arXiv research

Maps can transform non-homeomorphic manifolds into the same space.

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.

For each odd integer r greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise non-homeomorphic closed homogeneous spaces with fundamental group isomorphic to Z/r. As an application we construct the first examples of manifold…

No characterizing slopes for multi-component links.

Study groups of order 64 and non-homeomorphic double Kodaira fibrations with same invariants.

We show that in each dimension $n\ge 10$ there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian $n$-manifolds with $0\le \sec\le 1$, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type whic…

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic s…

New examples of non-homeomorphic foliation leaves found.

In this paper, we construct the first families of distinct Lagrangian ribbon disks in the standard symplectic 4-ball which have the same boundary Legendrian knots, and are not smoothly isotopic or have non-homeomorphic exteriors.

We prove the existence of lattice isomorphic line arrangements having $π_1$-equivalent or homotopy-equivalent complements and non homeomorphic embeddings in the complex projective plane. We also provide two explicit examples, one is formed by real-complexified arrangements while the second is not.

We propose a modification of the three-manifold invariant based on the use of Euclidean metric values ascribed to the elements of manifold triangulation. We thus obtain a nontrivial invariant that can, in particular, distinguish non-homeomorphic lens spaces.

Infinite knots have non-integer trace values.

Generalizes uniformization to algebraic correspondences.

By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, …

We construct examples of four dimensional manifolds with Spin$^c$-structures, whose moduli spaces of solutions to the Seiberg-Witten equations, represent a non-trivial bordism class of positive dimension, i.e. the Spin$^c$-structures are not induced by almost complex structures. As an application, we show the existence…

We give necessary and sufficient conditions for the existence of pin+, pin- and spin structures on Riemannian manifolds with holonomy group $Z_2^k$. For any n>3 (resp. n>5) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M_1, M_2, non homeomorphic to each other, that are Laplace isos…

As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well-defined since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up rays satisfying one of five hyperbolic-like propertie…

When a solenoid is embedded in three space, its complement is an open three manifold. We discuss the geometry and fundamental groups of such manifolds, and show that the complements of different solenoids (arising from different inverse limits) have different fundamental groups. Embeddings of the same solenoid can give…

Sapir, Birget and Rips showed how to construct groups from Turing machines. To achieve such a construction they introduced the notion of S-machine. Then considering a simplified S-machine Sapir and Olshanskii showed how to construct a group such that each of its asymptotic cone is non-simply connected. Still using the …

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of …

In this paper, we prove that (1) For any integers $n\geq 1$ and $g\geq 2$, there is a closed 3-manifold $M_{g}^{n}$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g, n)$ is $(2, 1)$. Furthermore, $M_{g}^{n}$ can be chosen to be hyperbolic except that the pair of $(g, n)$ is $(3, 1…

New invariant detects non-homeomorphic arrangements with similar coefficients.

Study shows bounds on knot groups and nonembeddability of certain open manifolds.

Constructs non-isometric iso-length-spectral surfaces.

This paper concerns the class of contractible open 3-manifolds which are ``locally finite strong end sums'' of eventually end-irreducible Whitehead manifolds. It is shown that whenever a 3-manifold in this class is a covering space of another 3-manifold the group of covering translations must be a free group. It follow…

New insights into Anosov representations of hyperbolic groups.

We study torsion properties of the twisted Alexander modules of the affine complement $M$ of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polyn…

In this paper we find smooth embeddings of solenoids in smooth foliations. We show that if a smooth foliation F of a manifold M contains a compact leaf L with H^1(L;R)= 0 and if the foliation is a product foliation in some saturated open neighbourhood U of L, then there exists a foliation F' on M which is C^1-close to …

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not Stein. (3) The circle bundle over a genus n>1 surface with euler number e=-1 ad…

We show that there exist infinitely many pairs of non-homeomorphic closed oriented SOL torus bundles with the same quantum (TQFT) invariants. This follows from the arithmetic behind the conjugacy problem in $SL(2,\Z)$ and its congruence quotients, the classification of SOL (polycyclic) 3-manifold groups and an elementa…

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic manifolds.…

An irreducible open 3-manifold $W$ is {\bf R}$^2$-irreducible if every proper plane in $W$ splits off a halfspace. In this paper it is shown that if such a $W$ is the universal cover of a connected, {\bf P}$^2$-irreducible open 3-manifold $M$ with finitely generated fundamental group, then either $W$ is homeomorphic to…

New findings show infinitely many non-homeomorphic manifolds with same proper homotopy type.

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed t…

Let $M$ be an orientable compact flat Riemannian manifold endowed with a spin structure. In this paper we determine the spectrum of Dirac operators acting on smooth sections of twisted spinor bundles of $M$, and we derive a formula for the corresponding eta series. In the case of manifolds with holonomy group $\Z_2^k$,…

Let T(γ) be the total space of the canonical line bundle γover CP^1 and r an integer which is greater than one and coprime to six. We prove that L_r^3\times T(γ) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L_r^3 is the standard 3-dimensional lens…

A curve $γ: [0,1] \rightarrow S^n$ of class $C^k$ ($k \geqslant n$) is locally convex if the vectors $γ(t), γ'(t), γ"(t), \cdots, γ^{(n)}(t)$ are a positive orthonormal basis to $R^{n+1}$ for all $t \in [0,1]$. Given an integer $n \geq 2$ and $Q \in SO_{n+1}$, let $LS^n(Q)$ be the set of all locally convex curves $γ: […

Given any connected, open 3-manifold $U$ having finitely many ends, a non-compact 3-manifold $M$ is constructed having the following properties: the interior of $M$ is homeomorphic to $U$; the boundary of $M$ is the disjoint union of finitely many planes; $M$ is not almost compact; $M$ is eventually end-irreducible; th…

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mat…

In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusp…