduction of representations from the orthogonal group to the identity representati…

We determine the contributions of isolated singularities of spin V 4-manifolds to the index of the Dirac operator over them. From these data we derive certain constraints on the intersection forms of spin 4-manifolds bounded by spherical 3-manifolds, and also on the embeddings of the real projective planes into 4-manif…

The paper studies invariant metrics with positive scalar curvature on 3-manifolds.

The study of universal links in 3-manifolds and their properties.

A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.

We prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two. This confirms a conjecture of R. Schoen for an infinite class of 3-manifolds.

In this paper we develop a method to compute the Burns-Epstein invariant of a spherical CR homology sphere, up to an integer, from its holonomy representation. As application, we give a formula for the Burns-Epstein invariant, modulo an integer, of a spherical CR structure on a Seifert fibered homology sphere in terms …

Defines state sum models with defects in 3-manifolds.

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

We prove global rigidity for compact hyperbolic and spherical cone-3-manifolds with cone-angles $\leq π$ (which are not Seifert fibered in the spherical case), furthermore for a class of hyperbolic cone-3-manifolds of finite volume with cone-angles $\leq π$, possibly with boundary consisting of totally geodesic hyperbo…

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes…

We show that open 3-manifolds that have a locally finite decomposition along 2-spheres are characterized by the existence of a Riemannian metric with respect to which the second homotopy group of the manifold is generated by small elements.

The so-called Mom-structures on hyperbolic cusped 3-manifolds without boundary were introduced by Gabai, Meyerhoff, and Milley, and used by them to identify the smallest closed hyperbolic manifold. In this work we extend the notion of a Mom-structure to include the case of 3-manifolds with non-empty boundary that does …

3-manifolds with hyperbolic handlebody complements are studied.

Short proof of Strong Haken Theorem for 3-manifolds.

This paper introduces quantum invariants for 3-alterfolds and proves their consistency with topological moves.

Let C be a spherical fusion category. We prove that the Turaev-Viro-Barrett-Westbury state sum invariant of 3-manifolds derived from C is equal to the Reshetikhin-Turaev surgery invariant of 3-manifolds derived from Z(C), where Z(C) is the Drinfeld-Joyal-Street center of C.

The paper studies relationships between twisted torsion and connected sums of 3-manifolds.

A spherical topological manifold of dimension n-1 forms a prototile on its cover, the (n-1)-sphere. The tiling is generated by the fixpoint-free action of the group of deck transformations. By a general theorem, this group is isomorphic to the first homotopy group. Multiplicity and selection rules appear in the form of…

Given a 3-manifold M with no spherical boundary components, and a primitive class φin H^1(M;Z), we show that the following are equivalent: (1) φis a fibered class, (2) the rank gradient of (M,φ) is zero, (3) the Heegaard gradient of (M,φ) is zero.

Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon-Vey invariant, the so-c…

The study proves manifolds with positive scalar curvature can be decomposed into spherical and toroidal pieces.

We compute a recently introduced geometric invariant of stricly pseudoconvex CR 3-manifolds for certain circle invariant spherical CR structures on Seifert manifolds. We give applications to the problem of filling the CR manifold by a complex hyperbolic manifold, and more generally by a Kaehler-Einstein or an Einstein …

Enhanced bounds on rho-invariants for 3-manifolds.

New findings link 3D shapes to group properties.

The famous Haken-Kneser-Milnor theorem states that every 3-manifold can be expressed in a unique way as a connected sum of prime 3-manifolds. The analogous statement for 3-orbifolds has been part of the folklore for several years, and it was commonly believed that slight variations on the argument used for manifolds wo…

Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…

Overview of 3D TQFTs and 3-manifold invariants.

In this paper, we study and almost completely classify contact structures on closed 3--manifolds which are totally geodesic for some Riemannian metric. Due to previously known results, this amounts to classifying contact structures on Seifert manifolds which are transverse to the fibers. Actually, we obtain the complet…

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the eighties by S. Lins for closed 3-manifolds, which …

We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifold…

We prove a homological version of a conjecture about the homotopy type of diffeomorphism spaces of reducible 3-manifolds.

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

Computations based on explicit 4-periodic resolutions are given for the cohomology of the finite groups G known to act freely on S^3, as well as the cohomology rings of the associated 3-manifolds (spherical space forms) M = S^3/G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies…

The paper constructs examples of coupled Dirac-Yang-Mills pairs on Riemannian manifolds.