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Hamiltonian action and grading shifts under recapping

Statement

Let $(M, ω, J_{*})$ be symplectic data with a chosen compatible structure, let $H_{t}$ be smooth and periodic, and let $x$ be a nondegenerate contractible one-periodic orbit with cap $w$ . For every recapping sphere $A$ , $A_{H} ([x, w # A]) - A_{H} ([x, w]) = - ω (A), μ_{C Z} (x, w # A) - μ_{C Z} (x, w) = - 2 c_{1} (A) .$ Hence, if $ω ∣_{π_{2} (M)} = 0$ and $c_{1} (T M, J_{*}) ∣_{π_{2} (M)} = 0$ , both action and integer grading are independent of the capping.

Statement references

Capped Hamiltonian orbits and recapping data

Facts & Assumptions

Given: The action, cap, recapping, Chern-number, and Conley–Zehnder conventions of the preceding definition.

[F1]

Capped Hamiltonian orbits and recapping data: $A_{H} ([x, w]) = \int H_{t} (x (t)) d t - \int_{w} ω$ , the vector-field convention is $ι_{X_{H_{t}}} ω = - d H_{t}$ , and the chosen index convention assigns a recapping loop with determinant winding $r$ the shift $2 r$ .

[F2]

Symplectic and almost-complex curve data: $d ω = 0$ , so the symplectic period is invariant under homotopy of a sphere representative.

[A1]

The spherical period maps $A \mapsto ω (A)$ and $A \mapsto c_{1} (A)$ vanish on $π_{2} (M)$ .

Proof

1.1givenF1F2algebra

The cap $w # A$ is obtained by gluing a representative sphere into $w$ , so additivity of integration gives $\int_{w # A} ω = \int_{w} ω + ω (A)$ . The Hamiltonian term is unchanged; substitution in the definition of $A_{H}$ gives the action shift $- ω (A)$ . Closedness in [F2] makes $ω (A)$ depend only on the sphere class.

1.2F1givenalgebra

Normalize the two cap-induced frames to agree at $t = 0$ . In the stated transition convention, the frame from $w # A$ has coordinate change $g_{A} (t)$ relative to the frame from $w$ , so the linearized paths are related by $Φ_{w # A} (t) = g_{A} (t) Φ_{w} (t)$ and $g_{A} (0) = g_{A} (1) = I$ . The loop-shift law gives $μ_{C Z} (g_{A} Φ_{w}) - μ_{C Z} (Φ_{w}) = 2 wind (\det_{C} g_{A}) = - 2 c_{1} (A)$ .

2.1A1step 1.1step 1.2∎

Under [A1], both displayed shifts vanish for every recapping sphere. Therefore the action and integer grading are capping-independent.

Depends on

Dependency tree · 2 levels

Sources

Dietmar Salamon, Lectures on Floer Homology