Compact Orientable Manifold Research Articles

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of \(F \times I\) has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes \(S^3\).

Higher eta invariants and the Novikov conjecture on manifolds with boundary

Let ( N,g) be a closed Riemannian manifold of dimension 2 m-1 and Γ → N ˜ → N be a Galois covering of N. We assume that Γ is of polynomial growth and that Δ N ˜ is L 2-invertible in degree m. By employing spectral sections which are symmetric with respect to the *-Hodge operator, we define the higher eta invariant associated to the signature operator on N ˜ , thus extending previous work of Lott. If π 1 ( M ) → M ˜ → M is the universal cover of a compact orientable even-dimensional manifold with boundary ( ∂M = N) then, under the above invertibility assumption on Δ ∂ M ˜ , we define a canonical Atiyah-Patodi-Singer signature-index class, in K 0 ( C r * (Γ)). Employing the higher APS index theory developed in [4] we express the Chern character of this index class in terms of a local integral and of the higher eta invariant defined above. We apply these results to the problem of the existence and homotopy invariance of higher signatures on manifolds with boundary.

Coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations

Let us consider a compact orientable manifold M and ( f, g) a pair of selfmaps of M. When M belongs to a nilpotent class of compact manifolds, so in particular if M is a nilmanifold. we establish relations among N( f, g), # R[f,g] , coin(f#,g#), and Λ( f, g). Such relations are well known for tori. Finally, we extend the results for spaces E which are locally trivial nilpotent fibrations which fibre over a nilmanifold with certain fibers.

Finite groups and flows on 2-manifolds

We consider flows on compact orientable two-dimensional manifolds all points of which are non-wandering. An inter-relation between so-called Conley-Lyapunov-Peixoto graphs of such flows and Cayley graphs of finite groups is clarified. We describe all finite groups that are `good candidates' to symmetry groups of the flows in question. An application of the above results to the integrability problem of Hamiltonian flows is suggested. We conclude with two questions which might be of interest to group theorists.

Isotopies of 3-manifolds

An isotopy of a manifold M that starts and ends at the identity diffeomorphism determines an element of π 1(Diff( M)). For compact orientable 3-manifolds with at least three nonsimply connected prime summands, or with one S 2 × S 1 summand and one other prime summand with infinite fundamental group, infinitely many integrally linearly independent isotopies are constructed, showing that π 1(Diff( M)) is not finitely generated. The proof requires the assumption that the fundamental group of each prime summand with finite fundamental group imbeds as a subgroup of SO(4) that acts freely on S 3 (conjecturally, all 3-manifolds with finite fundamental group satisfy this assumption). On the other hand, if M is the connected sum of two irreducible summands, and for each irreducible summand P of M, π 1(Diff( P)) is finitely generated, then results of Jahren and Hatcher imply that π 1(Diff( M)) is finitely generated. The isotopies are constructed on submanifolds of M which are homotopy equivalent to a 1-point union of two 2-spheres and some finite number of circles. The integral linear independence is proven by obstruction-theoretic methods.

Approximation of Lyapunov Exponents of Nonlinear Stochastic Differential Equations

We consider nonlinear stochastic differential systems defined either on a compact orientable manifold or on $\mathbb{R}^d $; under our hypotheses, their Lyapunov exponents are deterministic. We propose an efficient algorithm of numerical computation of these exponents, and we give a theoretical estimate for the approximation error.The method is based upon the discretization of the linearized stochastic flows of diffeomorphisms generated by the differential systems.Results of numerical experiments are also presented.

Relaxed Yang-Mills functional over 4-manifolds

We give suitable completions of the space of principal $G$-bundles over $M$ and the space of smooth connections on them, where $G$ is a compact, simple, simply connected Lie group and $M$ is a 4-dimensional compact orientable manifold. We also introduce a natural energy defined in such spaces and consider variational problems on them.

Roots of iterates of maps

Let f : X → X be a selfmap of a path-connected space and let a ϵ X. We call x ϵ X an irreducible root of, f n , at a if f n ( x) = a but f m ( x) ≠ a for all m < n, where f n denotes the nth iterate of f. A lower bound N I n ( f; a) for the number of irreducible roots of f n is defined that is homotopy invariant under suitable hypotheses and, in many cases, can be calculated algebraically from the homomorphism of the fundamental group of X induced by f. In particular, this is true when X is a compact orientable manifold without boundary and f has nonzero degree. The geometric content of these calculations is described in concrete examples in terms of the discrete semidynamical system determined by f. We apply the theory of roots of iterates to primitive roots of unity in H-spaces.

Hyperbolic volumes of Fibonacci manifolds

A. Yu. Vesnin and A. D. Mednykh UDC 515.16 + 512.817.7 This article is devoted to the study of three-dimensional compact orientable hyperbolic manifolds connected with the Fibonacci groups. The Fibonacci groups F(2, m) = (Zl, z2,..., z,n : ziZi+l = zi+2, { mod m) were introduced by J. Conway [1]. The first natural question connected with these groups was whether they are finite or not [1]. It is known from [2-6] that the group F(2, m) is finite if and only if m = 1,2,3, 4,5, 7. Some algebraic generalizations of the groups

A GEOMETRIC REALIZATION OF -GROUPS

It is shown that for each -group and each there exists an -dimensional compact orientable manifold without boundary such that . Furthermore, the well-known representation of Riemann surfaces () as a union of finitely many copies of the Riemann sphere with slits glued together is generalized to the -dimensional case.

Degree One Maps Between Geometric 3-Manifolds

Let M and N be two compact orientable 3-manifolds, we say that M ≥ N, if there is a degree one map from M to N. This gives a way to measure the complexity of 3-manifolds. The main purpose of this paper is to give a positive answer to the following conjecture: if there is an infinite sequence of degree one maps between Haken manifolds, then eventually all the manifolds are homeomorphic to each other. More generally, we prove a theorem which says that any infinite sequence of degree one maps between the so-called «geometric 3-manifolds» must eventually become homotopy equivalences

Affine conformal vector fields in semi-Riemannian manifolds

This paper is devoted to a systematic presentation of the essential results of research on affine conformal vector fields (ACV) and to exhibit the state of art as it now stands. Of particular interest is the new information on the existence of ACVs in compact orientable semi-Riemannian manifolds, their link with first integrals of the geodesics and the separability structures.

Dualite pour les feuilletages transversalement holomorphes

Let F be a transversely holomorphic codimension n foliation on a compact orientable manifold M and E a foliated holomorphic vector bundle on M. We prove that the cohomology of M with coefficients in the sheaf of E-valued holomorphic base-like differential forms satisfies Serre duality. Some computations are given. In case n=1 we show that the base-like cohomology H*(M/F,ℂ) is finite dimensional.

New generalizations of Poincar�'s geometric theorem

In this work we show that the number of closed trajectories of the field of kernels of a closed, nondegenerate, and center of gravity preserving 2-form on the total space of an oriented fiber bundle with fiber the circle over an orientable compact two-dimensional base is not less than the minimal number of critical points of a smooth function on the basis, under the assumption that the field of kernels is C1-close to a vertical field. Counting multiplicities, this number is not smaller than the minimal number of critical points of a Morse function on the base. We also give a lower estimate for the number of closed trajectories in the case of a higher-dimensional base. A form preserves.the center of gravity if its cohomology class is the lift of a class from the base. As an application we prove that the number of closed trajectories of a particle on a surface under the action of a strong and little varying magnetic field perpendicular to the surface is not smaller than the minimal number of critical points of a function on the given surface. The author is grateful to V. I. Arnol'd for formulating the problem on the closed trajectories of fields of kernels of differential forms preserving the center of gravity and for helping in work. I. Condition of Preservation of the Center of Gravity. Let S I + M ~ B be an orientable bundle over a closed (compact and boundaryless) orientable manifold B.

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.

On the existence of complete parallel vector fields

In 1966, Chern asked which compact orientable manifolds carry a Riemannian metric and a vector field which is parallel under this metric. An earlier paper answered this without assuming orientability. The present paper applies techniques which work without compactness to identify manifolds which can carry complete parallel vector fields

The extra dimensions of the eleven-dimensional simple supergravity are ricci-flat if they are compact

By using the de Rham-Yano-Bochner lemma on harmonic tensor fields in a compact orientable manifold, we show that the extra dimensions in eleven-dimensional simple supergravity will he Ricci-flat if they are compact.

Structure sets vanish for certain bundles over Seifert manifolds

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

Foliations with Nonorientable Leaves

We prove that a codimension one foliation of an orientable paracompact manifold has at most a countable number of nonorientable leaves. We also give an example of a codimension one foliation of a compact orientable manifold with an infinite number of nonorientable leaves.

Foliations with nonorientable leaves

We prove that a codimension one foliation of an orientable paracompact manifold has at most a countable number of nonorientable leaves. We also give an example of a codimension one foliation of a compact orientable manifold with an infinite number of nonorientable leaves.