Approximate fibrations and bundle maps on Hilbert cube manifolds

This paper is concerned with parameterized families of approximate fibrations from a compact Hilbert cube manifold M to a compact polyhedron B. The main result shows how to straighten out certain of these families to be nearly like a product. As an application of this technique, it is shown that an approximate fibration p:M → B can be approximated arbitrarily closely by bundle maps if and only if p is homotopic via approximate fibrations to a bundle map. Another result is that the space of bundle maps from M to B is locally n-connected for each n ? 0.     

Cohomology of GKM fiber bundles

The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray–Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.     

K-Orbit Closures on G/B as Universal Degeneracy Loci for Flagged Vector Bundles with Symmetric or Skew-Symmetric Bilinear Form

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of K-orbit closures on the flag variety G/B, where G = GL(n, $ \mathbb{C} $ ), and where K is one of the symmetric subgroups O(n, $ \mathbb{C} $ ) or Sp(n, $ \mathbb{C} $ ). We realize these orbit closures as universal degeneracy loci for a vector bundle over a variety equipped with a single flag of subbundles and a nondegenerate symmetric or skew-symmetric bilinear form taking values in the trivial bundle. We describe how our equivariant formulas can be interpreted as giving formulas for the classes of such loci in terms of the Chern classes of the various bundles.     

Equivariant cohomology of K-contact manifolds

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.     

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic γ : I → M in the total space of a semi-Riemannian submersion π : M → B by determining an explicit relation with the corresponding objects along the projected geodesic \({\pi\circ\gamma:I\to B}\) in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.     

The intersection of essential approximate point spectra of operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite-dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that there exists some operator C∈B(K,H) such that MC is upper semi-Fredholm and ind(MC)?0 if and only if there exists some left invertible operator C∈B(K,H) such that MC is upper semi-Fredholm and ind(MC)?0. A necessary and sufficient condition for MC to be upper semi-Fredholm and ind(MC)?0 for some C∈Inv(K,H) is given, where Inv(K,H) denotes the set of all the invertible operators of B(K,H). In addition, we give a necessary and sufficient condition for MC to be upper semi-Fredholm and ind(MC)?0 for all C∈Inv(K,H).     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

Harmonic morphisms and Riemannian geometry of tangent bundles

Let (TM, G) and \({(T_1 M,\tilde G)}\) respectively denote the tangent bundle and the unit tangent sphere bundle of a Riemannian manifold (M, g), equipped with arbitrary Riemannian g-natural metrics. After studying the geometry of the canonical projections π : (TM, G) → (M, g) and \({\pi_1:(T_1 M,\tilde G) \rightarrow (M,g)}\), we give necessary and sufficient conditions for π and π ₁ to be harmonic morphisms. Some relevant classes of Riemannian g-natural metrics will be characterized in terms of harmonicity properties of the canonical projections. Moreover, we study the harmonicity of the canonical projection \({\Phi:(TM-\{0\},G)\to (T_1 M,\tilde G)}\) with respect to Riemannian g-natural metrics \({G,\tilde G}\) of Kaluza–Klein type.     

Weierstrass points and Z2 homology

In a recent paper (1993), Lustig established a beautiful connection between the six Weierstrass points on a Riemann surface M₂ of genus 2 and intersection points of closed geodesics for the associated hyperbolic metric. As a consequence, he was able to construct an action of the mapping class group Out(π₁M₂) of M₂ on the set of Weierstrass points of M₂ and a virtual splitting of the natural homomorphism Aut(π₁M₂) → Out(π₁M₂). Our discussion in this paper begins with the observation that these two results of Lustig's are direct consequences of the work of Birman and Hilden (1973) on equivariant homotopies for surface homeomorphisms.It is well known that Γ₂ acts naturally on the Z₂ symplectic vector space of rank 4, H₁(M₂, Z₂). We identify this action with Lustig's action by constructing a natural correspondence between pairs of distinct Weierstrass points on M₂ and nonzero elements in H₁(M₂,Z₂). In this manner, the well-known exceptional isomorphism of finite group theory, S₆ ≅ Sp(4, Z₂), arises from a natural isomorphism of Γ₂ spaces.     

Smooth actions of finite Oliver groups on spheres

In this article, we deal with the following two questions. For smooth actions of a given finite group G on spheres S, which smooth manifolds F occur as the fixed point sets in S, and which real G-vector bundles ν over F occur as the equivariant normal bundles of F in S? We focus on the case G is an Oliver group and answer both questions under some conditions imposed on G, F, and ν. We construct smooth actions of G on spheres by making use of equivariant surgery, equivariant thickening, and Oliver's equivariant bundle extension method modified by an equivariant wegde sum construction and an equivariant bundle subtraction procedure.     

Direct injective modules

Throughout this paperR will denote a ring with idenity element andM a unitary right module overR. AnR-moduleM is said to be direct injective if and only if given direct summandN ofM with injectioni _N:N→M and a monomorphismg:N→M, there exists an endomorphismf ofR-moduleM such thatfg=i _N. In this paper we investigate properties of direct injective modules, and obtain the following results on direct injective modules.

1. We establish the necessary and sufficient condition for a module to be direct injective.
2. We show that the answer on problem of Krull-Schmidt-Matlis is in the affirmative in caseR-moduleM is extending direct injective.
3. We prove that extending direct injectivity of module implies same properties of its direct summands.

     

On the geometry of the space of smooth fibrations

We study geometrical aspects of the space of smooth fibrations between two given manifolds M and B, from the point of view of Fréchet geometry. As a first result, we show that any connected component of this space is the base space of a Fréchet-smooth principal bundle with the identity component of the group of diffeomorphisms of M as total space. Second, we prove that the space of fibrations is also itself the total space of a smooth Fréchet principal bundle with structure group the group of diffeomorphisms of the base B.     

Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

In [6, theorem IV.8.18], relatively norm compact sets K in L^p(μ) are characterized by means of strong convergence of conditional expectations, E_πf → f in L^p(μ), uniformly for f ∈ K, where (E_π) is the family of conditional expectations corresponding to the net of all finite measurable partitions.In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces $\text{L}{}_{\mathrm{E}}{}^{\mathrm{p}}\text{(μ)}$ of vector valued pth power Bochner integrable functions (and spaces M(Σ, E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in $\text{L}{}_{\mathrm{E}}{}^{\mathrm{p}}\text{(μ)}$ by means of uniform strong convergence E_πf → f, as well as relatively weak compact sets K by means of uniform weak convergence E_πf → f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.     

Sufficient Condition to Resolve Costa's First Conjecture

In this work, we give a sufficient condition to resolve Costa's first conjecture for each positive integer n and d with n ≥ 4. Precisely, we show that if there exists a local ring (A, M) such that λ _A (M) = n, and if there exists an (n + 2)-presented A-submodule of M ^m , where m is a positive integer (for instance, if M contains a regular element), then we may construct an example of (n + 4, d)-ring which is neither an (n + 3, d)-ring nor an (n + 4, d ? 1)-ring. Finally, we construct a local ring (B, M) such that λ _B (M) = 0 (resp., λ _B (M) = 1) and so we exhibit for each positive integer d, an example of a (4, d)-ring (resp., (5, d)-ring) which is neither a (4, d ? 1)-ring (resp., neither a (5, d ? 1)-ring) nor a (2, d′)-ring (resp., nor a (3, d′)-ring) for each positive integer d′.     

THE FUNDAMENTAL GROUP IN FIBRATIONS

A generalized Mayer-Vietoris sequence involving crossed homomorphisms is established and the construction is applied to the homotopy sequence of the CW-pair (X.X¹) to relate the homotopy sequences of (X.X¹) and the fibre bundle F → E → X in low dimensions. If there is a partial cross-section of E → X over X², the classical form, π₁ E ～ π₁ [xtilde] π₁ F as a semidirect product, results. In case there is no extension over X² of any cross-section of the restricted bundle χ:π₂ (x², x¹) → X¹ the corresponding obstruction map X_E:π₂(x²,x¹) → π₁F is non-trivial and in case F → E → X is an SO(n)-bundle (n ≥ 3), χ_E maps into a subgroup of the centre, Z(π₁ F), of order at most 2.