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

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

The authors outline combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an orientated 2 -manifold with out boundary, turn out that it truly is invariant lower than isotopy, and end up that it truly is isomorphic to the unique Lagrangian Floer homology. Their facts makes use of a formulation for the Viterbo-Maslov index for a tender lune in a 2 -manifold

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If Λ satisfies the arc condition then g A ∩ A = ∅ and g B ∩ B = ∅ for every g ∈ Γ \ {id}. In particular, for every g ∈ Γ \ {id}, we have gx, gy ∈ / A∪B and hence (21) holds by (i). 2 that the cancellation formula (20) holds for every g ∈ Γ \ {id}. 4. The next lemma deals with (α, β)-traces connecting a point x ∈ α ∩ β to itself. An example on the annulus is depicted in Figure 5. 5 (Isotopy Argument). Suppose Σ is not diffeomorphic to the 2sphere. 1. Suppose that there is a deck transformation g0 ∈ Γ \ {id} such that y = g0 x.

Then εx = −1 and εy = 1.

Hence 0 = μ(Λ0 ) − μ(Λ) = = mx0 (Λ0 ) + my0 (Λ0 ) mx (Λ) + my (Λ) − 2 2 1 mgk x (Λ) + mg−k y (Λ) 2 k=0 = mgx (Λ) + mg−1 y (Λ). Here the last equation follows from (21). 2. 3. Suppose Σ is not diffeomorphic to the 2-sphere. 1 and denote να := ∂ w|α\β and νβ := −∂ w|β\α . Choose smooth paths γα : [0, 1] → α, γβ : [0, 1] → β from γα (0) = γβ (0) = x to γα (1) = γβ (1) = y such that γα is an immersion when να ≡ 0 and constant when να ≡ 0, the same holds for γβ , and να (z) = deg(γα , z) for z ∈ α \ {x, y}, νβ (z) = deg(γβ , z) for z ∈ β \ {x, y}.

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