General capped-orbit and Novikov Hamiltonian Floer data

Let $(M,\omega )$ be a closed symplectic manifold and fix a compatible $J_0$, a smooth one-periodic Hamiltonian $H$, and a smooth one-periodic compatible family $J_t$. Write ${\mathrm{\Phi }}_{t,0}$ for the local Hamiltonian evolution. Let ${\mathcal{P}}_1^{\mathrm{c}\mathrm{t}\mathrm{r}}(H)$ be the set of smooth contractible loops $x:\mathbb{R}\mathrm{/}\mathbb{Z}\to M$ satisfying $\dot{x}(t)=X_{H_t}(x(t))$ and the nondegeneracy condition that $D{\mathrm{\Phi }}_{1,0}(x(0))-I_{T_{x(0)}M}$ is invertible; contractible means that $x$ extends to a smooth cap $w:D^2\to M$. These are the orbit and nondegeneracy conventions of the preceding capping definition, made explicit here for the generator set. For each connected component $M_{\alpha }$, define ${\mathrm{\Gamma }}_{\alpha }={\pi }_2(M_{\alpha })\mathrm{/}(\ker {\omega }_{\alpha }\cap \ker c_1(T{M}_{\alpha },J_0)),$ where ${\omega }_{\alpha }$ is the restriction of $\omega$. The induced maps are ${\nu }_{\alpha }([A])={\omega }_{\alpha }(A)$ and $d_{\alpha }([A])=-2c_1(A)$. The general capped generators are the capped orbits modulo precisely those recappings with both shifts zero.

For each underlying orbit $x$, choose one reference cap $w_x$. With coefficient field ${\mathbb{F}}_2$, set $\widehat{CF}(H)={⨁}_{\alpha }{⨁}_{x\in {\mathcal{P}}_1^{\mathrm{c}\mathrm{t}\mathrm{r}}(H)\cap M_{\alpha }}{\mathrm{\Lambda }}_{{\nu }_{\alpha },{\mathbb{F}}_2}⟨x⟩,$ where each coefficient is a Novikov series from the preceding definition and the direct sum means that only finitely many orbit labels occur in one chain. Identify the monomial generator $q^{[A]}x$ with $[x,w_x\mathrm{\#}A]$. Its action and grading are ${\mathcal{A}}_H(q^{[A]}x)={\mathcal{A}}_H([x,w_x])-\omega (A),\mathrm{\mid }{q}^{[A]}x\mathrm{\mid }={\mu }_{\mathrm{C}\mathrm{Z}}(x,w_x)-2c_1(A)\mathrm{.}$ Within each orbit label the coefficient support is finite below every energy cutoff. If the reference cap changes from $w_x$ to $w_x\mathrm{\#}{B}_x$, the same capped generator is relabeled by $q^{[A-B_x]}x$; multiplication by the fixed monomial $q^{-[B_x]}$ preserves the Novikov condition, so this gives the canonical deck relabeling on the direct sum. This definition does not supply a Floer differential, prove convergence of any differential, or assert a bubbling-corrected or virtual construction.

Symplectic and almost-complex curve data

Capped Hamiltonian orbits and recapping data

Hamiltonian action and grading shifts under recapping

Novikov convolution is a filtered ring operation