Free circle actions and positive Ricci curvature on manifolds with the cohomology ring of $S^2\times S^5$

Abstract: We classify which of the 672 oriented diffeomorphism types of closed, simply-connected spin 7-manifolds with the cohomology ring of $S^2\times S^5$ admit a free circle action. In addition, we show that whenever such an action exists, there exist infinitely many pairwise non-equivalent free circle actions. Finally, in almost all cases where such an action exists, we construct invariant Riemannian metrics of positive Ricci curvature.

• The paper classifies spin 7-manifolds with S²×S⁵ cohomology by leveraging Kreck-Stolz s-invariants to determine diffeomorphism, homeomorphism, and homotopy types.
• It shows that 462 of 672 diffeomorphism types admit free S¹ actions, with infinitely many non-equivalent actions and 441 types supporting invariant positive Ricci curvature.
• The research employs explicit constructions using principal circle bundles and advanced surgery techniques to bridge topological classification with geometric analysis.

Classification and Geometry of Spin Cohomology $S^2 \times S^5$ 7-Manifolds with Free Circle Actions

Overview

The paper "Free circle actions and positive Ricci curvature on manifolds with the cohomology ring of $S^2\times S^5$" (2603.29838) provides a thorough classification of simply-connected, spin 7-manifolds with the cohomology ring of $S^2 \times S^5$, focusing on the existence and properties of free $S^1$ actions and invariant metrics of positive Ricci curvature. Leveraging the Kreck-Stolz $s$-invariants, the article delivers complete diffeomorphism, homeomorphism, and homotopy classifications, quantifies the types admitting free circle actions, and establishes the existence of infinitely many non-equivalent such actions, often with invariant positive Ricci curvature.

Topological Classification

Spin cohomology $S^2 \times S^5$ manifolds are defined as smooth, closed, simply-connected spin 7-manifolds whose cohomology ring matches that of $S^2 \times S^5$. The Kreck-Stolz $s$-invariants $(s_1, s_2, s_3)$, derived from characteristic numbers and cobounding manifolds, uniquely determine the oriented diffeomorphism, homeomorphism, and homotopy types:

• Diffeomorphism types: $672$ distinct oriented types, parameterized by $S^2\times S^5$0.
• Homeomorphism types: $S^2\times S^5$1 oriented types, determined by $S^2\times S^5$2.
• Homotopy types: $S^2\times S^5$3 types, with a more intricate relation involving $S^2\times S^5$4 and $S^2\times S^5$5.

The comprehensive computation and realization of all allowed $S^2\times S^5$6-invariant values are supported by explicit constructions using principal circle bundles over classified 6-manifolds, and computational verification.

Existence of Free $S^2\times S^5$7 Actions

The existence of free circle actions is governed by the vanishing of the Euler characteristic and finer bundle-theoretic obstructions. The main result is as follows:

• 462 out of 672 diffeomorphism types admit a free $S^2\times S^5$8 action, covering $S^2\times S^5$9 of $S^2 \times S^5$0 homeomorphism types and $S^2 \times S^5$1 of $S^2 \times S^5$2 homotopy types.
• The classification is explicit in terms of the $S^2 \times S^5$3-invariants: certain values correspond to spin or non-spin orbit spaces, with additional parity conditions and bundle data precisely dictating realizability.
• Whenever such an action exists, infinitely many pairwise non-equivalent free circle actions are possible, arising from distinct principal bundle structures and Euler classes in the base manifold.

This result extends classical work on homotopy spheres, products, and connected sums, significantly broadening the landscape for free circle actions in high-dimensional topology.

Geometry: Invariant Positive Ricci Curvature

A central geometric question concerns the existence of invariant metrics of positive Ricci curvature for manifolds with a free circle action. Prior work established that all spin cohomology $S^2 \times S^5$4 admit metrics of positive Ricci curvature, but the invariant case is subtler and often constrained by the orbit space:

• 441 diffeomorphism types admit both infinitely many inequivalent free circle actions and, for each action, an invariant metric of positive Ricci curvature.
• This includes all diffeomorphism types homeomorphic to $S^2 \times S^5$5, i.e., the classical case, paralleling and generalizing results for products and plumbings.
• For the remaining $S^2 \times S^5$6 types admitting free circle actions but not covered, it is open whether invariant positive Ricci curvature metrics exist for the associated actions, as their orbit spaces are non-spin and present unresolved analytic obstructions.

The construction is backed by Gilkey-Park-Tuschmann's criterion: if the quotient of the free $S^2 \times S^5$7 action admits positive Ricci curvature, then so does the total space with an invariant metric, and this criterion is verified for large classes of 6-manifolds (plumbings and $S^2 \times S^5$8-bundles over $S^2 \times S^5$9).

Advanced Constructions and Suspension Operations

The results are extended to connected sums involving homotopy spheres and higher-dimensional factors using suspension operations and plumbing constructions. For manifolds of the form $S^1$0, with appropriate parity constraints, infinitely many inequivalent free circle actions are admitted, each supporting an invariant metric of positive Ricci curvature.

The suspension and surgery framework employed aligns with contemporary topological surgery techniques, exploiting bundle and cobordism invariants to transfer geometric properties across connected sums.

Implications and Future Directions

The explicit s-invariant classification unifies topological and geometric aspects of 7-manifolds with cohomology ring $S^1$1, resolving longstanding classification problems and providing a toolkit for constructing manifolds with prescribed topological and geometric features. The results have implications for:

• Transformation group theory: explicit classification of manifolds admitting free actions enhances understanding of symmetries in high dimensions and their relation to curvature and cobordism invariants.
• Riemannian geometry: bridging free actions and positive Ricci curvature strengthens connections between topology, geometric analysis, and bundle theory, especially in the context of principal circle bundles and their orbit spaces.
• Homotopy theory and exotic structures: the existence of infinitely many non-equivalent actions and the detailed homotopy classification supports the study of exotic smooth structures and bundle moduli in dimension 7.

Open questions remain regarding the analytic obstructions for invariant positive Ricci curvature on certain types, as well as potential extensions of the classification to broader classes of manifolds and bundle actions. Speculatively, analogous frameworks may be effective in dimensions beyond 7 and for other symmetry groups, contingent on further advances in surgery theory and metric foliation analysis.

Conclusion

This paper provides a definitive classification of spin cohomology $S^1$2 7-manifolds admitting free circle actions, establishes the profusion of inequivalent such actions, and characterizes the existence of invariant metrics of positive Ricci curvature for almost all such cases. The synthesis of topological invariants, bundle theory, and geometric analysis sets a foundation for further exploration of transformation groups and curvature conditions on high-dimensional manifolds (2603.29838).