Classical aspherical Hamiltonian Floer data

Let $(M,\omega )$ be a closed symplectic manifold, where closed means compact and without boundary. Fix a smooth one-periodic Hamiltonian $H:\mathbb{R}\mathrm{/}\mathbb{Z}\times M\to \mathbb{R}$ with ${\iota }_{X_{H_t}}\omega =-d{H}_t$, and a smooth one-periodic family of $\omega$-compatible almost-complex structures $J_t$. A contractible one-periodic orbit is a smooth loop $x:\mathbb{R}\mathrm{/}\mathbb{Z}\to M$ admitting a cap and satisfying $\dot{x}(t)=X_{H_t}(x(t))$. It is nondegenerate if $1$ is not an eigenvalue of the derivative of the time-one return map at $x(0)$.

Assume the spherical period maps vanish: $\omega (A)=0$ and $c_1(TM,J_0)(A)=0$ for every $A\in {\pi }_2(M)$. The local recapping result then makes the action ${\mathcal{A}}_H(x)$ and Conley–Zehnder grading ${\mu }_{\mathrm{C}\mathrm{Z}}(x)$ independent of the cap. Let ${\mathcal{P}}_1^{\mathrm{c}\mathrm{t}\mathrm{r}}(H)$ be the set of contractible nondegenerate one-periodic orbits. The classical aspherical chain module is the algebraic direct sum $C{F}_{\ast }(H;{\mathbb{F}}_2)={⨁}_{x\in {\mathcal{P}}_1^{\mathrm{c}\mathrm{t}\mathrm{r}}(H)}{\mathbb{F}}_2⟨x⟩,$ so each vector has finite support. The coefficient field ${\mathbb{F}}_2$ removes orientation signs. This is only the orbit, action, grading, and module data; no Floer differential, compactness, transversality, or continuation theorem is asserted.

Symplectic and almost-complex curve data

Capped Hamiltonian orbits and recapping data

Hamiltonian action and grading shifts under recapping