On string topology of three manifolds

Let M be a closed, oriented and smooth manifold of dimension d. Let LM be the space of smooth loops in M. In [String topology, preprint math.GT/9911159] Chas and Sullivan introduced the loop product, a product of degree -d on the homology of LM. We aim at identifying the three manifolds with “nontrivial” loop product. This is an application of some existing powerful tools in three-dimensional topology such as the prime decomposition, torus decomposition, Seifert fiber space theorem, torus theorem.     

A homotopy theoretic realization of string topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H _* (LM) of degree -d. They then investigated other structure that this product induces, including a Batalin -Vilkovisky structure, and a Lie algebra structure on the S¹ equivariant homology H _* ^{S 1} (LM). These algebraic structures, as well as others, came under the general heading of the ”string topology” of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the ” cactus operad” defined in [6] which is equivalent to operad of framed disks in . This operad action realizes the Chas - Sullivan BV structure on H _* (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH ^* (C ^* (M); C ^* (M)). Received: 31 July 2001 / Revised version: 11 September 2001 Published online: 5 September 2002     

Short Geodesic Segments between Two Points on a Closed Riemannian Manifold

Let M ⁿ be a closed Riemannian manifold homotopy equivalent to the product of S ² and an arbitrary (n–2)-dimensional manifold. In this paper we prove that given an arbitrary pair of points on M ⁿ there exist at least k distinct geodesics of length at most 20k!d between these points for every positive integer k. Here d denotes the diameter of M ⁿ .     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

Supports of fourier transforms of refinable frame functions and their applications to FMRA

Let M be a d × d expansive matrix, and FL ²(??) be a reducing subspace of L ²(? ^d ). This paper characterizes bounded measurable sets in ? ^d which are the supports of Fourier transforms of M-refinable frame functions. As applications, we derive the characterization of bounded measurable sets as the supports of Fourier transforms of FMRA (W-type FMRA) frame scaling functions and MRA (W-type MRA) scaling functions for FL ²(??), respectively. Some examples are also provided.     

Complexity of Finding Irreducible Components of a Semialgebraic Set

Let W ⊂ Rⁿ be a semialgebraic set defined by a quantifier-free formula with k atomic polynomials of the kind f ∈ Z[X₁, . . . , X_n] such that deg_{X1, . . . , Xn}(f) < d and the absolute values of coefficients of f are less than 2^M for some positive integers d, M. An algorithm is proposed for producing the complexification, Zariski closure, and also for finding all irreducible components of W. The running time of the algorithm is bounded from above by M^O(1)(kd)^nO(1). The procedure is applied to computing a Whitney system for a semialgebraic set and the real radical of a polynomial ideal.     

A generalization of finite dimensionality for modules

Let R be a ring with identity. Let C be a class of R-modules which is closed under submodules and isomorphic images. Define a submodule C of an R-module M to be a C-submodule of M if C ? C. An R-module M is said to be C-finite dimensional if it does not contain an infinite direct sum of non-zero C-submodules of M. Theorem: Let M be a C-finite dimensional R-module. Then there is a uniform bound (the C-dimension of M) on the number of non-zero C-submodules in a direct sum of submodules of M. When C = $\text{M}$_R, we recover the definition of dimension in the sense of Goldie. When C is the class of torsion-free modules relative to a kernel functor σ, we derive the formula: dim M = σ-dim M + dim (σ(M)) where for an R-module N, dim N is the dimension of N in the sense of Goldie and σ-dim N is the dimension of N relative to the class of σ-torsion- free modules. A special case gives a new interpretation of rank of a module as defined by Goldie.     

Hypersurfaces d'un fibré vectoriel Riemannien à courbure de Gauss prescriteHypersurfaces of a Riemannian vector bundle with prescribed Gaussian curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed Gaussian curvatures of various types. To cite this article: A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 927–930.     

Characterizing local rings via homological dimensions and regular sequences

Let (R,m) be a Noetherian local ring of depth d and C a semidualizing R-complex. Let M be a finite R-module and t an integer between 0 and d. If the G_C-dimension of M/aM is finite for all ideals a generated by an R-regular sequence of length at most d−t then either the G_C-dimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.     

Hypersurfaces compactes d'un fibré vectoriel riemannien à courbure moyenne prescriteCompact hypersurfaces of a Riemannian vector bundle with prescribed mean curvature

Let M be a compact Riemannian manifold, E a Riemannian vector bundle on M and Σ the sphere subbundle of E. We look for embeddings of Σ into E admitting prescribed mean curvatures of various types. To cite this article: P. Cherrier, A. Hanani, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 525–528.     

Loop coproducts in string topology and triviality of higher genus TQFT operations

Cohen and Godin constructed a positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree d homology group of the connected component of LM consisting of contractible loops, where d=dimM, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product.     

Divergence theorems in path space II: degenerate diffusions

Let x denote an elliptic diffusion process defined on a smooth compact manifold M. In a previous work, we introduced a class of vector fields on the path space of x and studied the admissibility of this class of vector fields with respect to the law of x. In the present Note, we extend this study to the case of degenerate diffusions. To cite this article: D. Bell, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the number of idempotent linear transformations

Let M and N be two subspaces of a finite dimensional vector space V over a finite field F. We can count the number of all idempotent linear transformations T of V such that R(T) ?M and N?N(T), where R(T) and N(T) denote the range space and the null space of T, respectively.     

A note on the independence number of triangle-free graphs

Let G be a triangle-free graph on n points with average degree d. Let α be the independence number of G. In this note we give a simple proof that α ? n (d ln d ? d + 1)/(d ? 1)². We also consider what happens when G contains a limited number of triangles.     

On the packing dimension of the Julia set and the escaping set of an entire function

Let f be a transcendental entire function. We give conditions which imply that the Julia set and the escaping set of f have packing dimension 2. For example, this holds if there exists a positive constant c less than 1 such that the minimum modulus L(r, f) and the maximum modulus M(r, f) satisfy log L(r, f) ≤ c logM(r, f) for large r. The conditions are also satisfied if logM(2r, f) ≥ d logM(r, f) for some constant d greater than 1 and all large r.     

On a Problem of Hasse for Certain Imaginary Abelian Fields

Let K be the composite field of an imaginary quadratic field Q(ω) of conductor d and a real abelian field L of conductor f distinct from the rationals Q, where (d,f)=1. Let Z_K be the ring of integers in K. Then concerning to Hasse's problem we construct new families of infinitely many fields K with the non-monogenic phenomena (1), (2) which supplement (J. Number Theory23 (1986), 347-353; Publ. Math. Fac. Sci Besançon, Theor. Nombres (1984) 25pp) and with monogenic (3).     

The effect of perturbations on the first eigenvalue of the p-LaplacianL'éffet des pérturbations sur la première valeur propre du p-Laplacien

Let $\text{Ω}$ be a domain with Lipschitzian boundary of a compact Riemannian manifold (M,g) and p>1. We prove that we can make the volume of M arbitrarily close to the volume of $\text{(Ω,g)}$ while the first eigenvalue of the p-Laplacian on M remains uniformly bounded from below in terms of the the first eigenvalue of the Neumann problem for the p-Laplacian on $\text{(Ω,g)}$. To cite this article: A.-M. Matei, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 255–258.     

Symplectic resolutions for coverings of nilpotent orbits

Let $\text{O}$ be a nilpotent orbit in a semisimple complex Lie algebra $\text{g}$. Denote by G the simply connected Lie group with Lie algebra $\text{g}$. For a G-homogeneous covering $\text{M→}\text{O}$, let X be the normalization of $\text{O}$ in the function field of M. In this Note, we study the existence of symplectic resolutions for such coverings X. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Addition requirements for matrix and transposed matrix products

Let M be an s × t matrix and let M^T be the transpose of M. Let x and y be t- and s-dimensional indeterminate column vectors, respectively. We show that any linear algorithm A that computes Mx has associated with it a natural dual linear algorithm denoted A^T that computes M^Ty. Furthermore, if M has no zero rows or columns then the number of additions used by A^T exceeds the number of additions used by A by exactly s − t. In addition, a strong correspondence is established between linear algorithms that compute the product Mx and bilinear algorithms that compute the bilinear form y^TMx.