On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO($n$) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.     

Invariant sets and measures of nonexpansive group automorphisms

We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link inS ³. Our main tool is a careful use of the Århus integral and the (now proven) “Wheels” and “Wheeling” conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give explicit formulas for the values of the Kontsevich integral on the Hopf link and on Hopf chains, and for the LMO invariant of lens spaces and Seifert fibered spaces. We find that the LMO invariant does not separate lens spaces, is far from separating general Seifert fibered spaces, but does separate Seifert fibered spaces which are integral homology spheres.     

On free $${\mathbb{Z}_{p}}$$ actions on products of spheres

In this paper we consider free actions of large prime order cyclic groups on the product of any number of spheres of the same odd dimension and on products of two spheres of differing odd dimensions. We require only that the action be free on the product as a whole and not each sphere separately. In particular we determine equivariant homotopy type, and for both linear actions and for even numbers of spheres the simple homotopy type and simple structure sets. The results are compared to the analysis and classification done for lens spaces. Similar to lens spaces, the first k-invariant generally determines the homotopy type of many of the quotient spaces, however, the Reidemeister torsion frequently vanishes and many of the homotopy equivalent spaces are also simple homotopy equivalent. Unlike lens spaces, which are determined by their ρ-invariant and Reidemeister torsion, the ρ-invariant here vanishes for even numbers of spheres and linear actions and the Pontrjagin classes become p-localized homeomorphism invariants for a given dimension. The cohomology classes, Pontrjagin classes, and sets of normal invariants are computed in the process.     

Analytic Rank of Graph Algebras and Quantum Spaces

We show that the analytic rank, as defined by Murphy, of a unitalgraph C*-algebra is either 1 or 0, depending on whether or notthe underlying graph possesses an initial loop. As a corollary,we show that the analytic rank of certain quantum spaces (includingsome quantum spheres, projective spaces and lens spaces) is1. 2000 Mathematics Subject Classification 46L05, 46L65, 46L87.     

Distributions of distances and volumes of balls in homogeneous lens spaces

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the kth moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.     

Configuration spaces are not homotopy invariant

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L_7,1 and L_7,2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products.     

Primary decomposition of the J-groups of complex projective and lens spaces

We determine the decomposition of J-groups of complex projective and lens spaces as a direct-sum of cyclic groups.     

OHTSUKI’S INVARIANT CANNOT DETERMINE THE FULL SO（3） QUANTUM INVARIANTS

Two lens spaces are given to show, that Ohtsuki’sτ for rational homology spheres does not determine Kirby-Melvin’s {τ r′, r odd ≥ 3}.     

On knot Floer homology and lens space surgeries

In an earlier paper, we used the absolute grading on Heegaard Floer homology HF⁺ to give restrictions on knots in S³ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising from knot Floer homology. One consequence is that the non-zero coefficients of the Alexander polynomial of such a knot are ±1. This information can in turn be used to prove that certain lens spaces are not obtained as integral surgeries on knots. In fact, combining our results with constructions of Berge, we classify lens spaces L(p,q) which arise as integral surgeries on knots in S³ with |p|?1500. Other applications include bounds on the four-ball genera of knots admitting lens space surgeries (which are sharp for Berge's knots), and a constraint on three-manifolds obtained as integer surgeries on alternating knots, which is closely to related to a theorem of Delman and Roberts.     

Determination of the J-groups of Complex Projective and Lens Spaces

We determine completely the J-groups of complex projective and lens spaces by means of a set of generators and a complete set of relations.     

Symmetry of Links and Classification of Lens Spaces

We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of 'symmetric' links in S ³.     

Cut loci in lens manifolds

Cut loci in geometric three-manifolds equipped with their natural metrics are an interesting source of spines with small number of vertices. An application of this principle to lens manifolds reveals an interplay between their geometry and topology, combinatorial types of convex hulls of group orbits, and estimates of rotation distance between certain triangulations. To cite this article: S. Anisov, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Minimal triangulations for an infinite family of lens spaces

The notion of a layered triangulation of a lens space was definedby Jaco and Rubinstein, and unless the lens space is L(3,1),a layered triangulation with the minimal number of tetrahedrawas shown to be unique and termed its minimal layered triangulation.This paper proves that for each n 2, the minimal layered triangulationof the lens space L(2n, 1) is its unique minimal triangulation.More generally, the minimal triangulations (and hence the complexity)are determined for an infinite family of lens spaces containingthe lens space of the form L(2n, 1). Received May 16, 2008.     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

Contact cuts

We describe a contact analog of the symplectic cut construction [L]. As an application we show that the group of contactomorphisms for certain overtwisted contact structures on lens spaces contains countably many non-conjugate two tori. Supported by the NSF grant DMS-980305.     

A Note on the Witten‘s Invariants of Three Manifolds

The purpose of this note is to study the three manifold invariants Qp(ML).For lens spaces L(7,1)and (7,2),we compute Q5(L(7,1) and Q5(L(7.2)) concretely,which enables us to prove that the in-variants can distinguish homotopy equivalence manifolds.     

A class of 4-pseudomanifolds similar to the lens spaces

A family of 4-dimensional pseudomanifolds is introduced using a standard graph-theoretical representation of lens spaces Some homeomorphisms between these lens-like spaces are established, the computation of their fundamental groups and of bounds for their genera are carried out     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.