In 1914 Lebesgue defined a ‘universal covering’ to be a convex subset of the plane that contains an isometric copy of any subset of diameter 1. His challenge of finding a universal covering with the least possible area has been addressed by various mathematicians: Pál, Sprague and Hansen have each created a smaller universal covering by removing regions from those known before. However, Hansen’...

We study the structures of partition functions of the large N generalized twodimensional Yang-Mills theories (gYM2) by recasting the higher Casimirs. We clarify the appropriate interpretations of them and try to extend the Cordes-MooreRamgoolam’s topological string model describing the ordinary YM2 [4] to those describing gYM2. The concept of ”deformed gravitational descendants” will be introdu...

In this note it will be shown that when A is a retract (cf. [3])1 of X, then the same relationship holds between the universal covering spaces of A and X (Theorem 1.1) and also between the fundamental groups of A and X (Theorem 2.2). We denote the universal covering space of X by X*, where pEX, and the fundamental group by iriX) [l]. All spaces are assumed to be arcwise connected, Hausdorff spa...

We give sufficient conditions for the existence of equivariant harmonic maps from the universal cover of a Riemann surface B to the Teichmüller space of a genus g ≥ 2 surface Σ. The condition is in terms of the representation of the fundamental group of B to the mapping class group of Σ. The metric on Teichmüller space is chosen to be the Kähler hyperbolic metric. Examples of such representatio...

Our main observation concerns closed geodesics on surfaces M with a smooth Finsler metric, i.e. a function F : TM → [0,∞) which is a norm on each tangent space TpM , p ∈ M , which is smooth outside of the zero section in TM , and which is strictly convex in the sense that Hess(F ) is positive definite on TpM \ {0}. One calls a Finsler metric F symmetric if F (p,−v) = F (p, v) for all v ∈ TpM . ...

A generalized solenoid is an inverse limit space with bonding maps that are (regular) covering maps of closed compact manifolds. We study the embedding properties of solenoids in linear space and in foliations. A compact and connected topological space is called a continuum. A space X is homogeneous if for every x, y ∈ X there exists a homeomorphism h : X → X such that h(x) = y. The space is bi...

All spaces are assumed to be locally path connected and semi-locally 1-connected. Let X be a connected topological group with identity e, and let p : X̃ → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ẽ in X̃ with pẽ = e there is a unique structure of topological group on X̃ such that ẽ is th...

A nilmanifold is a homogenous space of a nilpotent Lie group. If M is a compact symplectic nilmanifold, then any 1-periodic Hamiltonian system on M has at least dim(M) + 1 contractible periodic orbits with period 1. This provides an aarmative answer to the Arnold conjecture for such manifolds. The proof uses the techniques of rational homotopy theory, and in particular an extension of the ratio...

Given a 3-manifold M , there are generically infinitely many manifold which covers M . However, if we are restricted to the category of knot complements, the situation is quite different. It can be shown (see Lemma 1 and bellow) that if the complement E(K) of a knot K is n-fold covered by some knot complement, then the covering is cyclic, and K admits a cyclic surgery, i.e. a Dehn surgery such ...