Free Homotopy Class Research Articles

Combinatorial Lie bialgebras of curves on surfaces

Goldman (Invent. Math. 85(2) (1986) 263) and Turaev (Ann. Sci. Ecole Norm. Sup. (4) 24 (6) (1991) 635) found a Lie bialgebra structure on the vector space generated by non-trivial free homotopy classes of curves on a surface. When the surface has non-empty boundary, this vector space has a basis of cyclic reduced words in the generators of the fundamental group and their inverses. We give a combinatorial algorithm to compute this Lie bialgebra on this vector space of cyclic words. Using this presentation, we prove a variant of Goldman's result relating the bracket to disjointness of curve representatives when one of the classes is simple. We exhibit some examples we found by programming the algorithm which answer negatively Turaev's question about the characterization of simple curves in terms of the cobracket. Further computations suggest an alternative characterization of simple curves in terms of the bracket of a curve and its inverse. Turaev's question is still open in genus zero.

Some global properties of static spacetimes

We show that a static Lorentzian manifold satisfying a completeness assumption is geodesically connected. In particular, such condition is satisfied by all compact static manifolds; in the compact case, we also prove the existence of a closed geodesic in every free homotopy class determined by a finite number of deck transformations.

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central.

Geodesic Knots in the Figure-Eight Knot Complement

We address the problem of topologically characterising simple closed geodesics in the figure-eight knot complement. We develop ways of finding these geodesics up to isotopy in the manifold, and notice that many seem to have the lowest-volume complement amongst all curves in their homotopy class. However, we find that this is not a property of geodesics that holds in general. The question arises whether under additional conditions a geodesic knot has least-volume complement over all curves in its free homotopy class. We also investigate the family of curves arising as closed orbits in the suspension flow on the figure-eight knot complement, many but not all of which are geodesic. We are led to conclude that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.

The Palais-Smale Condition and Mañé's Critical Values

Let $ {\Bbb L} $ be a convex superlinear autonomous Lagrangian on a closed connected manifold N. We consider critical values of Lagrangians as defined by R. Mane in [23]. We define energy levels satisfying the Palais-Smale condition and we show that the critical value of the lift of $ {\Bbb L} $ to any covering of N equals the infimum of the values of k such that the energy level t satisfies the Palais-Smale condition for every t > k provided that the Peierls barrier is finite. When the static set is not empty, the Peierls barrier is always finite and thus we obtain a characterization of the critical value of $ {\Bbb L} $ in terms of the Palais-Smale condition.¶We also show that if an energy level without conjugate points has energy strictly bigger than c u ($ {\Bbb L} $) (the critical value of the lift of $ {\Bbb L} $ to the universal covering of N), then two different points in the universal covering can be joined by a unique solution of the Euler-Lagrange equation that lives in the given energy level. Conversely, if the latter property holds, then the energy of the energy level is greater than or equal to c u ($ {\Bbb L} $). In this way, we obtain a characterization of the energy levels where an analogue of the Hadamard theorem holds. We conclude the paper showing other applications such as the existence of minimizing periodic orbits in every non-trivial homotopy class with energy greater than c u ($ {\Bbb L} $) and homologically trivial periodic orbits such that the action of $ {\Bbb L} $ + k is negative if c u ($ {\Bbb L} $) < k < c a ($ {\Bbb L} $), where c a ($ {\Bbb L} $) is the critical value of the lift of $ {\Bbb L} $ the abelian covering of N. We also prove that given an Anosov energy level, there exists in each non-trivial free homotopy class a unique closed orbit of the Euler-Lagrange flow in the given energy level.

On Minimizing Problems with a Volume Constraint in Hyperbolic 3-Manifolds

This paper investigates the existence of an area (or Dirichlet integral) minimizing parametric surface in a hyperbolic 3-manifold subject to a volume constraint. The existence of a minimizing surface is proved, assuming some conditions on the prescribed free homotopy class. This result implies a non-existence result of minimizing surfaces of prescribed mean curvature. A criterion for the existence of surfaces of prescribed mean curvature, which turns out to be optimal in view of the non-existence result, is also obtained.

Closed curves and geodesics with two self-intersections on the Punctured torus

We classify the free homotopy classes of closed curves with minimal self intersection number two on a once punctured toms, T, up to homeomorphism. Of these, there are six primitive classes and two imprimitive. The classification leads to the topological result that, up to homeomorphism, there is a unique curve in each class realizing the minimum self intersection number. The classification yields a complete classification of geodesics on hyperbolic T which have self intersection number two. We also derive new results on the Markoff spectrum of diophantine approximation; in particular, exactly three of the impfimitive classes correspond to families of Markoff values below Hall's ray.

THE MOST REFINED VASSILIEV INVARIANT OF DEGREE ONE OF KNOTS AND LINKS IN ℝ1-FIBRATIONS OVER A SURFACE

As it is well-known, all Vassiliev invariants of degree one of a knot K ⊂ ℝ3 are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not ℝ3. Recently, T. Fiedler introduced such invariants of a knot in an ℝ1-fibration over a surface F. They take values in the free ℤ-module generated by all the free homotopy classes of loops in F. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described. We also construct a similar invariant of a two-component link in an ℝ1-fibration. It generalizes the linking number.

The correlation of length spectra of two hyperbolic surfaces

We derive an asymptotic formula for the number of free homotopy classes on a closed surface which have (approximately) the same length with respect to two different hyperbolic structures on the surface. The growth rate in the asymptotic formula is described in terms of the thermodynamic formalism for the geodesic flow.

A criterion for flatness in minimal area metrics that define string diagrams

It has been proposed that the string diagrams of closed string field theory be defined by a minimal area problem that requires that all nontrivial homotopy curves have length greater than or equal to 2π. Consistency requires that the minimal area metric be flat in a neighbourhood of the punctures. The theorem proven in this paper, yieds a criterion which if satisfied, will ensure this requirement. The theorem states roughly that the metric is flat in an open set,U if there is a unique closed curve of length 2π through every point inU and all of these closed curves are in the same free homotopy class.

H-Semidirect Products

AbstractThe concept of H-semidirect product structure on a grouplike space is introduced. It is shown that the loop space ΩX of any based CW-complex X is the H-semidirect product of the identity path-component of ΩX with π1,X. The set of free homotopy classes of maps into a Hsemidirect product inherits the structure of a semidirect product. This leads to new results concerning the nilpotency of homotopy classes of maps into a group-like space.

Detecting homotopy equivalences in base-point-free homotopy

Let ƒ:X→Y be a map of connected CW complexes, such that ƒ#:[K, X]→[K, Y] is a bijection for every finite complex K ([K, X] being the set of free homotopy classes). Examples are known where such ƒ is not a homotopy equivalence, but no example is known where Y is finite-dimensional. We prove that if Y has finite dimension d and if HdỸ ≠ 0 then ƒ is a homotopy equivalence. More generally, we give necessary and sufficient conditions in terms of the fundamental group of Y for ƒ to be a homotopy equivalence, and we show that these conditions are almost met whenever Y is finite-dimensional. An interesting sequence {Gm}m⩾2 of finitely presented ‘Artin groups’ appears naturally in the discussion.

On the modulus of the free homotopy class of a simple closed curve on an arbitrary Riemann surface

On the modulus of the free homotopy class of a simple closed curve on an arbitrary Riemann surface

Index theorem and equivariant cohomology on the loop space

In this paper, extending ideas of Witten and Atiyah, we describe some relations of equivariant cohomology on the loop space of a manifold to the path integral representation of the index of the Dirac operator on a twisted spin complex. In particular, the natural extension of the Chern character to the whole loop space is described. Also it is shown that the non-zero free homotopy classes of the loop space have 0 measure for the index measures associated to the index problem.

An Algorithm for Simple Curves on Surfaces

Let M be a compact orientable surface with non-empty boundary and with X{M) < 0, and let T = nlM. Let C be the free homotopy class of a closed loop on M and let W = W{C) be a word in a fixed set of generators T which represents C. In this paper we give an algorithm to decide, starting with W, whether C has a simple representative, that is a representative without self-intersections. Such a word will be said to be simple. As an application, we begin a study of simple words in F. Our results also apply to infinite geodesies on M, corresponding to biinfinite words in F, where now we ask which finite blocks appear in such a word when the corresponding infinite homotopy class has a simple representative. For finite words there are, of course, other such algorithms, see for example [8, 9, 2, 3, 4]. Our algorithm most resembles that in [2] in that it is purely mechanical and combinatorial. It is simpler than that in [2] but what is more important is that it reveals the underlying mechanism which determines whether self-intersections occur; the combinatorics of that mechanism seem quite interesting and non-trivial. We represent M as U/T where U ^ D is the universal covering space of M, and where D is the unit disc with the Poincare metric and F is a discrete group of hyperbolic isometries. Poincare showed in [7] that C contains a simple representative if and only if the unique smooth geodesic representative C of C is simple, and that C is simple if and only if for each lift y of C to D the curves in the infinite family {fy}fer a r e pairwise disjoint. Now, to see if geodesies yl5 y2 e {fy} are disjoint in D it is enough to know whether the ideal endpoints of yx on 3D separate those of y2. Crucial to our work is a scheme for parametrizing points on 3D by infinite words in F, first developed by Nielsen in [6]. The idea of this paper is to show how information on the order of the points 3yl53y2 on 3D is encoded in Nielsen's 'boundary expansion' (Theorem A) and then to examine consequences. When dM ± 0 the group F is a free group so that each conjugacy class has a unique shortest representative which is obtained by cyclic reduction of any word in the class. However, if dM = 0 the shortest word in the conjugacy class is in general not unique. If dM = 0 and W e T has a shortest representative which does not contain any pieces which are half of the defining relator in F, then the problem of deciding whether W is simple is identical with that on the surface with a disc removed, that is one simply regards F as if it were a free group. On the other hand, the exceptional cases when W contains half a relator involve some subtle points which are not without interest, but are somewhat tangential to the main idea in the paper. For that reason, we shall omit the case in which dM = 0 . Here is an outline of this paper. The tools we need are set up in §§2 and 3 where we prove Theorem A. The algorithm (Theorem B) is given in §4. In §5 we give

INTERSECTIONS OF LOOPS IN TWO-DIMENSIONAL MANIFOLDS. II. FREE LOOPS

In this paper the authors establish algebraic conditions necessary and sufficient for given free homotopy classes of loops in two-dimensional manifolds to be representable by simple nonintersecting loops.Bibliography: 9 titles.

On the set of free homotopy classes and Brown's construction

On the set of free homotopy classes and Brown's construction

On the Symplectic Geometry of Deformations of a Hyperbolic Surface

Let R be a Riemann surface. In this manuscript we consider a geometry on the moduli space X(R) for R, which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of R. Classically the space of flat metrics for a torus is the locally symmetric space 0(2) SL(2; R)/SL(2; Z). We shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative Euler characteristic. The Teichmiiller space T(R), a covering of the moduli space, is a complex Kihler manifold. A KUhler metric for T(R), defined in terms of the Petersson product for automorphic forms, was introduced by Weil, [1]. The Weil-Petersson metric is invariant under the covering transformations and so projects to the moduli space X(R). The metric provides a link between the function theory of R and the geometry of X(R). In the Fenchel-Nielsen manuscript [8] a deformation, based on an amalgamation construction for Fuchsian groups, is introduced. The deformation is defined geometrically by cutting the surface along a simple closed geodesic, rotating one side of the cut relative to the other, and attaching the sides in their new position. The hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface. Choose a free homotopy class [a] on the surface R; then for each marked surface R realize [a] by the closed geodesic aR. The Fenchel-Nielsen deformations for the athen define a 1-parameter group of diffeomorphisms of T(R), whose infinitesimal generator by definition is the Fenchel-Nielsen vector field ta. In [21] the Fenchel-Nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t * was begun. The Fenchel-Nielsen vector fields were found to be related to the geodesic length functions 1*, introduced by Fricke-Klein to provide coordinates for T(R).

Geodesic Rigidity in Compact Nonpositively Curved Manifolds

Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature $K \leqslant 0$. In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank $1$. Let $M$, $M\prime$ denote compact Riemannian manifolds with $K \leqslant 0$, and let $\theta :{\pi _1}(M, m) \to {\pi _1}(M\prime , m\prime )$ be an isomorphism. Let $\theta$ also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class $[\alpha ]$ in $M$ is, respectively, Clifford, bounded or rank $1$ if and only if the class $\theta [\alpha ]$ in $M\prime$ is of the same type. Theorem B. If $M$, $M\prime$ have dimension $3$ and do not have a rank $1$ free homotopy class then they have diffeomorphic finite covers of the form ${S^1} \times {M^2}$. The proofs of Theorems A and B use the fact that $\theta$ is induced by a homotopy equivalence $f:(M, m) \to (M\prime , m\prime )$. Theorem C. The manifold $M$ satisfies the Visibility axiom if and only if $M\prime$ satisfies the Visibility axiom.

Geodesic rigidity in compact nonpositively curved manifolds

Our goal is to find geometric properties that are shared by homotopically equivalent compact Riemannian manifolds of sectional curvature K ⩽ 0 K \leqslant 0 . In this paper we consider mainly properties of free homotopy classes of closed curves. Each free homotopy class is represented by at least one smooth periodic geodesic, and the nonpositive curvature condition implies that any two periodic geodesic representatives are connected by a flat totally geodesic homotopy of periodic geodesic representatives. By imposing certain geometric conditions on these periodic geodesic representatives we define and study three types of free homotopy classes: Clifford, bounded and rank 1 1 . Let M M , M ′ M\prime denote compact Riemannian manifolds with K ⩽ 0 K \leqslant 0 , and let θ : π 1 ( M , m ) → π 1 ( M ′ , m ′ ) \theta :{\pi _1}(M,\,m) \to {\pi _1}(M\prime ,\,m\prime ) be an isomorphism. Let θ \theta also denote the induced bijection on free homotopy classes. Theorem A. The free homotopy class [ α ] [\alpha ] in M M is, respectively, Clifford, bounded or rank 1 1 if and only if the class θ [ α ] \theta [\alpha ] in M ′ M\prime is of the same type. Theorem B. If M M , M ′ M\prime have dimension 3 3 and do not have a rank 1 1 free homotopy class then they have diffeomorphic finite covers of the form S 1 × M 2 {S^1} \times {M^2} . The proofs of Theorems A and B use the fact that θ \theta is induced by a homotopy equivalence f : ( M , m ) → ( M ′ , m ′ ) f:(M,\,m) \to (M\prime ,\,m\prime ) . Theorem C. The manifold M M satisfies the Visibility axiom if and only if M ′ M\prime satisfies the Visibility axiom.