By Vin De Silva, Joel W. Robbin, Dietmar A. Salamon

ISBN-10: 0821898868

ISBN-13: 9780821898864

The authors outline combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an orientated 2 -manifold with no boundary, turn out that it truly is invariant less than isotopy, and turn out that it's isomorphic to the unique Lagrangian Floer homology. Their evidence makes use of a formulation for the Viterbo-Maslov index for a tender lune in a 2 -manifold

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**Example text**

8 (Uniqueness). Assume (H). If two smooth (α, β)-lunes have the same trace then they are equivalent. Proof. See Chapter 8 page 55. 9. Assume (H) and let Λ = (x, y, w) be an (α, β)-trace. Choose a universal covering π : Σ → Σ, a point x ∈ π −1 (x), and lifts α and β of α and β such that x ∈ α ∩ β. Let Λ = (x, y, w) 6. LUNES AND TRACES 41 be the lift of Λ to the universal cover. (i) If Λ is a combinatorial (α, β)-lune then Λ is a combinatorial (α, β)-lune. (ii) Λ is a combinatorial (α, β)-lune if and only if there exists a smooth (α, β)-lune u such that Λu = Λ.

Deﬁne A := γα ([0, 1]), B := γβ ([0, 1]). Then, for every g ∈ Γ, we have gx ∈ A (23) (24) ⇐⇒ gx ∈ / A and gy ∈ /A gx ∈ A and gy ∈ A (25) g −1 y ∈ A, ⇐⇒ A ∩ g A = ∅, ⇐⇒ g = id. The same holds with A replaced by B. Proof. If α is a contractible embedded circle or not an embedded circle at all we have A ∩ g A = ∅ whenever g = id and this implies (23), (24) and (25). Hence assume α is a noncontractible embedded circle. Then we may also assume, without loss of generality, that π(R) = α, the map z → z + 1 is a deck transformation, π maps the interval [0, 1) bijectively onto α, and x, y ∈ R = α with x < y.

2, the cancellation formula (20) holds for every g ∈ Γ \ {id}. 1. 4 in the Non Simply Connected Case. 1. Then mx (Λ) + my (Λ) − mx (Λ) − my (Λ) = mgx (Λ) + mg−1 y (Λ) = 0. 1. 1, we have mx (Λ) + my (Λ) mx (Λ) + my (Λ) = . 2 2 This proves the trace formula in the case where Σ is not simply connected. μ(Λ) = μ(Λ) = Part II. Combinatorial Lunes CHAPTER 6 Lunes and Traces We denote the universal covering of Σ by π:Σ→Σ and, when Σ is not diﬀeomorphic to the 2-sphere, we assume Σ = C. 1 (Smooth Lunes).

### Combinatorial Floer homology by Vin De Silva, Joel W. Robbin, Dietmar A. Salamon

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