Foundational extensions of Pardon's construction of contact homology: integral coefficients, nonequivariant framed $E_2$-structure, DGA-level homotopies, Morse-Bott forms, and the full symplectic field theory package

Pardon (J. Amer. Math. Soc. 2019) constructed, via coherent virtual fundamental cycles on compactified moduli spaces of pseudo-holomorphic curves, the contact homology algebra of an arbitrary closed contact manifold with nondegenerate contact form: over $\mathbb{Q}$, at genus zero, with cobordism functoriality up to chain homotopy. His introduction (Section 1.8) explicitly lists what remains to complete the theory: (i) a natural lift of contact homology from $\mathbb{Q}$ to $\mathbb{Z}$ (an open question promoted by Abouzaid); (ii) the conjectural nonequivariant lift to a framed $E_2$-algebra whose differential encodes domain conformal types and rotation signs of multiply covered orbits; (iii) cobordism maps well-defined up to homotopies compatible with the algebra structure (DGA or $A_\infty$ homotopies); (iv) the full symplectic field theory (SFT) homology groups of Cieliebak-Latschev and Latschev-Wendl; (v) contact homology defined directly for Morse-Bott contact forms via holomorphic cascades; and (vi) an $A_\infty$-module (derived local system) structure on contact chains over the chains on the contactomorphism group. Hutchings-Nelson constructed integral and nonequivariant theories only in the hypertight special case; Chaidez (2024) settled the motivating invariance question for linearized contact homology without DGA homotopies, while the stronger homotopy statements remain open; Ishikawa's unpublished thesis (arXiv 2018; Kyoto degree granted March 2019) claims a complete rational SFT construction including the Morse-Bott case, but it remains uncertified by the community. The general integral lift, the framed $E_2$-lift, DGA-level homotopies, the derived-local-system structure, and a community-certified full SFT package remain open.

Background

A contact manifold is a smooth odd-dimensional manifold $Y^{2n-1}$ together with a maximally non-integrable hyperplane field $\xi$; a contact form $\lambda$ with $\ker\lambda=\xi$ determines the Reeb vector field, and its closed trajectories (Reeb orbits) carry the dynamical information of $(Y,\xi)$. Contact homology, proposed by Eliashberg, Givental, and Hofer as the genus-zero layer of symplectic field theory (SFT), is built as follows: one studies pseudo-holomorphic curves (solutions of a Cauchy-Riemann type equation) with cylindrical ends in the symplectization $\mathbb{R}\times Y$, asymptotic to Reeb orbits, and organizes the resulting counts into the differential of a differential graded algebra (DGA: a graded algebra with a degree-one square-zero derivation), freely generated by the good Reeb orbits (the Conley-Zehnder index of a closed Reeb orbit is the integer invariant measuring the total rotation of the linearized Reeb flow along the orbit; an orbit is good when the indices of all of its iterates have the same parity, while bad orbits, whose contributions cancel in pairs, are excluded). The full SFT of Eliashberg-Givental-Hofer additionally counts curves of all genera in symplectizations and symplectic cobordisms and organizes them into a Batalin-Vilkovisky-type algebraic hierarchy (a differential graded commutative algebra carrying an additional second-order bracket operator, mirroring the algebraic structure of the homology of a free loop space); the string-topology model of Cieliebak-Latschev and the algebraic-torsion framework of Latschev-Wendl specify the expected homology groups and their structure.

The foundational obstruction is transversality: the moduli spaces to be counted are generally not smoothly cut out, essentially because multiply covered curves carry reparametrization symmetry (an $S^1$ and finite-isotropy problem), and generic perturbation of the almost complex structure cannot cure this while preserving the algebraic identities the counts must satisfy. Several non-classical frameworks were developed to define virtual fundamental cycles (homology classes standing in for the missing fundamental classes) on such singular moduli spaces: Kuranishi structures (Fukaya-Ono), which cut the moduli space into finite-dimensional charts equipped with obstruction bundles; polyfolds (Hofer-Wysocki-Zehnder), a generalized infinite-dimensional manifold framework supporting perturbation theory; and implicit atlases (Pardon). An implicit atlas equips a compactified moduli space with an abstract perturbation package from which one extracts a rational virtual count satisfying the required compatibilities, including a master equation expressing that the count of the boundary vanishes.

Pardon implemented this program for genus-zero SFT: for every closed contact manifold with a nondegenerate contact form (one for which every closed Reeb orbit is a nondegenerate fixed point of the return map of the Reeb flow, meaning the linearized return map minus the identity is invertible) he constructed coherent virtual fundamental cycles and thereby defined the contact homology algebra $CC_\bullet(Y,\xi)$ over $\mathbb{Q}$, well-defined up to DGA quasi-isomorphism and canonical up to chain homotopy, functorial for exact symplectic cobordisms (cobordisms between contact manifolds carrying an exact symplectic form compatible with the contact forms at the two ends) at the chain-homotopy level. Bao-Honda later gave an independent construction of the same object using semi-global Kuranishi charts, with the same chain-level scope.

Pardon's introduction then states, as an explicit bulleted list of open directions, what his methods do not achieve. First, the virtual counts are rational, and it is an open question (promoted by Abouzaid) how to naturally lift contact homology from $\mathbb{Q}$ to $\mathbb{Z}$. Second, the maps induced by cobordisms are controlled only up to chain homotopy; the stronger homotopies compatible with the multiplication (DGA-level or $A_\infty$ homotopies, i.e. homotopies that themselves respect the algebra structure, for instance given by derivations or by a coherent hierarchy of higher maps), needed for example to define linearized contact homology (the homology of the complex obtained by linearizing the differential of the contact DGA at an augmentation, i.e. a morphism of DGAs from the contact DGA to the coefficient ring regarded as a DGA concentrated in degree zero) canonically for exact fillings (compact symplectic manifolds with boundary $(Y,\xi)$ on which the symplectic form is exact with a primitive restricting to the contact form), are not constructed. Third, ordinary contact homology is by default $S^1$-equivariant, its generators being unparametrized Reeb orbits; Pardon conjectures a nonequivariant lift in which the orbit generators generate a framed $E_2$-algebra (an algebra over the topological operad of framed little disks, i.e. the operad whose space of $n$-ary operations is the space of embeddings of a disjoint union of $n$ copies of $D^2$ into $D^2$), with the relation that rotating a $d$-fold multiply covered orbit by $2\pi/d$ acts by the good-orbit sign $\pm 1$, and with the differential of an orbit lifted by taking into account the conformal types of the punctured domains of the curves. Fourth, constructing the SFT homology groups in the sense of Cieliebak-Latschev and Latschev-Wendl would, by Pardon's analysis, follow from specifying the correct compactified moduli spaces and proving their stratifications locally cell-like (a tameness condition under which the stratified space admits local models of cone type, sufficient for defining coherent virtual counts). Fifth, the same sufficient conditions, together with the relevant gluing result, would allow defining contact homology directly for Morse-Bott contact forms (forms whose Reeb orbits come in smooth families, so that the orbit set is itself a manifold) by counting holomorphic cascades in the sense of Bourgeois (configurations of curves joined by gradient trajectory segments along the orbit families). Sixth, the action of the contactomorphism group on contact chains is expected to upgrade to an $A_\infty$-module structure over the chains on $\mathrm{Cont}(Y,\xi)$ (an action of the differential graded algebra of singular chains on the contactomorphism group, defined up to a coherent system of higher homotopies), making contact chains a derived local system (a homotopy-coherent local system of chain complexes) over the classifying space $B\,\mathrm{Cont}(Y,\xi)$ (the base of the universal principal $\mathrm{Cont}(Y,\xi)$-bundle). These six items share one technical core: compactified moduli spaces for the remaining layers of the theory whose stratifications are tame enough to support coherent virtual counts, plus gluing, and, for the coefficient and homotopy items, orientation-divisibility and homotopy-coherence control.

Problem Statement

The common objective is to complete the rigorous construction of the contact-homology/symplectic-field-theory package for general closed contact manifolds, in the full form expected by the theory, of which Pardon's theorem realizes only the genus-zero, rational, $S^1$-equivariant, chain-homotopy-level part. The requested results, all stated for general closed contact manifolds (with nondegenerate contact forms unless Morse-Bott is specified), are exactly the six directions Pardon's introduction leaves open:

1. (Integer coefficients) Construct a lift of contact homology from $\mathbb{Q}$ to $\mathbb{Z}$ that is natural and canonical, in the same sense as Pardon's rational construction (well-defined up to DGA quasi-isomorphism, canonical up to chain homotopy).

2. (Nonequivariant lift) Construct the conjectured lift of contact homology to a nonequivariant theory, namely a framed $E_2$-algebra: replace the free symmetric $\mathbb{Q}$-algebra on the good Reeb orbit generators with the free framed $E_2$-algebra on all orbit generators, impose the relations that rotating a $d$-fold multiply covered orbit by $2\pi/d$ acts by the corresponding sign ($\pm 1$ according to whether the orbit is good or bad), and lift the differential of each orbit generator by incorporating the conformal types of the domains of the pseudo-holomorphic curves, so that ordinary contact homology is recovered as the $S^1$-equivariant version of this theory, contact homology itself being an $S^1$-equivariant theory.

3. (Algebraic homotopies) Construct, for one-parameter families of exact symplectic cobordisms, homotopies between the induced contact-DGA maps that are compatible with the algebra structure (DGA-level or $A_\infty$ homotopies), sufficient in particular to define canonical linearized contact homology groups for exact fillings and functorial maps of linearized theories under cobordisms.

4. (Full SFT) Construct the symplectic field theory homology groups in the sense of Cieliebak-Latschev and Latschev-Wendl (all genera, with the full Batalin-Vilkovisky/string-topology algebraic structure).

5. (Morse-Bott) Define contact homology directly for Morse-Bott contact forms.

6. (Derived structures) Upgrade the natural action of the contactomorphism group on contact chains to an $A_\infty$-module structure of contact chains $CC_\bullet(Y,\xi)$ over $C_\bullet(\mathrm{Cont}(Y,\xi))$, making contact chains a derived local system over $B\,\mathrm{Cont}(Y,\xi)$, including the higher homotopies for families of cobordisms.

These six directions are jointly the target of the problem. Items 1, 2, and 6 the source poses as a question, conjecture, or expectation: each asks for the stated object in the stated generality, or else for a definitive obstruction to, or refutation of, the existence of such an object in the stated generality (for item 1, an obstruction to a lift that is simultaneously natural and canonical; a lift with only one of the two properties is not the lift item 1 asks for). Items 3-5 are construction problems; for items 4 and 5 the source itself states sufficient conditions, while for item 3 it records no positive result on the applicability of its methods. The six directions overlap and may be realized by a single construction, and a result confined to one direction, or to special classes of contact manifolds or forms, leaves the remaining directions open.

References

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