Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

We show that a closed simply connected 8-manifold (9-manifold) of positive sectional curvature on which a 3-torus (4-torus) acts isometrically is homeomorphic to a sphere, a complex projective space or a quaternionic projective plane (sphere). We show that a closed simply connected 2m-manifold (m5) of positive sectional curvature on which an (m–1)-torus acts isometrically is homeomorphic to a complex projective space if and only if its Euler characteristic is not 2. By [Wi], these results imply a homeomorphism classification for positively curved n-manifolds (n8) of almost maximal symmetry rank Supported by CNPq of Brazil, NSFC Grant 19741002, RFDP and Qiu-Shi Foundation of China.Supported partially by NSF Grant DMS 0203164 and a research grant from Capital normal university.     

The Hopf conjecture for positively curved manifolds with discrete abelian group actions

Let M be a closed even n-manifold of positive sectional curvature. The main result asserts that the Euler characteristic of M is positive, if M admits an isometric -action with prime p?p(n) (a constant depending only on n) and k satisfies any one of the following conditions: (i) and n≠12, 18 or 20; (ii) , and n≡0 mod 4 with n≠12 or 20; (iii) , and n≡0,4 or 12 mod 20 with n≠20. This generalizes some results in [T. Püttmann, C. Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002) 163-166; X. Rong, Positively curved manifolds with almost maximal symmetry rank, Geom. Dedicata 59 (2002) 157-182; X. Rong, X. Su, The Hopf conjecture for positively curved manifolds with abelian group actions, Comm. Cont. Math. 7 (2005) 121-136].     

Margulis lemma for compact lie groups

We improve Margulis lemma for a compact connected Lie group G: there is a neighborhood U of the identity such that for any finite subgroup , generates an abelian group. We show that for each n, there exists an integer , such that if H is a closed subgroup of a compact connected Lie group G of dimension n, then the quotient group, H/H ₀, has an abelian subgroup of index , where H ₀ is the identity component of H. As an application, we show that the fundamental group of the homogeneous space G/H has an abelian subgroup of index . We show this same property for the fundamental groups of almost non-negatively curved n-manifolds whose universal coverings are not collapsed. X. Rong: supported partially by NSF Grant DMS 0504534 and by a reach found from Beijing Normal University. Y. Wang: supported partially by LMAM of Peking University and by NSFC 10671018.     

Maximal transitive sets with singularities for genericC 1 vector fields

A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC ¹ generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC ¹ vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.     

Volume of Hyperbolic 3-Manifolds with Iterated Pseudo-Anosov Amalgamations (Dedicated to Professor Mitsuyoshi Kato on his sixtieth birthday)

Let : be a pseudo-Anosov homeomorphism. We will study an asymptotic behaviour of the volume of closed hyperbolic 3-manifolds N _n obtained from certain 3-manifolds M, M by attaching their boundaries by the n-th iteration ⁿ of .     

The mapping properties of filling radius and packing radius and their applications

Let be a map between closed, oriented Riemannian n-manifolds. It is shown that FillRad(W)?dil(f)⋅FillRad(V), if |deg(f)|=1. By this mapping property, we obtain an estimate from below for the filling radius of a closed, oriented, nonpositively curved manifold, or a manifold with sectional curvature bounded above by a positive constant. In addition, a similar mapping property of packing radius and a corollary are also obtained.     

Generalized Cn-Manifolds

This paper proves an important topological characterization of C-manifolds and especially of arbitrary finite dimensional Cⁿ-manifolds and arbitrary C∞-BANACH manifolds. Whereas differentiability structures in the usual sense may be proper classes this characterization always enables a definition of differentiability structures as sets. Further this characterization suggests a suitable generalization of the notion of differentiable manifold which we call Cⁿ-manifold. C ⁿ-manifolds have a lot of better properties than Cⁿ-manifolds and may be useful therefore in physics and technics. Some examples of C ⁿ-manifolds are given in the last two sections defined by means of certain geometric structures.     

On Morse--Smale Diffeomorphisms with Four Periodic Points on Closed Orientable Manifolds

We study Morse--Smale diffeomorphisms of n-manifolds with four periodic points which are the only periodic points. We prove that for n= 3 these diffeomorphisms are gradient-like and define a class of diffeomorphisms inevitably possessing a nonclosed heteroclinic curve. For n 4, we construct a complete conjugacy invariant in the class of diffeomorphisms with a single saddle of codimension one.     

On certain classes of hyperbolic 3-manifolds

We study the topological structure and the homeomorphism problem for closed 3-manifolds M(n,k) obtained by pairwise identifications of faces in the boundary of certain polyhedral 3-balls. We prove that they are (n/d)-fold cyclic coverings of the 3-sphere branched over certain hyperbolic links of d+1 components, where d= (n/k). Then we study the closed 3-manifolds obtained by Dehn surgeries on the components of these links. Received: 27 November 1998 / Accepted: 12 May 1999     

On the vertex group of n-manifolds

In this paper we obtain the decomposition of the vertex group of n-manifolds, extending the one given by Kauffman and Lins for dimension 3 and solving the related conjecture. The result is obtained in the more general category of gems: the vertex group of a gem , representing an n-manifold M, is the free product of n copies of the fundamental group of M and a free group F of rank N–n, where N is the number of n-residues of . In particular, for crystallizations FZ and consequently the vertex group is an invariant of M.     

G-symmetric products of the sphereS 2 and other closed surfaces

It is shown that a functor of G-symmetric degree maps closed 2-manifolds into manifolds if and only if it is isomorphic to the Cartesian product of functors of symmetric degree. We study the canonical stratification of the pseudomanifolds , where M is a closed orientable surface. We find the torsion subgroups of certain homology groups H_i(SP _G ⁿ M,).Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 9–14.     

Numerically regular hereditary classes of combinatorial geometries

A hereditary class of combinatorial geometries (or simple matroids) is a collection of geometries closed under minors and direct sums. A geometry G in is extremal if no proper extension of G of the same rank is in . The size function h(n) of is defined by h(n)=max {|G|: G and rank(G)=n}, where |G| is the number of points in G. A hereditary class is numerically regular if for every extremal geometry G in , |G|=h (rank(G)). We determine all the numerically regular hereditary classes for which the set {h(n)–h(n–1): 1n<} of positive integers does not have an upper bound: they are all varieties. We also give several examples of numerically regular hereditary classes which are not varieties.Partially supported by a North Texas State University Faculty Research Grant.     

Construction and Properties of the t-Invariant

Special spine theory is used for constructing a new invariant of compact 3-manifolds: the t-invariant. The behavior of the invariant under (boundary) connected sum is investigated. One of the TuraevViro invariants is expressed via the t-invariant. The t-invariant is interpreted from the point of view of TQFT. The values of the t-invariant are computed for lens spaces and for all closed oriented 3-manifolds of complexity at most six. It is proved that the set of values of the t-invariant on Seifert manifolds with fixed base (which is a closed surface) and fixed number of singular fibers is finite. Bibliography: 10 titles.     

Curvature, Harnack's Inequality, and a Spectral Characterization of Nilmanifolds

For closed n-dimensional Riemannian manifolds M with almostnonnegative Ricci curvature, the Laplacian on one-forms is known toadmit at most n small eigenvalues. If there are n small eigenvalues, or if M is orientable and has n – 1 small eigenvalues, then M isdiffeomorphic to a nilmanifold, and the metric is almost left invariant.We show that our results are optimal for n 4.     

On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen-Macaulay simplicial complexes admitting a proper action of the cyclic group of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper -action. Adin's upper bounds on the face numbers of Cohen-Macaulay complexes with symmetry are shown to hold for all (d−1)-dimensional Buchsbaum complexes with symmetry on n?3d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2k-manifolds are verified for all 2k-manifolds on n?6k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.     

Symmetries of the WDVV Equations

We investigate the symmetry structure of the WDVV equations. We obtain an r-parameter group of symmetries, where r= (n ²+7n+4)+n/2. Moreover, it is proved that for n=3 and n=4 these comprise all symmetries. We determine a subgroup, which defines an SL₂-action on the space of solutions. For the special case n=3 this action is compared to the SL₂-symmetry of the Chazy equation. We construct similar solutions in the cases n=4 and n=5.     

A survey on large finite group actions on graphs,surfaces and 3-manifolds

We give a survey on large finite group actions on low-dimensional objects such as graphs, surfaces, handlebodies and closed 3-manifolds, and also on large finite subgroups of the outer automorphism groups of their fundamental groups (free groups and surface groups) and of linear groups GL (n, ℤ); in particular we discuss recent results on maximal orders in various situations.     

The natural operators lifting 1-forms on manifolds to the bundles ofA-velocities

LetA be a Weil algebra withp variables. We prove that forn-manifolds (np+2) the set of all natural operatorsT ^*T ^* T ^A is a free finitely generated module over a ring canonically dependent onA. We construct explicitly the basis of this module.     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.