Special manifolds and shape fibrator properties

Our main interest in this paper is further investigation of the concept of (PL) fibrators (introduced by Daverman [R.J. Daverman, PL maps with manifold fibers, J. London Math. Soc. (2) 45 (1992) 180-192]), in a slightly different PL setting. Namely, we are interested in manifolds that can detect approximate fibrations in the new setting. The main results state that every orientable, special (a new class of manifolds that we introduce) PL n-manifold with non-trivial first homology group is a fibrator in the new category, if it is a codimension-2 fibrator (Theorem 8.2) or has a non-cyclic fundamental group (Theorem 8.4). We show that all closed, orientable surface S with χ(S)<0 are fibrators in the new category.     

Path and quasi-homotopy for Sobolev maps between manifolds

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic – and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere.     

Connected Sums of 4-Manifolds as Codimension-k Fibrators

The main result provides mild conditions under which a closed,orientable, PL 4-manifold N = N₁#N₂ with ₁(N_e) residually finite(e=1,2) is a codimension-5 PL fibrator. The paper also presentsa rich variety of conditions on a closed 4-manifold N⁴ underwhich every PL map between manifolds, where the domain is orientableand all point inverses are copies of N⁴, must be an approximatefibration.     

Fibrator properties of manifolds

Approximate fibrations form a useful class of maps, in part, because they provide computable relationships involving the domain, image and homotopy fiber. Fibrator properties, which pertain to and depend upon the homotopy fiber, allow instant recognition of approximate fibrations. This is a survey of results about those fibrator properties.     

Model aspherical manifolds with no periodic maps

A. Borel proved that, if the fundamental group of an aspherical manifold is centerless and the outer automorphism group of is torsion-free, then admits no periodic maps, or equivalently, there are no non-trivial finite groups of homeomorphisms acting effectively on . In the literature, taking off from this result, several examples of (rather complex) aspherical manifolds exhibiting this total lack of periodic maps have been presented.

In this paper, we investigate to what extent the converse of Borel's result holds for aspherical manifolds arising from Seifert fiber space constructions. In particular, for e.g. flat Riemannian manifolds, infra-nilmanifolds and infra-solvmanifolds of type (R), it turns out that having a centerless fundamental group with torsion-free outer automorphism group is also necessary to conclude that all finite groups of affine diffeomorphisms acting effectively on the manifold are trivial. Finally, we discuss the problem of finding (less complex) examples of such aspherical manifolds with no periodic maps.

     

On fundamental groups of symplectically aspherical manifolds

A smooth manifold M is called symplectically aspherical if it admits a symplectic form with |₂(M) = 0. It is easy to see that, unlike in the case of closed symplectic manifolds, not every finitely presented group can be realized as the fundamental group of a closed symplectically aspherical manifold. The goal of the paper is to study the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups ₁(M) of symplectically aspherical manifolds M. The first one consists of fundamental groups of such M with ₂(M)=0, while the second with ₂(M)0. Relations between these classes are discussed. We show that several important (classes of) groups can be realized in both classes, while some groups can be realized in the first class but not in the second one. Also, we notice that there are some interesting dimensional phenomena in the realization problem. The above results are framed by a general study of symplectically aspherical manifolds. For example, we find some conditions which imply that the Gompf sum of symplectically aspherical manifolds is symplectically aspherical, or that a total space of a bundle is symplectically aspherical.Mathematics Subject Classification (1991): 57R15, 53D05, 14F35     

Generalized Hopfian Property,a Minimal Haken Manifold,and Epimorphisms Between 3-Manifold Groups

We address the question that if π₁-surjective maps between closed aspherical 3-manifolds have the same rank on π₁ they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given. Received September 16, 1999, Revised August 31, 2000, Accepted March 29, 2001.     

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.     

Manifolds with Poly-Surface Fundamental Groups

We study closed topological 2n-dimensional manifolds M with poly-surface fundamental groups. We prove that if M is simple homotopy equivalent to the total space E of a Y-bundle over a closed aspherical surface, where Y is a closed aspherical n-manifold, then M is s-cobordant to E. This extends a well-known 4-dimensional result of Hillman in [14] to higher dimensions. Our proof is different from that of the quoted paper: we use Mayer-Vietoris techniques and the properties of the -theory assembly maps for such bundles.     

Holomorphic Maps from Sasakian Manifolds into K¨ahler Manifolds

The authors consider ±(Φ, J)-holomorphic maps from Sasakian manifolds into Ka¨hler manifolds, which can be seen as counterparts of holomorphic maps in Ka¨hler geometry. It is proved that those maps must be harmonic and basic. Then a Schwarz lemma for those maps is obtained. On the other hand, an invariant in its basic homotopic class is obtained. Moreover, the invariant is just held in the class of basic maps.     

Symplectically aspherical manifolds

This is a survey article on symplectically aspherical manifolds.     

Aspherical manifolds with relatively hyperbolic fundamental groups

We show that the aspherical manifolds produced via the relative strict hyperboli- zation of polyhedra enjoy many group-theoretic and topological properties of open finite volume negatively pinched manifolds, including relative hyperbolicity, nonvanishing of simplicial volume, co-Hopf property, finiteness of outer automorphism group, absence of splitting over elementary subgroups, and acylindricity. In fact, some of these properties hold for any compact aspherical manifold with incompressible aspherical boundary components, provided the fundamental group is hyperbolic relative to fundamental groups of boundary components. We also show that no manifold obtained via the relative strict hyperbolization can be embedded into a compact Kähler manifold of the same dimension, except when the dimension is two.     

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.     

A volume decreasing theorem for $$\mathbf{}$$-harmonic maps and applications

We establish a volume decreasing result for V-harmonic maps between Riemannian manifolds. We apply this result to obtain corresponding results for Weyl harmonic maps from conformal Weyl manifolds to Riemannian manifolds. We also obtain corresponding results for holomorphic maps from almost Hermitian manifolds to quasi-Kähler manifolds, which generalize or improve the partial results in Goldberg and Har’El (Bull Soc Math Grèce 18(1):141–148, 1977, J Differ Geom 14(1):67–80, 1979).     

Liouville theorem and coupling on negatively curved Riemannian manifolds

By using probabilistic approaches, Liouville theorems are proved for a class of Riemannian manifolds with Ricci curvatures bounded below by a negative function. Indeed, for these manifolds we prove that all harmonic functions (maps) with certain growth are constant. In particular, the well-known Liouville theorem due to Cheng for sublinear harmonic functions (maps) is generalized. Moreover, our results imply the Brownian coupling property for a class of negatively curved Riemannian manifolds. This leads to a negative answer to a question of Kendall concerning the Brownian coupling property.     

On compact aspherical homogeneous manifolds of dimension ≤7

In the paper, a classification of aspherical compact homogeneous manifolds of dimension less than or equal to 7 is discussed.     

Aspherical Products Which do not Support Anosov Diffeomorphisms

We show that the product of infranilmanifolds with certain aspherical closed manifolds do not support Anosov diffeomorphisms. As a special case, we obtain that products of a nilmanifold and negatively curved manifolds of dimension at least 3 do not support Anosov diffeomorphisms.     

The manifold of isospectral symmetric tridiagonal matrices and realization of cycles by aspherical manifolds

We consider the classical N. Steenrod’s problem of realization of cycles by continuous images of manifolds. Our goal is to find a class \(\mathcal{M}_n \) of oriented n-dimensional closed smooth manifolds such that each integral homology class can be realized with some multiplicity by an image of a manifold from the class \(\mathcal{M}_n \). We prove that as the class \(\mathcal{M}_n \) one can take a set of finite-fold coverings of the manifold M ⁿ of isospectral symmetric tridiagonal real (n + 1) × (n + 1) matrices. It is well known that the manifold M ⁿ is aspherical, its fundamental group is torsion-free, and its universal covering is diffeomorphic to ? ⁿ . Thus, every integral homology class of an arcwise connected space can be realized with some multiplicity by an image of an aspherical manifold with a torsion-free fundamental group. In particular, for any closed oriented manifold Q ⁿ , there exists an aspherical manifold that has torsion-free fundamental group and can be mapped onto Q ⁿ with nonzero degree.     

Invariant Hermitian structures and variational aspects of a family of holomorphic curves on flag manifolds

In this article, we study variational aspects for a special class of holomorphic maps on flag manifolds. We prove stability and non-stability results for such maps with respect to a large number of invariant Hermitian structures on maximal flag manifolds including the (1, 2)-symplectic structures, Einstein metrics, Cartan–Killing metrics, and so on. Some results in this article were announced without proofs in a past article by the first author.     

A note on <Emphasis Type="Italic">tt</Emphasis><Superscript>*</Superscript>-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt ^*-bundles over nearly Kähler manifolds. An application of the last two results is that any metric tt ^*-bundle over a compact nearly Kähler manifold is trivial (Theorem A). This result we apply to special Kähler manifolds to show that any compact special Kähler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kähler manifolds. Further we introduce harmonic bundles over nearly Kähler manifolds and study the implications of Theorem A for tt ^*-bundles coming from harmonic bundles over nearly Kähler manifolds.