Small Cover的L-S畴数

我们证明了如下结论:任意n维small cover的Lusternik-Schnirelmann畴数等于n;任意2n维toric-流形的Lusternik-Schnirelmann畴数等于n.我们的结果依赖于如下两个事实,一个是不等式cup(M)≤cat(M)≤dim(M)/r,它导致我们去计算cup(M).另一个事实是环面拓扑流形上同调环的明确表示,该表示使得cup(M)的计算变得容易.最后,我们进一步推广了相关结论.     

Stable systolic category of manifolds and the cup-length

It follows from a theorem of Gromov that the stable systolic category cat_stsys M{\rm cat}_{\rm stsys} M of a closed manifold M is bounded from below by cl_\mathbbQ M{\rm cl}_{\mathbb{Q}} M, the rational cup-length of M [Ka07]. We study the inequality in the opposite direction. In particular, combining our results with Gromov’s theorem, we prove the equality cat_stsys M = cl_\mathbbQ M{\rm cat}_{\rm stsys} M = {\rm cl}_{\mathbb{Q}} M for simply connected manifolds of dimension ≤ 7.     

Sobolev inequalities for differential forms andL q,p -cohomology

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold (M, g) and the L_q,p-cohomology of that manifold. The L_q,p-cohomology of (M,g) is defined to be the quotient of the space of closed differential forms in L^p(M) modulo the exact forms which are exterior differentials of forms in L^q (M).     

On the cohomology of one dimensional foliated manifolds

We show that the cohomology groupH ¹ (M, f) is an infinite dimensional vector space, for a dense set of one dimensional foliations on a closed manifold. In particular we compute this cohomology, for some foliations on the torus T².     

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.     

On the Generalized Geroch Conjecture for Complete Spin Manifolds*

Let W be a closed area enlargeable manifold in the sense of Gromov-Lawson and M be a noncompact spin manifold, the authors show that the connected sum M#W admits no complete metric of positive scalar curvature. When W = Tⁿ, this provides a positive answer to the generalized Geroch conjecture in the spin setting.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Fundamental group and contractible closed geodesics

We prove the existence of a nonempty class of finitely presented groups with the following property: If the fundamental group of a compact Riemannian manifold M belongs to this class, then there exists a constant c(M) > 1 such that for any sufficiently large x the number of contractible closed geodesics on M of length not exceeding x is greater than c(M)^x. In order to prove this result, we give a lower bound for the number of contractible closed geodesics of length ≤ x on a compact Riemannian manifold M in terms of the resource-bounded Kolmogorov complexity of the word problem for π₁ (M), thus answering a question posed by Gromov. © 1996 John Wiley & Sons, Inc.     

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S-Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,F) of S-Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe.If (X,F)∈S-Top, we define a transverse subset as a subspace A of X such that the intersection S∩A is at most countable for any S∈F. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C¹-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.     

Enveloppe d'holomorphie locale des variétés CR et élimination des singularités pour les fonctions CR intégrables

Let M be a locally embeddable CR manifold and Φ ⊂M be a closed set. We give sufficient conditions in order that L_loc¹ functions on M which are CR on M #x003A6; are CR on M.     

Fano contact manifolds and nilpotent orbits

A contact structure on a complex manifold M is a corank 1 subbundle F of T_M such that the bilinear form on F with values in the quotient line bundle L = T_M/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.?If is a simple Lie algebra, the unique closed orbit in (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.?In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M P(H ₀(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant. Received: July 28, 1997     

不动点集为$P(2m, 2l+1) \sqcup P(2m, 2n+1)$的对合

设$(M,\,T)$是一个带有光滑对合$T$的光滑闭流形, $T$在$M$上的不动点集为 $F=\{x\,|\,T(x)=x,\,x\in M\}$, 则$F$为$M$的闭子流形的不交并. 本文证明了： 当$F=P(2m,\,2l+1)\sqcup P(2m,\,2n+1)$时,其中$n>l\geq m,\,m\neq1,\,3$, $(M,\,T)$协边于零.     

On the boundedness of topological categories

Let K be a complete and cocomplete category with a given proper (E,M)-factorization. K is called well-bounded if K is moreover bounded with a generator and cowellpowered with respect to the given factorization. Freyd-Kelly proved the following theorem about well-bounded categories: Let K be a well-bounded category and let Γ be a class of cylinders in the small category C¹, and let all but a set of these cylinders be cones. Then Γ(C,K) is a reflective subcategory of [C,K]. The main results of this paper are: (I) If F: K→L is a Top-functor and L is well-bounded, then K is well-bounded. (II) If U is an E-reflective subcategory of a well-bounded category,then U is again wellbounded. As a corollary one obtains for instance that all coreflective and all epireflective subcategories of the category of topological spaces are well-bounded.     

不动点集为F=U_(i=1)~mRP_i(1)×HP_i(n)的对合

(M,T)是一个在r维光滑闭流形M上的不平凡光滑对合,它的不动点集为F．本文给出了F=U(i=1)~m RPi(1)×HPi(n)时对合的协边类,其中HP(n)表示n维四元数射影空间．     

論Понтрягин示性類Ⅴ

<正> 本文是這系列著作中Ⅱ的一個補充.在Ⅱ中(參閱Ⅱ的更正)我們證明了可微分閉流形的某些示性類特別是法3示性類的拓撲不變性.它的證明是隱合的(implicit).本文目的在進一步求得這些示性類用流形同調構造來表示的顧谿(explicit)公式,使我們能就任意可定向的可微分閉流形的這些示性類進行具體的計算.特別可以獲得下述結果:     

On killing characteristic classes by cobordism

Let M be an n-dimensional, differential, compact and closed manifold and let c be a characteristic class of degree greater or equal to (n+1)/2. We will prove that if the class c anihilates all the characteristic numbers of M, where it enters as a factor, then the manifold M is cobordant to a manifold in which the class c is zero. Also, we will examine the case of manifolds with an extra structure.     

A C-symplectic free -manifold with contractible orbits and

An interesting question in symplectic geometry concerns whether or not a closed symplectic manifold can have a free symplectic circle action with orbits contractible in the manifold. Here we present a c-symplectic example, thus showing that the problem is truly geometric as opposed to topological. Furthermore, we see that our example is the only known example of a c-symplectic manifold having non-trivial fundamental group and Lusternik-Schnirelmann category precisely half its dimension.

     

The Weyl functional near the Yamabe invariant

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y_C(M) and the L^n/2-norm W_C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y_C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W_C(M) is arbitrarily large. We study the image of the mapYW:C→(Y_C(M), W_C(M))∈R ² near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.     

The Riemannian <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> topology on the manifold of Riemannian metrics

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L ² Riemannian metric—so-called because it induces an L ² topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L ¹-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L ² metric. We also give a user-friendly criterion for convergence (with respect to the L ² metric) in the manifold of metrics.     

Nielsen theory on 3-manifolds covered by S 2 × ℝ

Let f: M → M be a self-map of a closed manifold M of dimension dim M ≥ 3. The Nielsen number N(f) of f is equal to the minimal number of fixed points of f among all self-maps f in the homotopy class of f. In this paper, we determine N(f) for all self-maps f when M is a closed 3-manifold with S²×R geometry. The calculation of N(f) relies on the induced homomorphisms of f on the fundamental group and on the second homotopy group of M.