The Poincar\'e-Hopf Theorem for relative braid classes

Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid classes is non-trivial, then additional closed integral curves of the Hamiltonian equations are forced via a Morse type theory. In this article we show that certain information contained in the braid Floer homology - the Euler-Floer characteristic - also forces closed integral curves and periodic points of arbitrary vector fields and diffeomorphisms and leads to a Poincar\'e-Hopf type Theorem. The Euler-Floer characteristic for any proper relative braid class can be computed via a finite cube complex that serves as a model for the given braid class. The results in this paper are restricted to the 2-disc, but can be extend to two-dimensional surfaces (with or without boundary).

A 2-periodic and 1-periodic closed integral curve represented as a 3-strand braids [left]. A closed integral curve go minimal period yields a -periodic point of the associated time-1 map [middle]. A relative braid with the skeleton (black) and one free strand (red) [right].
x i are integral curves of X and cannot intersect therefore! Multiple closed integral curves of various periods yield a multi-strand braid.
Let y be a geometric braid consisting of closed integral curves of X, which will be referred to as a skeleton. The strands y i (t), i = 1, · · · , m satisfy the periodicity condition y(0) = y(1) as point sets, i.e. y i (0) = y σ(i) (1) for some permutation σ ∈ S m . Let x = x 1 (t), · · · x n (t) be a geometric braid such that the 'union' x rel y := x 1 (t), · · · x n (t), y 1 (t), · · · , y m (t) is again a geometric braid, i.e. the strands in x do not intersect the strands in y. The pair x rel y is called a relative braid. Two relative braids x rel y and x rel y are equivalent if there exists a homotopy of relative braids connecting x rel y to x rel y . The equivalence class is denoted by [x rel y] and is called a relative braid class. The set of relative braids x rel y ∈ [x rel y], keeping y fixed, is denoted by [x ] rel y and is called a braid class fiber. A relative braid class [x rel y] is proper if components x c ⊂ x cannot be deformed onto (i) the boundary ∂D 2 , (ii) itself, 2 or other components x c ⊂ x, or (iii) components in y c ⊂ y, see [12] for details. In this paper we are concerned with relative braids x rel y for which x has only one strand, i.e. x(t + 1) = x(t). The central question is: given a skeleton y of integral curves, does there exist an integral curve x in the relative braid class [x rel y]. The theory also applies if x has more than one strand. To a braid y one can assign an integer Cross(y) which counts the number of crossings (with sign) of strands in the standard planar projection. In the case of a relative braid x rel y the number Cross(x rel y) is an invariant of the relative braid class [x rel y].

A brief summary of Braid Floer homology
In [12] a monotonicity lemma is proven, which states that along solutions u(s,t) of the nonlinear Cauchy-Riemann equations, the number Cross(u(s, ·) rel y) is nonincreasing (the jumps correspond to 'singular braids', i.e. 'braids' for which intersections occur). As a consequence an isolation property for proper relative braid classes exists: the set bounded solutions of the Cauchy-Riemann equations in a proper braid class fiber [x] rel y, denoted by M ([x] rel y; H), is compact and isolated with respect to the topology of uniform convergence on compact subsets of R 2 . These facts provide all the ingredients to use Floer's approach towards Morse Theory for the Hamiltonian action [7]. For generic Hamiltonians which satisfy (i) and (ii) above and for which y is a skeleton, the critical points in In [12] Braid Floer homology is used as Morse type theory for closed integral curves forced by a skeleton y. The Floer theory applies to the Hamilton action and therefore to Hamiltonian vector fields X H . It also provides a forcing theory for periodic points for area-preserving diffeomorphisms. Braid Floer homology cannot be applied to arbitrary vector fields X. The objective of this paper is twofold: (i) we extract an invariant from HB * ([x rel y]) -the Euler-Floer characteristic -that applies to arbitrary vector fields X and non-triviality of the Euler-Floer characteristic yields forcing of closed integral curves, and (ii) we develop an algorithm to compute the Euler-Floer characteristic.
The remainder of the introduction deals with formulating the main theorems and gives an extensive outline of the steps required to prove the main results.

The Euler-Floer characteristic and the Poincaré-Hopf Formula
The Euler-Floer characteristic of HB * [x rel y] is defined as follows: A 1-periodic function x ∈ C 1 (R/Z) is an isolated closed integral curve of X if there exists an ε > 0 such that x is the only solution of the differential equation in B ε (x) ⊂ C 1 (R/Z). For isolated, and in particular non-degenerate closed integral curves we can define an index as follows: Then η → F Θ (η) = d dt − R(t; η) defines a path in Fred 0 (C 1 ,C 0 ). Denote by Σ ⊂ Fred 0 (C 1 ,C 0 ) the set of non-invertible operators and by Σ 1 ⊂ Σ the non-invertible operators with a 1-dimensional kernel. If the end points of F are invertible one can choose the path η → R(t; η) such that F Θ (η) intersects Σ in Σ 1 and all intersections are transverse. Define γ = # intersections of F Θ (η) with Σ 1 , then This definition is independent of the choice of Θ, see Section 7. In Section 7 we also extend the definition of index to isolated closed integral curves, see Equation (32).
Theorem 1 (Poincaré-Hopf Formula) Let y be a skeleton of closed integral curves of a vector field X ∈ X (D 2 × R/Z) and let [x rel y] be a proper relative braid class. Suppose that all 1-periodic closed integral curves of X are isolated, then for all closed integral curves x 0 rel y in the fiber [x 0 ] rel y it holds that The index formula can be used to obtain existence results for closed integral curves in proper relative braid classes.
Theorem 2 Let y be a skeleton of closed integral curves of a vector field X ∈ X (D 2 × R/Z) and let [x rel y] be a proper relative braid class. If χ x rel y = 0, then there exist closed integral curves x 0 rel y in the fiber [x] rel y. Remark 1 In this paper we do not address the question whether the closed integral curves x rel y are non-constant, i.e. are not equilibrium points. However, closed integral curves in different relative braid classes correspond to different periodic points! By considering relative braid classes where x consists of more than one strand one can study non-constant closed integral curves. Braid Floer homology for relative braids with x consisting of n strands is defined in [12]. The ideas in this paper extend to relative braid classes with multi-strand braids x. In Section 11 we give an example of a multi-strand x in x rel y and explain how this yields the existence of non-trivial closed integral curves, which also provides detailed information about the linking of solutions.

Discretization and computability
The second part of the paper deals with the computability of the Euler-Floer characteristic. This is obtained through a finite dimensional model. The latter is constructed in three steps: (a) compose x rel y with ≥ 0 full twists ∆ 2 , such that (x rel y) · ∆ 2 is isotopic to a positive braid x + rel y + ; (b) relative braids x + rel y + are isotopic to Legendrian braids x L rel y L on R 2 , i.e. braids which have the form x L = (q t , q) and y L = (Q t , Q), where q = π 2 x and Q = π 2 y, and π 2 is the projection onto the q−coordinate; (c) discretize q and Q = {Q j } to q d = {q i }, with q i = q(i/d), i = 0, . . . , d and respectively, and consider the piecewise linear interpolations connecting the anchor points q i and Q j i for i = 0, . . . , d, see Figure 2. A discretization q D rel Q D is admissible if the linear interpolation is isotopic to q rel Q. All such discretizations form the discrete relative braid class [q D rel Q D ], for which each fiber is a finite cube complex, cf. [8]. Remark 2 If the number of discretization points is not large enough, then the discretizations may not be admissible and therefore not capture the topology of the braid class. See [8] and Section 10.1 for more details.
For d > 0 large enough there exists an admissible discretization q D rel Q D for any Legendrian representative x L rel y L ∈ [x rel y] and thus an associated discrete relative braid class [q D rel Q D ]. In [8] an invariant for discrete braid classes was introduced.
, which is a cube complex with a finite number of connected components and their closures are denoted by N j . The faces of the hypercubes N j can be co-oriented in direction of decreasing the number of crossings in q D rel Q D , and we define N − j as the closure of the set of faces with outward pointing co-orientation. Figure 3 below explains the sets N j and N − j for the example in Figure 2. The sets N − j are called exit sets. The invariant for a fiber is given by the Conley index This discrete braid invariant is well-defined for any d > 0 for which there exist admissible discretizations and is independent of both the particular fiber and the discretization size d. For the associated Euler characteristic we therefore write χ q D rel Q D . The Euler characteristic of the Braid Floer homology χ(x rel y) can be related to the Euler characteristic of the associated discrete braid class.
Theorem 3 Let [x rel y] a proper relative braid class and ≥ 0 is an integer such that (x rel y) · ∆ 2 is isotopic to a positive braid x + rel y + . Let q D rel Q D be an admissible discretization, for some d > 0, of a Legendrian representative x L rel y L ∈ [x + rel y + ]. Then D is an augmentation of Q D by adding the constant strands ±1 to Q D .

Outline of the paper
The first part of the paper, Sections 2 through 7, is concerned with proving Theorems 1 and 2. The second part of the paper, Sections 8 through 11, is concerned with proving Theorem 3, which deals with computing the Euler-Floer characteristic and applying it to specific braid classes. We will now give an outline of the main steps to prove Theorems 1, 2 and 3 and in which sections these steps are proved.
Degree Homotopy. For a skeleton y for X we can find a Hamilton vector field X H and thus y is a skeleton for the homotopy X α = (1 − α)X + αX H , for all α ∈ The Euler-Floer characteristic. In [12] it is proved that generically see Proposition 7. If we combine all the steps we obtain The index. In Section 7 we introduce the index ι(x) given in Equation (3) for closed integral curves x, which does not depend on the choices we used to define Φ(x).
The index relates directly to the Leray-Schauder degree. We use the index ι(x) to formulate the Poincaré-Hopf Formula and prove Theorems 1 and 2.
Legendrian braids. In order to compute the Euler-Floer characteristic we represent braid classes via Legendrian braids of the form x = (q t , q), where q is a 1periodic function, see Section 8. Legendrian braids can be realized via mechanical Hamiltonian systems for which the Conley-Zehnder index equals the Morse index of q, see Lemma 13. In Equation (40) we express the Euler-Floer characteristic by χ(x rel y) = ∑ q (−1) γ(q) , where γ(q) is the Morse index of a critical point.
Discrete braid classes. The final step toward proving Theorem 3 uses yet another representation of braids. Via the method of broken geodesics, Legendrian braids are discretized and represented via parabolic recurrence relations of conservative type, see [8]. For such braid classes the Conley index of proper relative braid classes is well-defined, see Section 10. In Lemma 15 we show that the Morse index of the discretized critical points is equal the the Morse index γ(q). The theory in [8]

Closed integral curves
In this section we rephrase the existence of a closed integral curves in terms of zeroes of appropriate mapping on C 0 (R/Z). Let X ∈ X (D 2 × R/Z), then closed integral curves of X of period 1 satisfy the differential equation Consider the unbounded operator L µ : The operator is invertible for µ = 2πk, k ∈ Z and the inverse L −1 µ : where K is a (non-linear) compact operator on C 0 (R/Z). Since X is a smooth vector field the mapping Φ is a smooth mapping on C 0 (R/Z).
Proposition 1 A function x ∈ C 0 (R/Z), with |x(t)| ≤ 1 for all t, is a solution of Φ(x) = 0 if and only if x ∈ C 1 (R/Z) and x satisfies Equation (5).
Note that the zero set Φ −1 (0) does not depend on the parameter µ. In order to apply the Leray-Schauder degree theory we consider appropriate bounded, open subsets Ω ⊂ C 0 (R/Z), which have the property that Φ −1 (0) ∩ ∂Ω = ∅. Let y be a skeleton for X consisting of closed integral curves and consider a proper relative braid class [x rel y]. Due to properness of [x rel y] all fibers [x] rel y are isolating neighborhoods for the Cauchy-Riemann equations, cf. [12], and in particular for Equation (5). Therefore, we consider Ω = [x] rel y.
Proposition 2 Let [x rel y] be a proper relative braid class and let Ω = [x] rel y be the fiber given by y. Then, there exists an 0 < r < 1 such that Let x n ∈ Φ −1 (0) ∩ Ω and assume that such an 0 < r < 1 does not exist. Then, by the compactness of Φ −1 (0) ∩ Ω, there is a subsequence x n k → x such that one, or both of the following two possibilities hold: (i) |x(t 0 )| = 1 for some t 0 . By the uniqueness of solutions of Equation (5) and the invariance of the boundary ∂D 2 (X(x,t) is tangent to the boundary), |x(t)| = 1 for all t, which is impossible since [x] rel y is proper; (ii) x(t 0 ) = y j (t 0 ) for some t 0 and some j. As before, by the uniqueness of solutions of Equation (5), then x(t) = y j (t) for all t, which again contradicts the fact that [x] rel y is proper.
By Proposition 2 the Leray-Schauder degree deg LS (Φ, Ω, 0) is well-defined. Consider the Hamiltonian vector field where H(x,t) is a smooth Hamiltonian satisfying (i)-(ii) in Section 1.1, and therefore, X H ∈ X (D 2 × R/Z). Assume that y is a skeleton for X H . Such Hamiltonians can always be constructed, see [12], and the set of Hamiltonians meeting these requirements is denote by H (y). Associate with the vector field X H we write Since y is a skeleton for both X and X H , it is also a skeleton for the linear homotopy X α = (1 − α)X + αX H , α ∈ [0, 1]. Associated with the homotopy X α of vector fields we define the homotopy Proposition 2 applies for all α ∈ [0, 1], i.e. by compactness there exists a uniform 0 < r < 1 such that for all t ∈ R, for all j and for all By the homotopy invariance of the Leray-Schauder degree we have where Φ 0 = Φ and Φ 1 = Φ H . Note that the zeroes of Φ H correspond to critical point of the functional and are denoted by Crit A H ([x] rel y). The Braid Floer homology groups HB * [x rel y] , defined [12], provide information about Φ −1 In the next section we examine spectral properties of the solutions of Φ −1 α (0) ∩ Ω in order to compute deg LS (Φ H , Ω, 0) and thus deg LS (Φ, Ω, 0).

Remark 3
There is obviously more room for choosing appropriate operators L µ and therefore functions Φ. In Section 7 this issue will be discussed in more detail.

The Leray-Schauder degree and parity
The Leray-Schauder degree of an isolated zero . If x is a non-degenerate zero, then it is an isolated zero and the degree can be determined from spectral information.

The Leray-Schauder degree
which will be referred to as the Morse index of x, or alternatively the Morse index of linearized operator D x Φ(x).
The functions Φ α (x) = x − K α (x) are of the form 'identity + compact' and Proposition 3 can be applied to non-degenerate zeroes of Φ α (x) = 0. If we choose the Hamiltonian 'generically', then the zeroes of . By compactness there are only finitely many zeroes in a fiber Ω = [x] rel y.
Then following criteria for non-degeneracy are equivalent: Proof A function ψ satisfies D x K H (x)ψ = ψ if and only if Bψ = 0, which shows the equivalence between (i) and (ii). The equivalence between (ii) and (iii) is proved in [12].
The generic choice of H follows from Proposition 7.1 in [12] based on criterion (iii). Hamiltonians for which the zeroes of Φ H are non-degenerate are denoted by H reg (y). Note that no genericity is needed for α ∈ [0, 1)! For the Leray-Schauder degree this yields The goal is to determine the Leray-Schauder degree deg LS (Φ, Ω, 0) from information contained in the Braid Floer homology groups HB * ([x rel y]). In order to do so we examen the Hamiltonian case. In the Hamiltonian case the linearized operator which is a bounded operator on C 0 (R/Z). The operator A extends to a bounded operator on L 2 (R/Z). Consider a path η → A(η), η ∈ I = [0, 1], given by where η → S(t; η) a smooth path of t-dependent symmetric matrices with the ends satisfying where θ = 2πk, for some k ∈ Z and D 2 x H(x(t),t) is the Hessian of H at a critical point in Crit A H ([x] rel y). The path of η → A(η) is a path bounded linear Fredholm operators on L 2 (R/Z) of Fredholm index 0, which are compact perturbations of the identity and whose ends are invertible. (10) is a smooth path of bounded linear Fredholm operators in H s (R/Z) of index 0, with invertible ends.
Proof By the smoothness of S(t; η) we have that S(t; η)x H m ≤ C x H m , for any x ∈ H m (R/Z) and any m ∈ N ∪ {0}. By interpolation the same holds for all x ∈ H s (R/Z) and the claim follows from the fact that L −1 µ : In the Hamiltonian case the Conley-Zehnder indices can be related to the spectral flow of self-adjoint operators. However, the paths A(η) are not self-adjoint however, and thus spectral flow cannot be used in general. We therefore need cruder tool to link the Morse indices β H (x) to the Conley-Zehnder indices. Parity is such an invariant for paths of Fredholm operators and is related to spectral flow.

Parity of paths of linear Fredholm operators
Let η → Λ(η) be a smooth path of bounded linear Fredholm operators of index 0 on a Hilbert space H . A crossing η 0 ∈ I is a number for which the operator Λ(η 0 ) is not invertible. A crossing is simple if dim ker Λ(η 0 ) = 1. A path η → Λ(η) between invertible ends can always be perturbed to have only simple crossings. Such paths are called generic. Following [4, 3, 5, 6], we define the parity of a generic path η → Λ(η) by where cross(Λ(η), I) = #{η 0 ∈ I : ker A(η 0 ) = 0}. The parity is a homotopy invariant with values in Z 2 . In [4, 3, 5, 6] an alternative characterization of parity is given via the Leray-Schauder degree. For any Fredholm path η → Λ(η) there exists a path η → M(η), called a parametrix, such that η → M(η)Λ(η) is of the form 'identity + compact'. For parity this gives: , H , 0 , for η = 0, 1, and the expression is independent of the choice of parametrix. The latter extends the above definition to arbitrary paths with invertible ends. For a list of properties of parity see [4, 3, 5, 6].
Proposition 4 Let η → A(η) be a path of bounded linear Fredholm operators on H s (R/Z) defined by (10). Then where β A(0) and β A(1) are the Morse indices of A(0) and A(1) respectively.
is a non-degenerate zero, then its local degree can be expressed in terms of the parity of A(η).
In the next subsection we establish yet another path of operators such that we can link the local degree to the Conley-Zehnder indices of the critical points.

Parity and spectral flow
The spectral flow for paths of selfadjoint operators is a more refined invariant than parity. Using the Fourier expansion x = ∑ k∈Z e 2πJkt x k we can characterize we define the selfadjoint operators For µ > 0 and µ = 2πk, k ∈ Z, the operators P µ are isomorphisms on H s (R/Z), for all s ≥ 0. 3 Consider the path which is a path of operators of Fredholm index 0. The constant path η → M µ (η) = P −1 µ is a parametrix for η → C(η) (see [5,6]) and since M µ C(η) = A(η), the parity of C(η) is given by parity(C(η), I) = parity(A(η), I).
Proof From the functional calculus we derive that where n µ (k) = 2π|k| + µ and p µ (k) = 2πk+µ 2π|k|+µ . For s = 1/2 we have that For a path η → Λ(η) of selfadjoint operators on a Hilbert space H , which is continuously differentiable in the (strong) operator topology we define the crossing oper- with invertible ends can always be chosen to be generic by a small perturbation. At a simple crossing η 0 , there exists a C 1 -curve λ(η), for η near η 0 , and λ(η) is an eigenvalue of Λ(η), with λ(η 0 ) = 0 and λ (η 0 ) = 0, see [10,11]. The spectral flow for a generic path is defined by For a simple crossing η 0 the crossing operator is simply multiplication by λ (η 0 ) and where ψ(η 0 ) is normalized in H , and The spectral flow is defined any for continuously differentiable path η → Λ(η) with invertible ends. From the theory in [6] there is a connection between the spectral flow of Λ(η) and its parity: which in view of Equation (11) follows since cross(Λ(η), I) = specflow(Λ(η), η) mod 2 in the generic case. The path η → C(η) defined in (15) is a continuously differentiable path of operators on H = H 1/2 (R/Z) with invertible ends, and therefore both parity and spectral flow are well-defined. If we combine Equations (13) and (16) with Equation (20) we obtain In the next section we link the spectral flow of C(η) to the Conley-Zehnder indices of non-degenerate zeroes and therefore to the Euler-Floer characteristic.

The Conley-Zehnder index
For a non-degenerate 1-periodic solution x(t) of the Hamilton equations the Conley-Zehnder index can be defined as follows. The linearized flow Ψ is given by By Lemma 1(iii), a 1-periodic solution is non-degenerate if Ψ(1) has no eigenvalues equal to 1. The Conley-Zehnder index is defined using the symplectic path Ψ(t). Following [11], consider the crossing form Γ(Ψ,t), defined for vectors ξ ∈ ker(Ψ(t)− Id), A crossing t 0 > 0 is defined by det(Ψ(t 0 ) − Id) = 0. A crossing is regular if the crossing form is non-singular. A path t → Ψ(t) is regular if all crossings are regular. Any path can be approximated by a regular path with the same endpoints and which is homotopic to the initial path, see [10] for details. For a regular path t → Ψ(t) the Conley-Zehnder index is given by and at simple crossings η 0 , after normalizing φ(η 0 ) in L 2 (R/Z). As before the derivative of at η 0 is given by Proposition 6 Let η → B(η), η ∈ I, as defined above, be a generic path of unbounded self-adjoint operators with invertible endpoints, and let η → Ψ(η;t) be the associated path of symplectic matrices defined by where µ CZ B(0) = µ CZ (Ψ(t; 0)), µ CZ B(1) = µ CZ (Ψ(t; 1)).
Proof The expression for the spectral flow follows from [11] and [12].
In the case η = 0, the Conley-Zehnder index µ CZ B(0) can be computed explicitly.
The zeroes  (1), is given by χ x rel y = ∑ k∈Z (−1) k dim HB k ([x rel y]), which is also the Euler-Floer characteristic for HB * ([x] rel y). In [12] the following analogue of the Poincaré-Hopf formula is proved.
Since the Conley-Zehnder index is linked to spectral flow we obtain: Proposition 8 For a proper braid class fiber [x] rel y and a generic Hamiltonian H ∈ H reg (y), we have that Proof By Proposition 6 and Lemma 5 the spectral flow satisfies, This implies which completes the proof.
In order to prove that the Euler-Floer characteristic χ(x rel y) and deg LS (Φ H , Ω, 0) are related we need to investigate the relation between the spectral flows of B(η) and C(η).
6 The spectral flows are the same In order to show that the spectral flows are the same we use the fact that the paths η → C(η) and η → B(η) for a non-degenerate zero x i ∈ Φ −1 H (0) ∩ Ω are chosen to have only simple crossings for their crossing operators, i.e. zero eigenvalues are simple. In this case the spectral flows are determined by the signs of the derivatives of the eigenvalues at the crossings. For η → B(η) the expression given by Equation (26) and from Equation (19) a similar expression for η → C(η) can be derived and is given by Lemma 7 For all µ > 0, with µ = 2πk, k ∈ Z, sgn λ (η 0 ) = sgn (η 0 ) for all crossings at η 0 .
Lemma 7 implies that for any non-degenerate where B(η; x) and C(η; x) are the above described path associated with x. Therefore which completes the proof.

The proof of Theorems 1 and 2
We start with the proof of Theorem 2. Since HB * ([x rel y]) is an invariant of the proper braid class [x rel y] it does not depend on a particular fiber [x] rel y. Recall the homotopy invariance of the Leray-Schauder degree as expressed in Equation (7) deg By Proposition 9 we have that and χ(x rel y) = 0, then implies that Φ −1 (0) ∩ Ω = ∅. Therefore there exist a closed integral curves in any relative braid class fiber [x] rel y, whenever χ(x rel y) = 0. This completes the proof of Theorem 2. The remainder of this section is to prove the Poincaré-Hopf Formula in Theorem 1 for closed integral curves in proper relative braid fibers. The mapping is smooth (nonlinear) Fredholm mapping of index 0. Let M ∈ GL(C 0 ,C 1 ) be an isomorphism such that ME (x) is of the form ME ( Let x ∈ C 1 (R/Z) be a non-degenerate zero of E and recall, using the theory in [4, 3, 5, 6], that the index ι(x) defined in (3) is equivalent to: where Θ ∈ M 2×2 (R), with σ(Θ) ∩ 2πkiR = ∅, k ∈ Z and β M (Θ) is the Morse index of Id −K M (0).

Lemma 8
The index ι(x) for a non-degenerate zero of E is well-defined, i.e. independent of the choices of M ∈ GL(C 0 ,C 1 ) and Θ ∈ M 2×2 (R).
Lemmas 8 shows that the index of a non-degenerate zero of E is well-defined. We now show that the same holds for isolated zeroes.

Lemma 9
The index ι(x) for an isolated zero of E is well-defined and for a fixed choice of M and Θ the index is given by where ε > 0 is small enough such that x is the only zero of E in B ε (x).
Proof By the Sard-Smale Theorem one can choose an arbitrarily small h ∈ C 0 (R/Z), h C 0 < ε , such that h is a regular value of E and E −1 (h) ∩ B ε (x) consists of finitely many non-degenerate zeroes in x h . Set E (x) = E (x) − h and define We now show that ι(x) is well-defined. Choose a fixed parametrix M (for E ) and fixed Θ ∈ M 2×2 (R), and let Φ M = M E , then which proves the lemma.
Theorem 1 now follows from the Leray-Schauder degree. Suppose all zeroes of E in Ω = [x] rel y are isolated, then Lemma 9 implies that

Legendrian braids
In this section we prove Theorem 3 and show that the Euler-Floer characteristic can be determined via a discrete topological invariant.

Hyperbolic Hamiltonians on R 2
Consider Hamiltonians of the form where h satisfies the following hypotheses: Since h is smooth we can rewrite the equations as If x(t) is a 1-periodic solution to the Hamilton equations, and suppose there exists an interval I = [t 0 ,t 1 ] ⊂ [0, 1] such that |q(t)| > R on int(I) and |q(t)| ∂I = R. The function q| I satisfies the equation q tt −q = 0, and obviously such solutions do not exist. Indeed, if q| I ≥ R, then q t (t 0 ) ≥ 0 and q t (t 1 ) ≤ 0 and thus 0 ≥ q t | ∂I = I q ≥ R|I| > 0, a contradiction. The same holds for q| I ≤ −R. We conclude that |q(t)| < R, for all t ∈ R/Z.
We now use the a priori q-estimate in combination with Equation (35) and Hypothesis (h3). Multiplying Equation (35) by q and integrating over [0, 1] gives: which implies that 1 0 q 2 t ≤ C(R). The L 2 -norm of the right hand side in (35) can be estimated using the L ∞ estimate on q and the L 2 -estimate on q t , which yields 1 0 q 2 tt ≤ C(R). Combining these estimates we have that q H 2 (R/Z) ≤ C(R) and thus |q t (t)| ≤ C(R), for all t ∈ R/Z. From the Hamilton equations it follows that |p(t)| ≤ |q t (t)| +C, which proves the lemma. Proof The a priori H 2 -estimates in Lemma 10 hold with uniform constants with respect to α ∈ [0, 1]. This then proves the lemma.

Braids on R 2 and Legendrian braids
In Section 1 we defined braid classes as path components of closed loops in LC n (D 2 ), denoted by [x]. If we consider closed loops in C n (R 2 ), then the braid classes will be denoted by [x] R 2 . The same notation applies to relative braid classes [x rel y] R 2 . A relative braid class is proper if components x c ⊂ x cannot be deformed onto (i) itself, or other components x c ⊂ x, or (ii) components y c ⊂ y. A fiber [x] R 2 rel y is not bounded! In order to compute the Euler-Floer characteristic of [x rel y] we assume without loss of generality that x rel y is a positive representative. If not we compose x rel y with a sufficient number of positive full twists such that the resulting braid is positive, i.e. only positive crossings, see [12] for more details. The Euler-Floer characteristic remains unchanged. We denote a positive representative x + rel y + again by x rel y.
Define an augmented skeleton y * by adding the constant strands y − (t) = (0, −1) and y + (t) = (0, 1). For proper braid classes it holds that [x rel y] = [x rel y * ]. For notational simplicity we denote the augmented skeleton again by y. We also choose the representative x rel y with the additional the property that π 2 x rel π 2 y is a relative braid diagram, i.e. there are no tangencies between the strands, where π 2 the projection onto the q-coordinate. We denote the projection by q rel Q, where q = π 2 x and Q = π 2 y. Special braids on R 2 can be constructed from (smooth) positive braids. Define x L = (q t , q) and y L = (Q t , Q), where the subscript t denotes differentiating with respect to t. These are called Legendrian braids with respect to θ = pdt − dq.
Lemma 12 For positive braid x rel y with only transverse, positive crossings, the braids x L rel y L and x rel y are isotopic as braids on R 2 . Moreover, if x L rel y L and x L rel y L are isotopic Legrendrian braids, then they are isotopic via a Legendrian isotopy.
Proof By assumption x rel y is a representative for which the braid diagram q rel Q has only positive transverse crossings. Due to the transversality of intersections the associated Legendrian braid x L rel y L is a braid [x rel y] R 2 . Consider the homotopy ζ j (t, τ) = τp j (t) + (1 − τ)q j t , for every strand q j . At q-intersections, i.e. times t 0 such that q j (t 0 ) = q j (t 0 ) for some j = j , it holds that p j (t 0 ) − p j (t 0 ) and q j t (t 0 ) − q j t (t 0 ) are non-zero and have the same sign since all crossings in x rel y are positive! Therefore, ζ j (t 0 , τ) = ζ j (t 0 , τ) for any intersection t 0 and any τ ∈ [0, 1], which shows that x rel y and x L rel y L are isotopic. Since x L rel y L and x L rel y L have only positive crossings, a smooth Legendrian isotopy exists.
The associated equivalence class of Legendrian braid diagrams is denoted by [q rel Q] and its fibers by [q] rel Q.

Mechanical Hamiltonian systems
Legendrian braids can be described via Hamiltonian of mechanical type, i.e. H(x,t) = 1 2 p 2 +V (q,t). Due to the special form we can also use the Lagrangian formalism for such systems. In the next subsection we investigate the relation between the Conley-Zehnder index and the Lagrangian Morse index of closed integral curves.

The Lagrangian Morse index
A mechanical system is defined as the Euler-Lagrange equations of the Lagrangian density L(q,t) = 1 2 q 2 t − V (q,t). The linearization at a critical points q(t) of the Lagrangian action is given by the unbounded opeartor For a mechanical system the Hamiltonian is given by H(x,t) = 1 2 p 2 +V (q,t). As such the Conley-Zenhder index of a critical point q can be defined as the Conley-Zehnder index of x = (q t , q), see also [1] and [2]. The Morse index of a critical point is defined as Lemma 13 Let q be a critical point of the mechanical Lagrangian action, then the associated Conley-Zehnder index µ CZ (x) is well-defined, and µ CZ (x) = γ(q).

The Poincaré-Hopf Furmula and the Morse index
Legendrian braids can be described with Lagrangian systems and Hamiltonians of the form H L (x,t) = 1 2 p 2 − 1 2 q 2 + g(q,t). On the potential functions g we impose the following hypotheses: In order to have a straightforward construction of a mechanical Lagrangian we may consider a special representation of y. The Euler-Floer characteristic χ x rel y does not depend on the choice of the fiber [x] rel y and therefore also not on the skeleton y. We assume that y has linear crossings in y L . Let t = t 0 be a crossing and let I(t 0 ) be the set of labels defined by: i, j ∈ I(t 0 ), if i = j and Q i (t 0 ) = Q j (t 0 ). A crossing at t = t 0 is linear if Q i t (t) = constant, ∀i ∈ I(t 0 ), and ∀t ∈ (−ε + t 0 , ε + t 0 ), for some ε = ε(t 0 ) > 0. Every skeleton Q with transverse crossings is isotopic to a skeleton with linear crossings via a small local deformation at crossings. For Legendrian braids x L rel y L ∈ [x rel y] R 2 with linear crossings the following result holds: Lemma 14 Let y L be a Legendrian skeleton with linear crossings. Then, there exists a Hamiltonian of the form H L (x,t) = 1 2 p 2 − 1 2 q 2 +g(q,t), with g satisfying Hypotheses (g1)-(g2), and R > 0 sufficiently large, such that y L is a skeleton for X H L (x,t).
Proof Due to the linear crossings in y L we can follow the construction in [12]. For each strand Q i we define the potentials g i (t, x) = −Q i tt (t)q. By construction Q i is a solution of the equation Q i tt = −g i q (t, Q i ). Now choose small tubular neighborhoods of the strands Q i and cut-off functions ω i that are equal to 1 near Q i and are supported in the tubular neighborhoods. If the tubular neighborhoods are narrow enough, then supp(ω i g i ) ∩ supp(ω j g j ) = ∅, for all i = j, due to the fact that at crossings the functions g i in question are zero. This implies that all strands Q i satisfy the differential equation t) is compactly supported. The latter follows from the fact that for the constant strands Q i = ±1, the potentials g i vanish. Let R > 1 and definẽ where m = #Q, which yields smooth functionsg i on R × R/Z. Now define By construction supp(g) ⊂ [−R, R] × R/Z, for some R > 1 and the strands Q i all satisfy the Euler-Lagrange equations Q i tt = Q i − g qq (Q i ,t), which completes the proof.
The Hamiltonian H L given by Lemma 14 gives rise to a Lagrangian system with the Lagrangian action given by The braid class [q] rel Q is bounded due to the special strands ±1 and all free strands q satisfy −1 ≤ q(t) ≤ 1. Therefore, the set of critical points of L in [q] rel Q is a compact set. The critical points of L in [q] rel Q are in one-to-one correspondence with the zeroes of the equation in the set Ω R 2 = [x L ] R 2 rel y L , which implies that Φ H L is a proper mapping on Ω R 2 . From Lemma 10 we derive that the zeroes of Φ H L are contained in ball in R 2 with radius R > 1, and thus Φ −1 H L (0) ∩ Ω R 2 ⊂ B R (0) ⊂ C 1 (R/Z). Therefore the Leray-Schauder degree is well-defined and in the generic case Lemma 13 and Equations (21), (27) and (30) yield We are now in a position to use a homotopy argument. We can scale y to a braid ρy such that the rescaled Legendrian braid ρy L is supported in D 2 . By Lemma 12, y is isotopic to y L and scaling defines an isotopy between y L and ρy L . Denote the isotopy from y to ρy L by y α . By Proposition 9 we obtain that for both skeletons y and ρy L it holds that where Ω ρ = [ρx L ] rel ρy L ⊂ [x rel y] and H ρ ∈ H (ρy L ). Now extend H ρ to R 2 × R/Z, such that Hypotheses (h1)-(h3) are satisfied for some R > 1. We denote the Hamiltonian again by H ρ . By construction all zeroes of Φ H ρ in [ρx L ] rel ρy L are supported in D 2 and therefore the zeroes of Φ H ρ in [ρx L ] R 2 rel ρy L are also supported in D 2 . Indeed, any zero intersects D 2 , since the braid class is proper and since ∂D 2 is invariant for the Hamiltonian vector field, a zero is either inside or outside D 2 . Combining these facts implies that a zero lies inside D 2 . This yields where Ω ρ,R 2 = [ρx L ] R 2 rel ρy L . For the next homotopy we keep the skeleton ρy L fixed as well as the domain Ω ρ,R 2 . Consider the linear homotopy of Hamiltonians where H ρ,L (t, x) = 1 2 p 2 − 1 2 q 2 + g ρ (q,t) given by Lemma 14. This defines an admissible homotopy since ρy L is a skeleton for all α ∈ [0, 1]. The uniform estimates are obtained, as before, by Lemma 11, which allows application of the Leray-Schauder degree: Finally, we scale ρy L to y L via y α,L = (1−α)ρy L +αy L and we consider the homotopy between H L and H ρ,L , where g(q,t; α) is found by applying Lemma 14 to y α,L . The uniform estimates from Lemma 11 allows us to apply the Leray-Schauder degree: Combining the equalities for the various Leray-Schauder degrees with (39) yields: 10 The proof of Theorem 3 The Lagrangian setting introduced in the previous section allows for another simplification via finite dimensional systems.

Discretized braid classes
The Lagrangian problem (38) can be treated by using a variation on the method of broken geodesics. If we choose 1/d > 0 sufficiently small, the integral has a unique minimizer , and τ i = i/d. Moreover, if 1/d is small, then the minimizers are non-degenerate and S i is a smooth function of q i and q i+1 . Critical points q of L with |q(t)| ≤ 1 correspond to sequences q D = (q 0 , · · · , q d ), with q 0 = q d , which are critical points of the discrete action A concatenation # i q i of minimizers q i is continuous and is an element in the function space H 1 (R/Z), and is referred to as a broken geodesic. The set of broken geodesics # i q i is denoted by E(q D ) and standard arguments using the non-degeneracy of min- and yields the following commuting diagram: In the above diagram # i is regarded as a mapping q D → # i q i , where the minimizers q i are determined by q D . The tangent space to E(q D ) at a broken geodesic # i q i is identified by and # i q i + T # i q i E(q D ) is the tangent hyperplane at # i q i . For H 1 (R/Z) we have the following decomposition for any broken geodesic # i q i ∈ E(q D ): where E = {η ∈ H 1 (R/Z) | η(τ i ) = 0, ∀i}. To be more specific the decomposition is orthogonal with respect to the quadratic form Indeed, let η ∈ E and ψ ∈ T # i q i E(q D ), then by the above orthogonality. By construction the minimizers q i are non-degenerate and therefore D 2 L | E is positive definite. This implies that the Morse index of a (stationary) broken geodesic is determined by D 2 L | T # i q i E(q D ) . By the commuting diagram for W this implies that the Morse index is given by quadratic form D 2 W (q D ). We have now proved the following lemma that relates the Morse index of critical points of the discrete action W to Morse index of the 'full' action L .
Lemma 15 Let q be a critical point of L and q D the corresponding critical point of W , then the Morse indices are the same i.e. γ(q) = γ(q D ).
For a 1-periodic function q(t) we define the mapping and q D is called the discretization of q. The linear interpolation reconstructs a piecewise linear 1-periodic function. For a relative braid diagram q rel Q, let q D rel Q D be its discretization, where Q D is obtained by applying D d to every strand in Q. A discretization q D rel Q D is admissible if q D rel Q D is homotopic to q rel Q, i.e. q D rel Q D ∈ [q rel Q]. Define the discrete relative braid class [q D rel Q D ] as the set of 'discrete relative braids' q D rel Q D , such that q D rel Q D ∈ [q rel Q]. The associated fibers are denoted by [q D ] rel Q D . It follows from [8], Proposition 27, that [q D rel Q D ] is guaranteed to be connected when d > #{ crossings in q rel Q}, i.e. for any two discrete relative braids q D rel Q D and q D rel Q D , there exists a homo- Note that fibers are not necessarily connected! For a braid classes [q rel Q] the associated discrete braid class [q D rel Q D ] may be connected for a smaller choice of d.
We showed above that if 1/d > 0 is sufficiently small, then the critical points of L , with |q| ≤ 1, are in one-to-one correspondence with the critical points of W , and their Morse indices coincide by Lemma 15. Moreover, if 1/d > 0 is small enough, then for all critical points of L in [q] rel Q, the associated discretizations are admissible and [q D rel Q D ] is a connected set. The discretizations of the critical points of L in [q] rel Q are critical points of W in the discrete braid class fiber [q D ] rel Q D . Now combine the index identity with (40), which yields The invariant is well-defined for any d > 0 for which there exist admissible discretizations and is independent of both the fiber and the discretization size. From [8] we have for any Morse function W on a proper braid class fiber The latter can be computed for any admissible discretization and is an invariant for [q rel Q]. Combining Equations (44) and (45) gives In this section we assumed without loss of generality that x rel y is augmented and since the Euler-Floer characteristic is a braid class invariant, an admissible discretization is construction for an appropriate augmented, Legendrian representative x L rel y L . Summarizing Since χ q D rel Q * D is the same for any admissible discretization, the Euler-Floer characteristic can be computed using any admissible discretization, which proves Theorem 3.

Remark 5
The invariant χ q D rel Q D is a true Euler characteristic of a topological pair. To be more precise where [q D ] − rel Q D is the exit set a described above. A similar characterization does not a priori exist for [x] rel y. Firstly, it is more complicated to designate the equivalent of an exit set [x] − rel y for [x] rel y, and secondly it is not straightforward to develop a (co)-homology theory that is able to provide meaningful information about the topological pair [x] rel y, [x] − rel y . This problem is circumvented by considering Hamiltonian systems and carrying out Floer's approach towards Morse theory (see [7]), by using the isolation property of [x] rel y. The fact that the Euler characteristic of Floer homology is related to the Euler characteristic of topological pair indicates that Floer homology is a good substitute for a suitable (co)-homology theory.

Examples
We will illustrate by means of two examples that the Euler-Floer characteristic is computable and can be used to find closed integral curves of vector fields on the 2-disc. Figure 4[left] shows the braid diagram q rel Q of a positive relative braid x rel y. The discretization with q D rel Q D , with d = 2, is shown in Figure 4[right]. The chosen discretization is admissible and defines the relative braid class [q D rel Q D ]. There are five strands, one is free and four are fixed. We denote the points on the free strand by q D = (q 0 , q 1 ) and on the skeleton by Q D = {Q 1 , · · · , Q 4 }, with Q i = (Q i 0 , Q i 1 ), i = 1, · · · , 4. In Figure 5[left] the braid class fiber [q D ] rel Q D is depicted. The coordinate q 0 is allowed to move between Q 3 0 and Q 2 0 and q 1 remains in the same braid class if it varies between Q 1 1 and Q 4 1 . For the values q 0 = Q 3 0 and q 0 = Q 2 0 the relative braid becomes singular and if q 0 crosses these values two intersections are created. If q 1 crosses the values Q 1 1 or Q 4 1 two intersections are destroyed. This provides the desired co-orientation, see Figure 5 From Theorem 2 we derive that any vector field for which y is a skeleton has at least 1 closed integral curve x 0 rel y ∈ [x] rel y. Theorem 2 also implies that any orientation preserving diffeomorphism f on the 2-disc which fixes the set of four points A 4 , whose mapping class [ f ; A 4 ] is represented by the braid y has an additional fixed point.

Example
The theory can also be used to find additional closed integral curves by concatenating the skeleton y. As in the previous example y is given by Figure 4. Glue copies of the skeleton y to its -fold concatenation and a reparametrize time by t → ·t. Denote the rescaled -fold concatenation of y by # y. Choose d = 2 and discretize # y as Fig. 6 A discretization of a braid class with a 5-fold concatenation of the skeleton y. The number of odd anchor points in middle position is µ = 3.
in the previous example. For a given braid class [x rel # y], Figure 6 below shows a discretized representative q D rel # Q D , which is admissible. For the skeleton # Q D we can construct 3 − 2 proper relative braid classes in the following way: the even anchor points of the free strand q D are always in the middle and for the odd anchor points we have 3 possible choices: bottom, middle, top (2 braids are not proper). We now compute the Conley index of the 3 − 2 different proper discrete relative braid classes and show that the Euler-Floer characteristic is non-trivial for these relative braid classes.
The configuration space N = cl [q D ] rel # Q D in this case is given by a cartesian product of 2 closed intervals, and therefore a 2 -dimensional hypercube. We now proceed by determining the exit set N − . As in the previous example the co-orientation is found by a union of faces with an outward pointing co-orientation. Due to the simple product structure of N, the set N − is determined by the odd anchor points in the middle position. Denote the number of middle positions at odd anchor points by µ. In this way N − consists of opposite faces at at odd anchor points in middle position, see Figure 6. Therefore Let X(x,t) be a vector field for which y is a skeleton of closed integral curves, then # y is a skeleton for the vector field X (x,t) := X(x, t). From Theorem 2 we derive that there exists a closed integral curve in each of the 3 − 2 proper relative classes [x] rel y described above. For the original vector field X this yields 3 − 2 distinct closed integral curves. Using the arguments in [13] one can find a compact invariant set for X with positive topological entropy, which proves that the associated flow is 'chaotic' whenever y is a skeleton of given integral curves

Example
So far we have not addressed the question whether the closed integral curves x rel y are non-trivial, i.e.not equilibrium points of X. The theory can also be extended in order to find non-trivial closed integral curves. This paper restricts to relative braids where x consists of just one strand. Braid Floer homology for relative braids with x consisting of n strands is defined in [12]. To illustrate the importance of multi-strand braids we consider the discrete braid class in Figure 7. The braid class depicted in Figure 7[right] is discussed in the previous example and the Euler-Floer characteristic is equal to 1. By considering all translates of x on the circle R/Z, we obtain the braids in Figure 7[left]. The latter braid class is proper and encodes extra information about q D relative to Q D . The braid class fiber is a 6dimensional cube with the same Conley index as the braid class in Figure 7[right]. Therefore, χ(q D rel Q D ) = (−1) 2 = 1.
As in the 1-strand case, the discrete Euler characteristic can used to compute the associated Euler-Floer characteristic of x rel y and χ(x rel y) = 1. The skeleton y thus forces solutions x rel y of the above described type. The additional information we obtain this way is that for braid classes [x rel y], the associated closed integral curves for X cannot be constant and therefore represent non-trivial closed integral curves.