This article contains a new argument which proves vanishing of the first cohomology for negative vector bundles over a complex projective variety if the rank of the bundle is smaller than the dimension of the base. Similar argument is applied to the construction of holomorphic functions on the universal covering of the complex projective variety .

This paper describes certain properties of the fundamental group of the shrinking wedge of circles. Covering space theory is an inadequate tool since the shrinking wedge of circles is not semi-locally simply connected. We conclude with the fact that the shrinking wedge of circles is not homotopy equivalent to the wedge product of a countable number of circles.

in this paper, we verify the solvability of free product of finite cyclic groups with topological methods. we use cayley graphs and everitt methods to construct suitable 2-complexes corresponding to the presentations of groups and their commutator subgroups. in particular, using these methods, we prove that the commutator subgroup of $mathbb{z}_{m}*mathbb{z}_{n}$ is free of rank $(m-1)(n-1)$ fo...

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their unive...

This paper gives an upper bound for the first eigenvalue of the universal cover of a complete, stable minimal surface in hyperbolic space, and a sharper one for least area disks.

We give a general structure theory for reconstructing group actions on sets without any assumptions on the group, the action, or the set on which the group acts. Using certain ‘local data’ D from the action we build a group G(D) of the data and a space X (D) with an action of G(D) on X (D) that arise naturally from the data D. We use these to obtain an approximation to the original group G, the...

This is a survey on known results and open problems about closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. Many examples come from certain kinds of non-positive curvature conditions. The property aspherical, which is a purely homotopy theoretical condition, has many striking implications about the geometry and analysis of the manifold or ...

See [1] for proofs and terminology.

̂ M of the connection ∇M to the universal cover ̂ M of M is a locally flat connection. The affine structure of ̂ M is defined by a local diffeomorphism DM : ̂ M → R called the developing map. The developing map gives rise to a representation hM : π1 M → Aff R called the holonomy. The linear part L hM of hM is the linear holonomy. The affine manifold M,∇M is complete if and only if DM is a diffeomor...

In the paper we prove a factorization theorem for representations of fundamental groups of compact Kähler manifolds (Kähler groups) into solvable matrix groups. We apply this result to prove that the universal covering of a compact Kähler manifold with a residually solvable fundamental group is holomorphically convex.