A gerbe-like construction in gauge theory II: the case of homology tori

Abstract: In the previous paper, the author showed that for a smooth family $X \to \mathbb{X} \to B$ of a homotopy $K3$ surface, the obstruction for the tangent bundle along the fibers $T_B \mathbb{X}$ to have a spin structure is canonically isomorphic to the obstruction for $\mathcal{H}<sup>+(\mathbb{X})$, the vector bundle over $B$ consisting of self-dual harmonic 2-forms, to have a spin structure. In this paper, we show an analogous result for homology tori with odd determinant. The strategy for proof is similar to the case of homotopy $K3$ surfaces: take the determinant line bundle of the $K$-theoretic Seiberg--Witten invariant and construct an anti-linear $\mathbb{Z}/4$-action on it at the representative level. We also see that the anti-linear $\mathbb{Z}/4$-action possesses the information of the ordinary mod 2 Seiberg--Witten invariant. This recovers part of the result by Baraglia(2023) which computes the mod 2 Seiberg--Witten invariants for any closed spin 4-manifold.

• The paper establishes a gerbe-theoretic framework that identifies spin structure obstructions in homology tori using finite-dimensional Seiberg–Witten approximations.
• It employs Pin(2)-equivariant techniques and anti-linear Z/4 actions to relate characteristic class obstructions with mod 2 Seiberg–Witten invariants.
• The approach demonstrates a canonical isomorphism between obstructions in vertical tangent bundles and self-dual harmonic forms, influencing isotopy and index theory.

Gerbe-Like Obstructions in Gauge Theory on Homology Tori

Introduction and Context

This paper develops a gerbe-theoretic framework for analyzing obstructions to spin structures in families of 4-manifolds, extending previous work on homotopy $K3$ surfaces to the class of homology 4-tori with odd determinant. The interaction between gauge theory—specifically, Seiberg–Witten theory—and the topology of 4-manifolds underlies the analysis. The work leverages finite-dimensional approximations of the Seiberg–Witten equations, coupled with a careful construction of anti-linear $\mathbb{Z}/4$-actions, to relate characteristic class obstructions for tangent and harmonic bundles.

Main Results and Methodology

The paper proves that for a smooth fiber bundle $X \to \mathcal{X} \to B$, with $X$ a homology 4-torus of odd determinant, the primary obstruction to a spin structure on the vertical tangent bundle $T_B \mathcal{X}$ pulls back (over a certain Albanese torus bundle $\Alb(\mathcal{X})$) to the same class as the obstruction for the bundle $\mathcal{H}^+(\mathcal{X})$ of self-dual harmonic 2-forms to admit a spin structure. This is an analogue of the $K3$ result, but with subtle differences due to the nontrivial first Betti number in the torus setting.

The proof constructs, for a universal principal bundle $E \to B$ with structure group a natural extension $\mathrm{Diff}^{\mathrm{spin}}(X)$ (those diffeomorphisms preserving a fixed spin structure up to isomorphism), a canonical lift to a principal $\mathbb{Z}/4$0-bundle over $\mathbb{Z}/4$1. This lifting is equivalent, in terms of characteristic classes, to equating the primary obstruction for a spin structure on $\mathbb{Z}/4$2 with the second Stiefel–Whitney class $\mathbb{Z}/4$3 of $\mathbb{Z}/4$4. Functoriality of the construction is established with respect to maps of families, and the effect of automorphisms is controlled at the level of spin structures.

The technical heart of the paper is an explicit finite-dimensional approximation (FDA) scheme for the families Seiberg–Witten map in the case $\mathbb{Z}/4$5, including precise control of the $\mathbb{Z}/4$6-equivariant structures and the role of harmonic 1-forms, which necessitates working over the non-canonically trivial Albanese torus bundle. The FDA is used to produce a complex line bundle with an anti-linear $\mathbb{Z}/4$7 action, from which a spin structure on $\mathbb{Z}/4$8 is extracted using determinant line bundle formalism.

A key result, supporting earlier work by Baraglia (2023), is that the anti-linear $\mathbb{Z}/4$9-action encodes the mod 2 Seiberg–Witten invariants, providing a topological constraint that connects the FDA, spin geometry on harmonic bundles, and low-dimensional mod 2 invariants.

Quantitative Outcomes and Claims

• The pullback via $X \to \mathcal{X} \to B$0 of the primary characteristic class $X \to \mathcal{X} \to B$1 for spin structures on $X \to \mathcal{X} \to B$2 agrees with $X \to \mathcal{X} \to B$3.
• The construction is canonical up to contractible choices; independence is established for auxiliary choices like the FDA model and splittings.
• The anti-linear $X \to \mathcal{X} \to B$4-action on the determinant line bundle has trace encoding the mod 2 Seiberg–Witten invariant of $X \to \mathcal{X} \to B$5: for homology tori with odd determinant and $X \to \mathcal{X} \to B$6, this invariant is nontrivial and equals 1 (mod 2), generalizing classic results for $X \to \mathcal{X} \to B$7-like manifolds.

Implications and Applications

The identification of gerbe-theoretic obstructions and their canonical isomorphism has several implications:

• Topological applications: The result is directly applied to isotopy problems: the nontriviality of the boundary Dehn twist on punctured homology tori (with odd determinant) is demonstrated, advancing earlier results by Qiu (2025) and Kronheimer–Mrowka (2020) by an independent method not relying solely on Bauer–Furuta-type invariants.
• Functorial obstruction theory: The compatibility of the FDA-based construction with bundle maps and automorphisms provides a robust formalism for families-index theory in 4-manifold topology, particularly where the first Betti number is positive and spin structures are not homologically unique.
• Relation to mod 2 invariants: The connection with Baraglia’s formulae for the mod 2 Seiberg–Witten invariant supports the interpretation of anti-linear structures as underlying topological constraints, and further systematizes the relations between gauge-theoretic and classical characteristic class obstructions.

Limitations and Scope

The FDA construction applies specifically to families where $X \to \mathcal{X} \to B$8 and the mod 2 Seiberg–Witten invariant is nonzero, which constrains the generalizability of the method. The extension from the Albanese torus bundle to the base space $X \to \mathcal{X} \to B$9—i.e., establishing the isomorphism over $X$0 itself rather than its subtorsor—remains open. The framework relies on the existence of a global section in the family, and does not cover all possible 4-manifold families or all values of Betti numbers.

Future Directions

This development suggests lines of inquiry into:

• Generalization to broader 4-manifold families where $X$1, and classification of possible FDA gerbe constructions in higher-dimensional parameter spaces.
• Refinements of families Seiberg–Witten theory, particularly with twisted or nontrivial Albanese bundles, and the interplay with diffeomorphism group invariants.
• Detailed computation of isotopy and mapping class group obstructions for more exotic classes of 4-manifolds, using the gerbe-theoretic machinery established here.

Conclusion

By extending the gerbe-based analysis of spinorial obstructions from homotopy $X$2 surfaces to homology tori with odd determinant, this paper clarifies the intrinsic relation between characteristic class obstructions and mod 2 Seiberg–Witten invariants in the context of families and positive first Betti number. The fusion of finite-dimensional approximation, anti-linear symmetry, and characteristic class calculation provides powerful tools for current research in 4-manifold topology, establishing a solid foundation for further advances in both the gauge-theoretic and differential-topological investigation of families of manifolds.

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