Tangent Fibration of a Poincare Complex

The unstable tangent fibration of a Poincaré complexis defined so that it is consistent with the manifold case.It exists uniquely for each Poincaré complex X up tofibrewise homotopy equivalence and, furthermore, if a Poincaréembedding structure exists on the diagonal XXxX, its normalfibration is the tangent fibration.     

Weak curvature conditions and functional inequalities

We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N². The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.     

Equivariant cohomology of real flag manifolds

Let P=G/K be a semisimple non-compact Riemannian symmetric space, where G=I₀(P) and K=Gp is the stabilizer of p∈P. Let X be an orbit of the (isotropy) representation of K on Tp(P) (X is called a real flag manifold). Let K₀⊂K be the stabilizer of a maximal flat, totally geodesic submanifold of P which contains p. We show that if all the simple root multiplicities of G/K are at least 2 then K₀ is connected and the action of K₀ on X is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning H^∗(X). In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.     

Varieties fibered by good minimal models

Let f : X → Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map with relative dimension no more than three, then X has a good minimal model.     

A solution to de Groot’s absolute cone conjecture

A compactum X is an ‘absolute cone’ if, for each of its points x, the space X is homeomorphic to a cone with x corresponding to the cone point. In 1971, J. de Groot conjectured that each n-dimensional absolute cone is an n-cell. In this paper, we give a complete solution to that conjecture. In particular, we show that the conjecture is true for n≤3 and false for n≥5. For n=4, the absolute cone conjecture is true if and only if the 3-dimensional Poincaré Conjecture is true.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

Pointwise Lipschitz functions on metric spaces

For a metric space X, we study the space D^∞(X) of bounded functions on X whose pointwise Lipschitz constant is uniformly bounded. D^∞(X) is compared with the space LIP^∞(X) of bounded Lipschitz functions on X, in terms of different properties regarding the geometry of X. We also obtain a Banach-Stone theorem in this context. In the case of a metric measure space, we also compare D^∞(X) with the Newtonian-Sobolev space N1,∞(X). In particular, if X supports a doubling measure and satisfies a local Poincaré inequality, we obtain that D^∞(X)=N1,∞(X).     

A counter-example to a conjecture of Félix

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L_X. The radical of L_X, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite.Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dimL_X<∞, i.e., if X is elliptic, then L_X is its own radical, and therefore the total dimension of the radical of L_X in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.     

F-structure on collapsed orbifolds

In this paper, we prove the existence of nilpotent Killing structures and F-structures on collapsed Riemannian orbifolds. Therefore a sufficiently collapsed orbifold X is the union of orbits, each orbit is an infranil orbifold of positive dimension; in particular, the F-structure provides a decomposition of X into compact flat orbifolds.     

The representation type of Hecke algebras of type B

This paper determines the representation type of the Iwahori-Hecke algebras of type B when q≠±1. In particular, we show that a single parameter non-semisimple Iwahori-Hecke algebra of type B has finite representation type if and only if q is a simple root of the Poincaré polynomial, confirming a conjecture of Uno's (J. Algebra 149 (1992) 287).     

On an existence theorem of Wagner manifolds

In their common paper [An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 43 (1997) 307-321] the authors give a condition for a 1-form β to be the perturbation of a Riemannian manifold (M, α) such that the manifold equipped with any (α, β) -metric is a Wagner manifold with respect to the Wagner connection induced by β. The condition shows that its covariant derivative with respect to the Lévi-Civita connection must be of a special form

(∇β)(X,Y)=∥β^#∥2α(X,Y)−β(X)β(Y)     

A class of Riemannian manifolds that pinch when evolved by Ricci flow

The purpose of this paper is to construct a set of Riemannian metrics on a manifold X with the property that will develop a pinching singularity in finite time when evolved by Ricci flow. More specifically, let , where N ⁿ is an arbitrary closed manifold of dimension n≥ 2 which admits an Einstein metric of positive curvature. We construct a (non-empty) set of warped product metrics on the non-compact manifold X such that if , then a smooth solution , t∈[0,T) to the Ricci flow equation exists for some maximal constant T, 0<T<∞, with initial value , and

Topological structure of complete Riemannian manifolds with cyclic holonomy groups

Let M be a complete m-dimensional Riemannian manifold with cyclic holonomy group, let X be a closed flat manifold homotopy equivalent to M, and let L→X be a nontrivial line bundle over X whose total space is a flat manifold with cyclic holonomy group. We prove that either M is diffeomorphic to X×Rm-dimX or M is diffeomorphic to L×Rm-dimX−1.     

Fisher information metric and Poisson kernels

A complete Riemannian manifold X with negative curvature satisfying −b²?KX?−a²<0 for some constants a,b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.     

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M→X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points of X: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.     

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH ⁱ (X) is finitely generated for i ≤ 4.     

Detecting codimension one manifold factors with topographical techniques

We prove recognition theorems for codimension one manifold factors of dimension n?4. In particular, we formalize topographical methods and introduce three ribbons properties: the crinkled ribbons property, the twisted crinkled ribbons property, and the fuzzy ribbons property. We show that X×R is a manifold in the cases when X is a resolvable generalized manifold of finite dimension n?3 with either: (1) the crinkled ribbons property; (2) the twisted crinkled ribbons property and the disjoint point disk property; or (3) the fuzzy ribbons property.     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Characteristic classes of complex hypersurfaces

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincaré dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schürmann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of vanishing cycles and singular strata in a Whitney stratification of X. Our approach is based on Schürmann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schürmann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces.