Rational characteristic classes of bundles with fibre a product of spheres

Abstract: We prove the existence of many non-trivial characteristic classes of smooth oriented bundles with fibre a product $S^{n}\times S^{n}$ of odd-dimensional spheres. We do so by proving injectivity of the map from the ring of rational characteristic classes of oriented fibrations with fibre $S^{n}\times S^{n}$; the latter is proven by Berglund--Zeman to be isomorphic to the group cohomology of the symmetric powers of the standard representation of a certain finite-index subgroup $Γ$ of $\mathrm{SL}_{2}(\mathbb{Z})$. These characteristic classes of smooth bundles are not generalised Miller--Morita--Mumford classes, and they exist in arbitrarily large cohomological degrees. Inspired by an example given by Morita, we provide a collection of smooth oriented $S^{n}\times S^{n}$-bundles, indexed by cyclic subgroups of $Γ$, which detect any given non-zero characteristic class of such fibrations.

• The paper introduces new rational characteristic classes beyond the classical Miller–Morita–Mumford framework by studying bundles with fibre a product of odd spheres.
• It employs homotopy and differential-topological methods to establish an injective map from homotopy-theoretic to smooth bundle characteristic classes.
• Explicit constructions using arithmetic group actions and modular forms detect nontrivial odd-degree classes, providing fresh geometric invariants.

Rational Characteristic Classes of Bundles with Fibre a Product of Spheres

Overview and Motivation

The paper "Rational characteristic classes of bundles with fibre a product of spheres" (2604.27948) rigorously investigates the structure of rational characteristic classes associated to smooth, oriented fibre bundles whose fibre is a product $\mathbb{S}^n \times \mathbb{S}^n$ of odd-dimensional spheres. The work analyzes the interplay between the homotopy and diffeomorphism classification of such bundles, identifies new classes outside the paradigm of Miller–Morita–Mumford (MMM) classes, and establishes injectivity results relating homotopy-theoretic and differential-topological characteristic classes. Furthermore, the author describes explicit detection results for these classes by constructing bundles indexed by cyclic subgroups of certain arithmetic groups.

Theoretical Framework and Classification Results

The classification of smooth, oriented $\mathbb{S}^n \times \mathbb{S}^n$-bundles is recast in terms of homotopy classes of maps into classifying spaces; specifically, there is a correspondence

$$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$

and

$$\mathrm{Fibrations\ up\ to\ homotopy} \ \leftrightarrow\ [B, B\mathrm{aut}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$

with $\mathrm{Diff}^+$ and $\mathrm{aut}^+$ denoting orientation-preserving diffeomorphisms and homotopy equivalences, respectively.

A central result is that the natural map from the ring of rational characteristic classes of homotopy-theoretic fibrations to that of smooth bundles,

$$H^*(B\mathrm{aut}^+(\mathbb{S}^n \times \mathbb{S}^n); \mathbb{Q}) \to H^*(B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n); \mathbb{Q}),$$

is injective. This result extends to products of more than two spheres and is derived from a detailed analysis of the relevant arithmetic group actions on (co)homology.

Description of the Ring of Characteristic Classes

Utilizing and extending results from Berglund–Zeman, the cohomology ring $$H^*(B\mathrm{aut}^+(\mathbb{S}^n \times \mathbb{S}^n); \mathbb{Q})$$ is calculated as the group cohomology

$H^*(\Gamma; \operatorname{Sym}^\bullet(V^*)),$

where $\Gamma$ is an explicit arithmetic group (being $\mathbb{S}^n \times \mathbb{S}^n$0 or a specific theta subgroup depending on $\mathbb{S}^n \times \mathbb{S}^n$1), and $\mathbb{S}^n \times \mathbb{S}^n$2 is the standard (degree $\mathbb{S}^n \times \mathbb{S}^n$3) homology representation. The image of these classes in the cohomology of $\mathbb{S}^n \times \mathbb{S}^n$4 does not intersect the subring generated by generalized MMM classes, as the latter are concentrated in even degrees whereas the former are concentrated in odd degrees.

Notably, these rings contain non-trivial classes in arbitrarily high cohomological degree, providing natural examples of bundles admitting higher odd degree characteristic classes not detected by traditional MMM theory.

Moreover, the ring structure is shown to be trivial due to degree constraints, and the characteristic classes are identified with modular forms via the Eichler–Shimura isomorphism when tensored with $\mathbb{S}^n \times \mathbb{S}^n$5.

Detection and Explicit Evaluation

Inspired by Morita's example for $\mathbb{S}^n \times \mathbb{S}^n$6, the paper constructs, for each cyclic subgroup $\mathbb{S}^n \times \mathbb{S}^n$7, a bundle $\mathbb{S}^n \times \mathbb{S}^n$8 whose characteristic classes correspond (under the restriction map) to the cohomology of $\mathbb{S}^n \times \mathbb{S}^n$9 with coefficients in appropriate symmetric powers. Using results of Goldman–Millson on decomposable classes in group cohomology, it is proven that for any nonzero characteristic class, there exists a $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$0 such that the class evaluates nontrivially on the corresponding bundle $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$1.

For parabolic $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$2, explicit Eisenstein-type classes $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$3 are constructed, and it is shown that restriction to the parabolic subgroup is nonzero. These classes, under the Eichler–Shimura correspondence, are directly related to Eisenstein series of modular forms, and their detection by explicit geometric bundles is established for all relevant $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$4.

An explicit geometric model for bundles detecting these classes is given, showing—for example—that when $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$5 the construction recovers Morita's classical example, and for higher $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$6, reflection maps and their associated mapping tori serve as the required diffeomorphism representatives.

Contrasts with Connected Sum Manifold Theory

The results are placed in the context of recent advances in the study of characteristic classes of bundles whose fibres are connected sums $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$7 (studied by Galatius–Randal-Williams, Berglund–Madsen, Krannich et al.), noting that the known stable range theorems for the cohomology of diffeomorphism groups in higher genus do not apply to the genus one case treated here. In particular, the new classes found are not explained by the algebra generated by generalized MMM classes, and previous stability and vanishing results for the tautological ring do not apply to these odd dimensional, genus one cases.

Implications and Prospects

This work demonstrates the existence of a large supply of nontrivial rational characteristic classes for smooth fibre bundles with fibre $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$8, including highly explicit classes outside the stable range and far beyond the reach of classical tautological constructions. The explicit injectivity and detection statements confirm that these classes are genuine geometric invariants of smooth bundles—despite their origin in the homotopy-theoretic classification.

Theoretical implications include:

• Separation from MMM Classes: The existence of nontrivial classes in odd cohomological degrees, disjoint from the subring generated by MMM classes, indicates rich secondary structure in the cohomology of diffeomorphism groups and their arithmetic group analogues.
• Modular and Arithmetic Model: The identification of the cohomology in terms of arithmetic group representations and modular forms, via the Eichler–Shimura isomorphism, opens further connections to number theory, particularly the explicit construction of geometric cycles corresponding to Eisenstein series.

From a practical point of view, the results provide new invariants for detecting nontrivial $$\mathrm{Bundles\ up\ to\ isomorphism} \ \leftrightarrow\ [B, B\mathrm{Diff}^+(\mathbb{S}^n \times \mathbb{S}^n)],$$9-bundles, potentially relevant for classification questions in high-dimensional topology and in the explicit computation of characteristic classes for construction and classification in geometric topology.

Potential future developments could include:

• The extension to bundles with fibre a larger product or more general highly connected manifolds,
• Investigation of integral and torsion phenomena beyond the rational setting,
• Deeper analysis of the interaction between diffeomorphism and homotopy group actions in the presence of exotic smooth structures,
• Connections with higher chromatic and structured cohomology theories.

Conclusion

The paper systematically identifies, constructs, and detects a wealth of rational characteristic classes in the cohomology of smooth bundles with (odd sphere) product fibre, outside the established Miller–Morita–Mumford paradigm, and provides both algebraic and geometric mechanisms for their evaluation. These contributions clarify a previously opaque region of manifold bundle theory and open new avenues for the algebraic and arithmetic investigation of diffeomorphism groups and their classifying spaces.