Vanishing structure set of Haken 3-manifolds

In this article we prove that the surgery groups of the fundamental group of a certain class of Haken 3-manifolds can be computed in terms of a generalized homology theory even if the manifolds do not support any nonpositively curved Riemannian metric. A consequence of this result is that the integral Novikov conjecture is true for the fundamental group of this class of manifolds. Received October 2, 1998 / in revised form February 10, 2000 / Published online July 20, 2000     

Macroscopic dimension and essential manifolds

M. Gromov asked whether the macroscopic dimension of rationally essential n-dimensional manifolds equals n. We show that the answer depends only on the corresponding group homology class and give an affirmative answer for certain classes. In particular, the answer is positive for manifolds with amenable fundamental groups.     

Aspherical Kähler manifolds with solvable fundamental group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\mathbb{C}^{n}We survey recent developments which led to the proof of the Benson-Gordon conjecture on K?hler quotients of solvable Lie groups. In addition, we prove that the Albanese morphism of a K?hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K?hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of \mathbbCⁿ\mathbb{C}^{n} by a discrete group of complex isometries.     

Polar manifolds and actions

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. We show how to construct a manifold admitting a polar group action by prescribing its isotropy groups along a fundamental domain in the section. This generalizes a classical construction for cohomogeneity-one manifolds.We give many examples showing the richness of this class of group actions and relate the topology of the section to the topology of the manifold.     

Closed geodesics in homology classes on hyperbolic manifolds with cusps

On a class of hyperbolic manifolds with infinite volume, we give an asymptotic estimate for the number of closed geodesics in a given homology class. We show that, in certain cases, the existence of parabolic transformations in the fundamental group Γ of these manifolds has an effect on this estimate. This happens when the Hausdorff dimension of the limit set of Γ is less than 3/2. The geometrical meaning of this critical value remains to be understood.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

PL fibrator properties of partially aspherical manifolds

Approximate fibrations form a useful class of maps. By definition fibrators provide instant detection of maps in this class, and PL fibrators do the same in the PL category. This paper formalizes a natural concept of partial asphericity and establishes fibrator properties of certain partially aspherical closed manifolds. One consequence is that any connected sum of aspherical PL manifolds with residually finite fundamental groups is a codimension-(2n−2) PL fibrator.     

Fundamental groups,homology equivalences and one-sided h-cobordisms

We give a necessary and sufficient condition for the fundamental group homomorphism of a map between CW-complexes(manifolds) to induce partial homology equivalences. As applications, we obtain characterizations of fundamental groups of homology spheres and Moore manifolds. Moreover, a classification of one-sided h-cobordism of manifolds up to diffeomorphisms is obtained, based on Quillen's plus construction with Whitehead torsions.     

On L-spaces and left-orderable fundamental groups

Examples suggest that there is a correspondence between L-spaces and three-manifolds whose fundamental groups cannot be left-ordered. In this paper we establish the equivalence of these conditions for several large classes of manifolds. In particular, we prove that they are equivalent for any closed, connected, orientable, geometric three-manifold that is non-hyperbolic, a family which includes all closed, connected, orientable Seifert fibred spaces. We also show that they are equivalent for the twofold branched covers of non-split alternating links. To do this we prove that the fundamental group of the twofold branched cover of an alternating link is left-orderable if and only if it is a trivial link with two or more components. We also show that this places strong restrictions on the representations of the fundamental group of an alternating knot complement with values in $\text{ Homeo}_+(S^1)$ .     

Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent torus manifolds which are not homeomorphic. Moreover, we characterize those groups which appear as the fundamental groups of locally standard torus manifolds. In the second part we give a classification of quasitoric manifolds and certain six-dimensional torus manifolds up to equivariant diffeomorphism. In the third part we enumerate the number of conjugacy classes of tori in the diffeomorphism group of torus manifolds. For torus manifolds of dimension greater than six there are always infinitely many conjugacy classes. We give examples which show that this does not hold for six-dimensional torus manifolds.     

A theory of extended lie transformation groups

Summary The theory of Lie transformation groups is extended to a theory of extended Lie transformation groups by extending the group parameters to functions of coordinates in the base manifolds. The result is global both in the group manifolds (the O. Schreier's fundamental theorems being not taken into account) as well as in the base differentiable manifolds owing to the introduction of the author'sII-geodesic parallel coordinates. The Lie's fundamental theorems are extremely simplified.     

Growth of fundamental group for Finsler manifolds with integral Ricci curvature bound

In this paper, an upper bound on the growth of fundamental group for a class of Finsler manifolds with integral Ricci curvature bound is given. This generalizes the corresponding results with pointwise Ricci curvature in literature.     

Symplectically aspherical manifolds with nontrivial Flux groups

We investigate the fundamental group of symplectically aspherical manifolds with nontrivial Flux groups and conclude that such manifolds cannot be symplectically hyperbolic.     

On the fundamental group of some open manifolds

We study fundamental groups of noncompact Riemannian manifolds. We find conditions which ensure that the fundamental group is trivial, finite or finitely generated.     

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.     

On the full holonomy group of Lorentzian manifolds

The classification of restricted holonomy groups of \(n\) -dimensional Lorentzian manifolds was obtained about ten years ago. However, up to now, not much is known about the structure of the full holonomy group. In this paper we study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. Based on the classification of the restricted holonomy groups of such manifolds, we prove several structure results about the full holonomy. We establish a construction method for manifolds with disconnected holonomy starting from a Riemannian manifold and a properly discontinuous group of isometries. This leads to a variety of examples, most of them being quotients of pp-waves with disconnected holonomy, including a non-flat Lorentzian manifold with infinitely generated holonomy group. Furthermore, we classify the full holonomy groups of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with a parallel null spinor. Finally, we construct examples of globally hyperbolic manifolds with complete spacelike Cauchy hypersurfaces, disconnected full holonomy and a parallel spinor.     

Invariant group orderings and Galois conjugates

This paper investigates conditions under which a given automorphism of a residually torsion-free nilpotent group respects some ordering of the group. For free groups and surface groups, this has relevance to ordering the fundamental groups of three-dimensional manifolds which fiber over the circle.     

Topological methods for complex-analytic Brauer groups

Using methods from algebraic topology and group cohomology, I pursue Grothendieck's question on equality of geometric and cohomological Brauer groups in the context of complex-analytic spaces. The main result is that equality holds under suitable assumptions on the fundamental group and the Pontrjagin dual of the second homotopy group. I apply this to Lie groups, Hopf manifolds, and complex-analytic surfaces.     

On fundamental groups of Finsler manifolds In memory of the centenary birthday of Professor S.S.Chern

In this paper,we study the growth of fundamental groups of Finsler manifolds.Some relationships between the growth of fundamental groups and the volume growth of universal covers of Finsler manifolds are found.Some estimates of entropies and the number of generators of fundamental groups of Finsler manifolds are given.Moreover,the quasi-isometry and the geometric norm in Finsler geometry are considered.