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On the existence of universal covering spaces for metric spaces and subsets of the Euclidean plane

G. R. Conner, J. W. Lamoreaux (2005)

Fundamenta Mathematicae

We prove several results concerning the existence of universal covering spaces for separable metric spaces. To begin, we define several homotopy-theoretic conditions which we then prove are equivalent to the existence of a universal covering space. We use these equivalences to prove that every connected, locally path connected separable metric space whose fundamental group is a free group admits a universal covering space. As an application of these results, we prove the main result of this article,...

Polynomial invariants of 2-component links.

Kunio Murasugi (1985)

Revista Matemática Iberoamericana

Let L = X U Y be an oriented 2-component link in S3. In this paper we will define two different types of polynomials which are ambient isotopic invariants of L. One is associated with a cyclic cover branched along one of their components, an the other is associated with a metabelian cover of L. This invariants are defined for any link unless the linking number lk(X,Y), is ±1.

Rank gradient, cost of groups and the rank versus Heegaard genus problem

Miklós Abért, Nikolay Nikolov (2012)

Journal of the European Mathematical Society

We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the ‘rank vs. Heegaard genus’ conjecture on hyperbolic 3-manifolds is incompatible with the ‘fixed price problem’ in topological dynamics.

Sur le groupe fondamental de l'espace des noeuds

André Gramain (1977)

Annales de l'institut Fourier

On déduit par générateurs et relations, pour chaque composante de cet espace, un sous-groupe du groupe fondamental π 1 ( P l ( S 1 , S 3 ) ) . Les générateurs ont été trouvés à partir de considérations géométriques ; cependant les démonstrations sont de caractère algébrique.

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