Pardon's implicit-atlas machinery constructs virtual fundamental cycles for moduli spaces of pseudo-holomorphic curves only as ordinary (Čech) homology classes; Remark 1.3.2 of that work notes this is inadequate for applications requiring the fundamental class in smooth or framed bordism (Abouzaid's Lagrangian embeddings of exotic spheres; Ekholm–Smith's exact Lagrangian immersions) and records that, by working at the space level, the construction should upgrade to a virtual fundamental cycle in Steenrod framed bordism twisted by a “virtual spherical normal bundle”. The rival global-Kuranishi-chart program now yields complex-oriented refinements (complex cobordism, Morava K-theory, K-theory) in general and framed refinements only under additional hypotheses, while no route yet produces framed/sphere-spectrum-valued virtual fundamental classes for general moduli spaces of pseudo-holomorphic curves, and the implicit-atlas upgrade itself remains unimplemented. The problem — the space-level framed-bordism upgrade proposed in Remark 1.3.2 — remains open.

Background

A pseudo-holomorphic curve is a map uu from a Riemann surface into an almost-complex manifold (typically a symplectic manifold with a compatible almost-complex structure JJ) satisfying a Cauchy–Riemann type equation ∂ˉJu=0\bar\partial_J u=0. Symplectic topology produces many invariants by “counting” such curves, but the relevant moduli spaces are generally not manifolds: they arise as zero sets of nonlinear Fredholm sections and can be singular, non-transverse, or of excess dimension. To count curves one therefore needs a virtual fundamental cycle/class (VFC): a homology class of the “expected” dimension carried by the singular moduli space. Constructing such classes rigorously is a central foundational problem of symplectic topology.

Pardon (2016) developed an algebraic approach. A compact Hausdorff moduli space XX is endowed with an implicit atlas: roughly, a system of local finite-dimensional reductions — charts in which XX is presented as the zero locus of a finite-dimensional obstruction-bundle section on a thickened space, with compatible transition data. From an implicit atlas, the paper's “VFC package”, a functorial formalism of chain complexes and sheaves on XX, produces a canonical virtual fundamental cycle [X]vir[X]^{\mathrm{vir}} in Čech homology of XX (with local coefficient systems), requiring only topological (not smooth) gluing theorems as analytic input. Two steps are setting-specific: domain stabilization, meaning the existence of a codimension-two divisor meeting each curve transversally so that adding the preimage points as marked points stabilizes the Riemann-surface domain (needed for the covering axiom of the atlas); and “formal regularity implies topological regularity”, the analytic gluing input ensuring the regular loci of the thickened moduli spaces are topological manifolds of the expected dimension.

Ordinary homology is a weak output for some applications. Steenrod framed bordism is the bordism theory of manifolds equipped with a stable framing (a trivialization of the stable normal bundle); by the Pontryagin–Thom construction it coincides with stable homotopy, i.e. homology with coefficients in the sphere spectrum, the initial ring spectrum of stable homotopy theory. Cohen–Jones–Segal (1995) proposed that Floer-type counts should lift to such stable homotopy types (“Floer homotopy theory”). Two works made the gap concrete: Abouzaid (2012) counted pseudo-holomorphic disks in framed bordism to obstruct Lagrangian embeddings of exotic spheres, and Ekholm–Smith (2016) used framed bordism counts of holomorphic curves to study exact Lagrangian immersions; in both, the bordism classes had to be constructed by hand in special low-dimensional settings. Pardon's machinery, producing only ordinary homology classes, could not support such applications. Remark 1.3.2 of the paper records (from discussions with Abouzaid) that “by working on the space level, one can probably upgrade our construction to yield a virtual fundamental cycle in the (Steenrod) framed bordism group of XX, twisted by a certain ‘virtual spherical normal bundle’” — the twisting being a stable spherical fibration (a fibration with spherical fibers, considered up to suspension, serving as the homotopy-theoretic substitute for a normal bundle) that encodes virtual normal data the singular moduli space does not carry canonically. Remark 1.3.4 adds that the natural language for the VFC package, and for a good theory of derived manifolds and their virtual fundamental cycles, is that of ∞-categories (higher categories whose composition laws are coherent up to a hierarchy of homotopies, as developed by Lurie), and that a more intrinsic formulation could ultimately yield more flexible calculational tools. The introduction further expresses the hope that the framework extend to moduli spaces of solutions of other nonlinear elliptic PDEs that give rise to interesting invariants.

Since 2016 a rival technique has transformed the area. A global Kuranishi chart presents an entire moduli space as a quotient of the zero set of a section of an equivariant vector bundle over a finite-dimensional manifold by a compact group action. Because the obstruction data in curve moduli carry natural stable complex structures, this route yields VFCs valued in complex-oriented generalized cohomology theories — theories such as complex K-theory, Morava K-theories, and complex cobordism, equipped with a compatible theory of Thom classes for complex vector bundles: Abouzaid–McLean–Smith computed Gromov–Witten-type invariants in Morava K-theory; Bai–Xu and Rezchikov extended global charts to Hamiltonian Floer trajectories and proved the Arnold conjecture (bounding Hamiltonian fixed points) over the integers; Abouzaid–Blumberg built a stable ∞-category of flow categories (categories enriched in spaces encoding moduli of Floer trajectories, from which stable homotopy types are assembled) over derived-orbifold bordism theories, where a derived orbifold is a space presented as the zero locus of an orbifold vector-bundle section. What these constructions provide in general are complex-oriented refinements; framed bordism (the sphere spectrum) is not complex-oriented, and framed refinements are so far available only under additional hypotheses. The implicit-atlas machinery itself has not been upgraded to the space level.

Problem Statement

Construct virtual fundamental cycles for moduli spaces of pseudo-holomorphic curves at the space level, valued in Steenrod framed bordism rather than ordinary homology, as proposed in Remark 1.3.2 of Pardon (2016).

Specifically: given a compactified moduli space XX of pseudo-holomorphic curves in a general closed symplectic manifold, equipped with its implicit atlas (or an equivalent finite-dimensional-reduction structure), construct a canonical virtual fundamental cycle in the framed bordism group of XX, twisted by the “virtual spherical normal bundle” (the stable spherical fibration encoding the virtual normal data). Remark 1.3.2 does not define this fibration, so constructing it — a canonical stable spherical fibration over XX, well-defined up to coherent homotopy from the implicit-atlas data — is part of the task. The framed-bordism construction must satisfy: (i) it is independent of the auxiliary choices entering the framed lift, such as thickening and perturbation data; and (ii) its image in ordinary homology recovers the virtual fundamental cycle [X]vir[X]^{\mathrm{vir}} produced by the algebraic VFC package. As Remark 1.3.2 records, the motivation for the upgrade is adequacy for bordism-valued curve counting, exemplified by the hand-constructed framed bordism counts of pseudo-holomorphic disks and curves used by Abouzaid (2012) for Lagrangian embeddings of exotic spheres and by Ekholm–Smith (2016) for exact Lagrangian immersions. The current frontier is exactly the non-complex-oriented case: framed bordism (the sphere spectrum), as opposed to the complex-oriented refinements now available via global Kuranishi charts.

The same paper records two further programmatic hopes: an intrinsic ∞-categorical formulation of the VFC package (Remark 1.3.4), and extension of the framework to moduli spaces of solutions of other nonlinear elliptic PDEs that give rise to interesting invariants. These are related background directions of the source's extension program, not components of the open question posed here, which is the Remark 1.3.2 framed-bordism upgrade.

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