On fundamental groups of quotient spaces

In classical covering space theory, a covering map induces an injection of fundamental groups. This paper reveals a dual property for certain quotient maps having connected fibers, with applications to orbit spaces of vector fields and leaf spaces in general.     

On the fundamental groups of one-dimensional spaces

We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective.     

On fundamental domains of arithmetic Fuchsian groups

Let be a totally real algebraic number field and an order in a quaternion algebra over . Assume that the group of units in with reduced norm equal to is embedded into as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of as well as a complete system of generators of .

     

On fundamental groups of compact Hausdorff spaces

We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space' is consistent with the Zermelo-Fraenkel-Choice set theory.

     

On fundamental groups of symplectically aspherical manifolds

A smooth manifold M is called symplectically aspherical if it admits a symplectic form with |₂(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups ₁(M) of symplectically aspherical manifolds M. The first one consists of fundamental groups of such M with ₂(M)=0, while the second with ₂(M)0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.Mathematics Subject Classification (1991): 57R15, 53D05, 14F35     

On the geometry of spheres in L-spaces

It is shown that any two points on the surface of the unit ball ofL ¹(μ), where the measureμ is non-atomic, may be joined in the surface by a curve whose length is equal to the straight-line distance between its endpoints. This property is contrasted with the metric properties of the unit sphere in other L-spaces. This work was supported in part by NSF grant GP-19126.     

On fundamental groups of symplectically aspherical manifolds II: Abelian groups

We describe all abelian groups which can appear as the fundamental groups of closed symplectically aspherical manifolds. The proofs use the theory of symplectic Lefschetz fibrations.      

On Calabi-Yau threefolds with large nonabelian fundamental groups

In this short note we construct Calabi-Yau threefolds with nonabelian fundamental groups of order as quotients of the small resolutions of certain complete intersections of quadrics in that were first considered by M. Gross and S. Popescu.

     

L-spaces and the P-ideal dichotomy

We extend a theorem of Todor?evi?: Under the assumption ( $ \mathcal{K} $ ) (see Definition 1.11), $$ \boxtimes \left\{ \begin{gathered} any regular space Z with countable tightness such that \hfill \\ Z^n is Lindel\ddot of for all n \in \omega has no L - subspace. \hfill \\ \end{gathered} \right. $$ We assume $ \mathfrak{p} $ > ω ₁ and a weak form of Abraham and Todor?evi?’s P-ideal dichotomy instead and get the same conclusion. Then we show that $ \mathfrak{p} $ > ω ₁ and the dichotomy principle for P-ideals that have at most ?₁ generators together with ? do not imply that every Aronszajn tree is special, and hence do not imply (ie1-4). So we really extended the mentioned theorem.     

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C~2 or in CP~2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.     

On the fundamental groups of manifolds with integral curvature bounds

In this paper, we give an upper bound on the growth of π₁(M) for a class of manifolds with integral Ricci curvature bounds. This generalizes the main theorem of [8] to the case where the negative part of Ricci curvature is small in an averaged L¹- sense.Received: 19 July 2004     

On the existence of non-geometric sections of arithmetic fundamental groups

We show the existence of group-theoretic sections of the “étale-by-geometrically abelian” quotient of the arithmetic fundamental group of hyperbolic curves over $p$ -adic local fields relative to a proper and flat model which are non-geometric, i.e., which do not arise from rational points.     

On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds

We apply the concept of asymptotic cone to distinguish quasi-isometry classes of fundamental groups of 3-manifolds. We prove that the existence of a Seifert component in a Haken manifold is a quasi-isometry invariant of its fundamental group.This research was partially supported by the grant SFB 256 Nichtlineare partielle Differentialgleichungen and the NSF grant DMS-9306140 (Kapovich).     

Finite graphs of groups with isomorphic fundamental groups

Translated from Algebra i Logika, Vol. 30, No. 5, pp. 595–623, September–October, 1991.     

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined overQ there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. This research was partially supported by NSF grant DMS-96-26633 at University of Utah. This research was partially supported by NSF grant DMS-95-04193 at University of Maryland.