This paper gives a new proof of a result of Geoghegan and Mihalik which states that whenever a contractible open n-manifold W which is not homeomorphic to Rn is a covering space of an n-manifold M and either n ≥ 4 or n = 3 and W is irreducible, then the group of covering translations injects into the homeotopy group of W .

We consider closed topological 4-manifolds M with universal cover S × S and Euler characteristic χ(M) = 1. All such manifolds with π = π1(M) ∼= Z/4 are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If π ∼= Z/2×Z/2, we show that there are three homotopy types (a...

We prove the homotopy invariance of L torsion for covering spaces, whenever the covering transformation group is either residually finite or amenable. In the case when the covering transformation group is residually finite and when the L cohomology of the covering space vanishes, the homotopy invariance was established by Lück [Lu]. We also give some applications of our results.

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The ...

We find the largest ǫ for which any simple closed path α in the universal cover R̃2 \ Z2 of R2 \ Z2, equipped with the natural lifted metric from the Euclidean two dimensional plane, satisfies L(α) ≥ ǫA(α). Where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Gomis, and provides an alternative proof for the same isoperimetric inequality in R2...

Stable, holomorphic vector bundles are constructed on an torus fibered, non-simply connected Calabi-Yau threefold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequen...

In order to classify digital spaces in terms of digital-homotopic theoretical tools, a recent paper by Han (2006b) (see also the works of Boxer and Karaca (2008) as well as Han (2007b)) established the notion of regular covering space from the viewpoint of digital covering theory and studied an automorphism group (or Deck’s discrete transformation group) of a digital covering. By using these to...

We settle a number of questions about the possible behaviour of the harmonic map heat flow at finite-time singularities. In particular, we show that a type of nonuniqueness of bubbles can occur at finite time, we show that the weak limit of the flow at the singular time can be discontinuous, we determine exactly the (polynomial) rate of blow-up in one particular example, and we show that ‘windi...

Let (M, g) be a complete non compact Kähler manifold with non-negative and bounded holomorphic bisectional curvature. Extending our techniques developed in [8], we prove that the universal cover M̃ of M is biholomorphic to Cn provided either that (M, g) has average quadratic curvature decay, or M supports an eternal solution to the Kähler-Ricci flow with non-negative and uniformly bounded holomo...

The double torus provides a relativistic model for a closed 2D cosmos with topology of genus 2 and constant negative curvature. Its unfolding into an octagon extends to an octagonal tesselation of its universal covering, the hyperbolic space H. The tesselation is analysed with tools from hyperbolic crystallography. Actions on H of groups/subgroups are identified for SU(1, 1), for a hyperbolic C...