Trivial Fiber Bundle Research Articles

On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows

We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^{n}$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].

The fiber of persistent homology for simplicial complexes

We study the inverse problem for persistent homology: For a fixed simplicial complex K , we analyze the fiber of the continuous map PH on the space of filters that assigns to a filter f : K → R the total barcode of its associated sublevel set filtration of K . We find that PH is best understood as a map of stratified spaces. Over each stratum of the barcode space the map PH restricts to a (trivial) fiber bundle with fiber a polyhedral complex. Amongst other we derive a bound for the dimension of the fiber depending on the number of distinct endpoints in the barcode. Furthermore, taking the inverse image PH − 1 can be extended to a monodromy functor on the (entrance path) category of barcodes. We demonstrate our theory on the example of the simplicial triangle giving a complete description of all fibers and monodromy maps. This example is rich enough to have a Möbius band as one of its fibers.

Limits of an increasing sequence of complex manifolds

Let M be a complex manifold which admits an exhaustion by open subsets $$M_j$$ each of which is biholomorphic to a fixed domain $$\Omega \subset \mathbb C^n$$ . The main question addressed here is to describe M in terms of $$\Omega $$ . Building on work of Fornaess–Sibony, we study two cases, namely M is Kobayashi hyperbolic and the other being the corank one case in which the Kobayashi metric degenerates along one direction. When M is Kobayashi hyperbolic, its complete description is obtained when $$\Omega $$ is one of the following domains—(i) a smoothly bounded Levi corank one domain, (ii) a smoothly bounded convex domain, (iii) a strongly pseudoconvex polyhedral domain in $$\mathbb C^2$$ , or (iv) a simply connected domain in $$\mathbb C^2$$ with generic piecewise smooth Levi-flat boundary. With additional hypotheses, the case when $$\Omega $$ is the minimal ball or the symmetrized polydisc in $$\mathbb C^n$$ can also be handled. When the Kobayashi metric on M has corank one and $$\Omega $$ is either of (i), (ii) or (iii) listed above, it is shown that M is biholomorphic to a locally trivial fibre bundle with fibre $$\mathbb C$$ over a holomorphic retract of $$\Omega $$ or that of a limiting domain associated with it. Finally, when $$\Omega = \Delta \times \mathbb B^{n-1}$$ , the product of the unit disc $$\Delta \subset \mathbb C$$ and the unit ball $$\mathbb B^{n-1} \subset \mathbb C^{n-1}$$ , a complete description of holomorphic retracts is obtained. As a consequence, if M is Kobayashi hyperbolic and $$\Omega = \Delta \times \mathbb B^{n-1}$$ , it is shown that M is biholomorphic to $$\Omega $$ . Further, if the Kobayashi metric on M has corank one, then M is globally a product; in fact, it is biholomorphic to $$Z \times \mathbb C$$ , where $$Z \subset \Omega = \Delta \times \mathbb B^{n-1}$$ is a holomorphic retract.

Symmetry and Relative Equilibria of a Bicycle System

In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients of these equations appeared to depend on the lean and steer angles only, and in one such equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would always vanish. These properties indeed arise from the symmetry of the bicycle system. From the point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian and constraint distribution invariant. We analyze the dimension relationship between the space of admissible velocities and the tangent space to the group orbit, and then employ the reduced nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local representative of the reduced dynamic system on the reduced constraint space, whose relative equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several families of one-parameter solutions. Finally, we prove that these relative equilibria are Lyapunov (but not asymptotically) stable under certain conditions. However, an isolated asymptotically stable equilibrium may be achieved by restricting the system to an invariant manifold, which is the level set of the reduced constrained energy.

A structure theorem for semi-parabolic Hénon maps

Consider the parameter space Pλ⊂C2 of complex Hénon mapsHc,a(x,y)=(x2+c+ay,ax),a≠0, which have a semi-parabolic fixed point with one eigenvalue λ=e2πip/q. In this paper we provide a complete topological description of the Julia set and forward Julia set within a family of semi-parabolic Hénon maps, i.e. those Hénon maps from the curve Pλ which are perturbations of a quadratic polynomial p with a parabolic fixed point of multiplier λ. We prove that there is an open disk of parameters in Pλ for which the semi-parabolic Hénon map has connected Julia set J and is structurally stable on J and J+. The forward Julia set J+ has a nice local description: inside a certain bidisk it is a trivial fiber bundle over Jp, the Julia set of the polynomial p, with fibers biholomorphic to a disk. The Julia set J is homeomorphic to the quotient of a solenoid by an equivalence relation and J=J⁎, the closure of the saddle periodic points.

Some remarks on the parametrized Borsuk–Ulam theorem

Given a locally trivial fibre bundle $E \to B$ (with fibres and base finite complexes), an orthogonal real line bundle $\lambda$ over $E$ and a real vector bundle $\xi$ over $B$, we consider a fibrewise map $f : S(\lambda ) \to \xi$ over $B$ defined on the unit sphere bundle of $\lambda$. Following the fundamental work of Jaworowski and Dold on the parametrized Borsuk-Ulam theorem, we investigate lower bounds on the cohomological dimension of the set of points $v$ in $S(\lambda )$ such that $f(v) = f(-v)$.

Geometry of locomotion of the generalized Purcell’s swimmer: Modelling, controllability and motion primitives

Micro-robotics at low Reynolds number has been a growing area of research over the past decade. We propose and study a generalized 3-link robotic swimmer inspired by the planar Purcell’s swimmer. By incorporating out-of-plane motion of the outer limbs, this mechanism generalizes the planar Purcell’s swimmer, which has been widely studied in the literature. Such an evolution of the limbs’ motion results in the swimmer’s base link evolving in a 3 dimensional space. The swimmer’s configuration space admits a trivial principal fiber bundle structure, which along with the slender body theory at the low Reynolds number regime, facilitates in obtaining a principal kinematic form of the equations. We derive a coordinate-free expression for the local form of the kinematic connection. A novel approach for local controllability analysis of this 3 dimensional swimmer in the low Reynolds number regime is presented by employing the controllability results of the planar Purcell’s swimmer. This is followed by control synthesis using the motion primitives approach. We prove the existence of motion primitives based control sequence for point to point maneuver of the swimmer’s base link whose motion evolves on a Lie group. A set of control sequences for translational and rotational maneuvers is then provided along with numerical simulations.

The Teichmüller space of the Hirsch foliation

We prove that the Teichmüller space of the Hirsch foliation (a minimal foliation of a closed 3-manifold by non-compact hyperbolic surfaces) is homeomorphic to the space of closed curves in the plane. This allows us to show that the space of hyperbolic metrics on the foliation is a trivial principal fiber bundle. And that the structure group of this bundle, the arc-connected component of the identity in the group of homeomorphisms which are smooth on each leaf and vary continuously in the smooth topology in the transverse direction of the foliation, is contractible.

A generalization of the Gauss–Bonnet–Hopf–Poincaré formula for sections and branched sections of bundles

Abstract For a two-dimensional compact oriented Riemannian manifold ( M , g ) , and a vector field V on M , the Hopf–Poincare theorem combined with the Gauss–Bonnet theorem gives the Gauss–Bonnet–Hopf–Poincare (GBHP) formula : ∑ z ∈ Z ( V ) i n d z ( V ) = 1 2 π ∫ M K d σ , where Z ( V ) is the set of zeros of V , i n d z ( V ) is the index of V at z ∈ Z ( V ) , and K is the curvature of g . We consider a locally trivial fiber bundle π : E → M over a compact oriented two-dimensional manifold M , and a section s of this bundle defined over M ∖ Σ , where Σ is a discrete subset of M called the set of singularities of the section. We assume that the behavior of the section s at the singularities is controlled in the following way: s ( M ∖ Σ ) coincides with the interior part of a surface S ⊂ E with boundary ∂ S , and ∂ S is π − 1 ( Σ ) . For such sections s we define an index of s at a point of Σ , which generalizes in the natural way the index of zero of a vector field, and then prove that the sum of these indices at the points of Σ can be expressed as an integral over S of a 2 -form constructed via a connection in E , thus we obtain a generalization of the GBHP formula. Also we consider branched sections with singularities, define an index of a branched section at a singular point, and find a generalization of the GBHP formula for the branched sections.

A note on parallelizable dynamical systems

Hajek [ 3 ] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber \begin{document}$\mathbb{R}$\end{document} . The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.

On the Projection of Stein Domains in Holomorphic Fiber Bundles

If \(\pi :E\rightarrow X\) is a locally trivial holomorphic fiber bundle, with positive dimensional connected fiber and with the total space E being a Stein manifold, and \(D\subset X\) is an arbitrary connected open subset, then there exists an open, connected Stein subset \(\widetilde{D}\subset E\) such that \(\pi (\widetilde{D})=D\).

Wieler solenoids, Cuntz–Pimsner algebras and -theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated$K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one$K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The$K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

IN MEMORIAM OF ALEXANDER GROTHENDIECK. THE MAN.

Reflection symmetry in higher dimensional black hole spacetimes

In four spacetime dimensions there is a well known proof that for any asymptotically flat, stationary, and axisymmetric vacuum solution of Einstein’s equation there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of the axial Killing vector field. However, this proof does not generalize to higher spacetime dimensions. Here we consider asymptotically flat, stationary, and axisymmetric (i.e., having one or more commuting rotational isometries) black hole spacetimes in vacuum general relativity in spacetime dimensions such that the action of the isometry group is trivial. (Here ‘trivial’ means that if the ‘axes’—i.e., the points where the axial Killing fields are linearly dependent—are removed, the action of the isometry group is that of a trivial principal fiber bundle. This excludes actions like that found in the Sorkin monopole.) We prove that there exists a ‘t-ϕ’ reflection isometry that reverses the direction of the timelike Killing vector field and the direction of each axial Killing vector field. The proof relies in an essential way on the first law of black hole mechanics.

Crystal Spheres as the World

A geometric concept of the world (W ) is considered where the manifold W is identified with a locally trivial fibre bundle pr: W ! U of so–called crystal spheres over a manifold U called the universal time. For every point p 2 U, Mn = pr 1 (p) is a n–dimensional crystal sphere and close crystal spheres are called the parallel universes. There exists a geometric black hole on the smooth manifold M n. Tensor fields, fibre bundles, operators (physical structures and equations) can be deformed towards the black hole into continuous and sectionally smooth those, further, they can be retracted together with the black hole into a small black ball to initiate the Big Bang.

FAMILIES OF GROUP ACTIONS, GENERIC ISOTRIVIALITY, AND LINEARIZATION

Our base field is the field ℂ of complex numbers. We study families of reductive group actions on $$ {\mathbb A} $$ 2 parametrized by curves and show that every faithful action of a non-finite reductive group on $$ {\mathbb A} $$ 3 is linearizable, i.e., G-isomorphic to a representation of G. The difficulties arise for non-connected groups G. We prove a Generic Equivalence Theorem which says that two affine morphisms 𝑝: S ⟶ Y and q : Τ ⟶ Y of varieties with isomorphic (closed) fibers become isomorphic under a dominant étale base change φ: U ⟶ Y . A special case is the following result. Call a morphism φ: X ⟶ Y a fibration with fiber F if φ is at and all fibers are (reduced and) isomorphic to F. Then an affine fibration with fiber F admits an étale dominant morphism μ: U ⟶ Y such that the pull-back is a trivial fiber bundle: U × Y X ≅ U × F. As an application we give short proofs of the following two (known) results: (a) Every affine A1-_bration over a normal variety is locally trivial in the Zariskitopology (see [KW85]). (b) Every affine A2-_bration over a smooth curve is locally trivial in the ZariskiTopology (see [KZ01]).

Generic structure of the Lagrangian manifold in chattering problems

This paper studies the structure of the singularity of Lagrangian manifolds in a neighborhood of the surface of singular extremals of second order in optimal control problems. For the Fuller classical problem, the structure of the Lagrangian manifold is explicitly constructed: it is shown that it has a singularity of conic type at the origin of coordinates. In the general case, it is proved that the Lagrangian manifold is a locally trivial fiber bundle over the surface of singular extremals with each fiber having a singularity of a similar conic type at the point of exit of the singular extremals.

On the geometry of flat pseudo-Riemannian homogeneous spaces

Let $M$ be complete flat pseudo-Riemannian homogeneous manifold and $\Gamma\subset\Iso(\RR^n_s)$ its fundamental group. We show that $M$ is a trivial fiber bundle $G/\Gamma\to M\to\RR^{n-k}$, where $G$ is the Zariski closure of $\Gamma$ in $\Iso(\RR^n_s)$. Moreover, we show that the $G$-orbits in $\RR^n_s$ are affinely diffeomorphic to $G$ endowed with the (0)-connection. If the induced metric on the $G$-orbits is non-degenerate, then $G$ (and hence $\Gamma$) has linear abelian holonomy. If additionally $G$ is not abelian, then $G$ contains a certain subgroup of dimension 6. In particular, for non-abelian $G$ orbits with non-degenerate metric can appear only if $\dim G\geq 6$.

Entire curves, integral sets and principal bundles

We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.

Obstructions to fibering a manifold

Given a map f : M → N of closed topological manifolds we define torsion obstructions whose vanishing is a necessary condition for f being homotopy equivalent to a projection of a locally trivial fiber bundle. If N = S 1, these torsion obstructions are identified with the ones due to Farrell (Indiana Univ Math J 21:315–346, 1971/1972).