Universal Novikov completion and energy filtration

Let $k$ be a field, let $\mathrm{\Gamma }$ be an additive abelian group, let $\nu :\mathrm{\Gamma }\to \mathbb{R}$ be a group homomorphism, and optionally let $d:\mathrm{\Gamma }\to \mathbb{Z}$ be a grading homomorphism. The universal Novikov completion for these data is the set of formal sums ${\mathrm{\Lambda }}_{\nu ,k}=\{{\sum }_{\gamma \in \mathrm{\Gamma }}{a}_{\gamma }{q}^{\gamma }:a_{\gamma }\in k,\mathrm{\#}\{\gamma :a_{\gamma }\mathrm{\ne }0,\nu (\gamma )<C\}<\mathrm{\infty }\text{ for every }C\in \mathbb{R}\}\mathrm{.}$ For a sum set $v(\sum a_{\gamma }{q}^{\gamma })=\inf \{\nu (\gamma ):a_{\gamma }\mathrm{\ne }0\}$, with $v(0)=+\mathrm{\infty }$; the following local result proves that this infimum is finite and attained for every nonzero sum. The energy filtration is $F^c{\mathrm{\Lambda }}_{\nu ,k}=\{a:v(a)\ge c\}$. The monomial $q^{\gamma }$ has degree $d(\gamma )$ when a grading is declared. Formal addition is coefficientwise, the proposed monomial multiplication is $q^{\gamma }{q}^{\eta }=q^{\gamma +\eta }$, and a local lemma verifies that convolution makes this set a filtered ring.

The positive-energy subring is ${\mathrm{\Lambda }}_{\nu ,k}^0$, the subset supported on $\{\gamma :\nu (\gamma )\ge 0\}$. The FOOO energy-only convention is the case $\mathrm{\Gamma }=\mathbb{R}$, $\nu (\lambda )=\lambda$, $d=0$, with coefficient field $\mathbb{Q}$; its $\nu \ge 0$ subring has the source's completed energy filtration.

For the Hamiltonian Floer specialization, fix a smooth symplectic manifold $(M,\omega )$, a basepoint $m_0\in M$, and a chosen smooth compatible almost-complex structure $J$: $J^2=-\mathrm{i}\mathrm{d}$ and $\omega (\cdot ,J\cdot )$ is a symmetric positive-definite metric. Regard $(TM,J)$ as a complex vector bundle, with multiplication by $\mathrm{i}$ given by $J$. The abelian group ${\pi }_2(M,m_0)$ consists of based homotopy classes of spheres in $M$, with addition by the pinch map. Its spherical period homomorphisms are $\omega :{\pi }_2(M,m_0)\to \mathbb{R}$, $\omega ([u])={\int }_{S^2}{u}^{\ast }\omega$, and $c_1:{\pi }_2(M,m_0)\to \mathbb{Z}$, the first Chern number of $u^{\ast }(TM,J)$, using smooth representatives $u$ and the standard orientation of $S^2$. Explicitly, choose complex frames $e_N,e_S$ over the northern and southern hemispheres, write $e_S=e_{N}g$ on the equator oriented as the northern boundary, and set $c_1([u])=\mathrm{wind}(\det g)$, where $\mathrm{wind}(f)$ is the degree of $f\mathrm{/}\mathrm{\mid }f\mathrm{\mid }:S^1\to S^1$; the determinant is $1$ in rank zero. Changes of hemisphere frames have zero boundary winding. Homotopies preserve both periods, by closedness of $\omega$ and homotopy invariance of winding, and the pinch sum adds them. With the capped-generator convention used here, take $\mathrm{\Gamma }={\pi }_2(M,m_0)\mathrm{/}(\ker \omega \cap \ker c_1)$, $\nu ([A])=\omega (A)$, $d([A])=-2c_1(A)$, and $k={\mathbb{F}}_2$. These maps descend because both periods vanish on the subgroup being quotiented out. The sign of $d$ is a chosen coefficient convention, not a recapping-index theorem.

The finiteness condition is part of the definition; no operation count, differential, or Floer homology group is asserted here.

Symplectic form and symplectic manifold