Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 094, 28 pages      arXiv:2603.29838      https://doi.org/10.3842/SIGMA.2026.094

Free Circle Actions and Positive Ricci Curvature on Manifolds with the Cohomology Ring of $S^2\times S^5$

Philipp Reiser
Karlsruhe, Germany

Received April 21, 2026, in final form September 15, 2026; Published online October 02, 2026

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.

Key words: positive Ricci curvature; free circle action; 6-manifold; $s$-invariant; principal circle bundle.

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References

  1. Antieau B., Williams B., The topological period-index problem over 6-complexes, J. Topol. 7 (2014), 617-640, arXiv:1208.4430.
  2. Bauer T., Quigley J.D., Infinite families of very exotic spheres with free ${S}^1$- and ${S}^3$-actions, arXiv:2603.23241.
  3. Bérard-Bergery L., Certains fibrés à courbure de Ricci positive, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A929-A931.
  4. Bérard-Bergery L., Scalar curvature and isometry group, in Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, 9-28.
  5. Bredon G.E., Introduction to compact transformation groups, Pure Appl. Math., Vol. 46, Academic Press, New York, 1972.
  6. Burdick B.L., Metrics of positive Ricci curvature on connected sums: projective spaces, products, and plumbings, Ph.D. Thesis, University of Oregon, 2019, available at https://www.proquest.com/docview/2293989895.
  7. Burdick B.L., Ricci-positive metrics on connected sums of projective spaces, Differential Geom. Appl. 62 (2019), 212-233, arXiv:1705.05055.
  8. Burdick B.L., Metrics of positive Ricci curvature on the connected sums of products with arbitrarily many spheres, Ann. Global Anal. Geom. 58 (2020), 433-476, arXiv:1811.11906.
  9. Davis J.F., Kirk P., Lecture notes in algebraic topology, Grad. Stud. Math., Vol. 35, American Mathematical Society, Providence, RI, 2001.
  10. Duan H., Circle actions and suspension operations on smooth manifolds, Math. Proc. Cambridge Philos. Soc. 180 (2026), 643-662, arXiv:2202.06225.
  11. Duan H., Liang C., Circle bundles over 4-manifolds, Arch. Math. (Basel) 85 (2005), 278-282, arXiv:math/0502469.
  12. Eells Jr. J., Kuiper N.H., An invariant for certain smooth manifolds, Ann. Mat. Pura Appl. 60 (1962), 93-110.
  13. Escher C., Ziller W., Topology of non-negatively curved manifolds, Ann. Global Anal. Geom. 46 (2014), 23-55, arXiv:1206.5997.
  14. Escher C.M., Montagantirud P., Classifying seven dimensional manifolds of fixed cohomology type, Differential Geom. Appl. 49 (2016), 312-325.
  15. Galaz-García F., Reiser P., Free torus actions and twisted suspensions, Forum Math. Sigma 13 (2025), e3, 31 pages, arXiv:2305.06068.
  16. Gilkey P.B., Park J., Tuschmann W., Invariant metrics of positive Ricci curvature on principal bundles, Math. Z. 227 (1998), 455-463.
  17. Goldstein R.Z., Lininger L., A classification of $6$-manifolds with free $S^1$ actions, in Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I, Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, 316-323.
  18. Gromoll D., Walschap G., Metric foliations and curvature, Progr. Math., Vol. 268, Birkhäuser, Basel, 2009.
  19. Hsiang W.-C., A note on free differentiable actions of $S^1$ and $S^{3}$ on homotopy spheres, Ann. of Math. 83 (1966), 266-272.
  20. Huang R., Sphere bundles over 4-manifolds are trivial after looping, Math. Z. 310 (2025), 48, 12 pages, arXiv:2210.17352.
  21. Husemoller D., Fibre bundles, 3rd ed., Grad. Texts in Math., Vol. 20, Springer, New York, 1994.
  22. Jiang Y., Regular circle actions on 2-connected 7-manifolds, J. Lond. Math. Soc. 90 (2014), 373-387, arXiv:1302.0923.
  23. Jiang Y., Su Y., Free circle actions on $(n-1)$-connected $(2n+1)$-manifolds, Pacific J. Math. 338 (2025), 1-17, arXiv:2409.03194.
  24. Jupp P.E., Classification of certain $6$-manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293-300.
  25. Kobayashi S., Fixed points of isometries, Nagoya Math. J. 13 (1958), 63-68.
  26. Kreck M., Stolz S., A diffeomorphism classification of $7$-dimensional homogeneous Einstein manifolds with ${\rm SU}(3)\times{\rm SU}(2)\times{\rm U}(1)$-symmetry, Ann. of Math. 127 (1988), 373-388.
  27. Kreck M., Stolz S., Some nondiffeomorphic homeomorphic homogeneous $7$-manifolds with positive sectional curvature, J. Differential Geom. 33 (1991), 465-486.
  28. Kreck M., Stolz S., A correction on ''Some nondiffeomorphic homeomorphic homogeneous $7$-manifolds with positive sectional curvature'', J. Differential Geom. 49 (1998), 203-204.
  29. Kreck M., Su Y., Mapping class group of manifolds which look like 3-dimensional complete intersections, Duke Math. J. 174 (2025), 501-574, arXiv:2009.08054.
  30. Kruggel B., A homotopy classification of certain $7$-manifolds, Trans. Amer. Math. Soc. 349 (1997), 2827-2843.
  31. Kruggel B., Kreck-Stolz invariants, normal invariants and the homotopy classification of generalised Wallach spaces, Quart. J. Math. Oxford Ser. (2) 49 (1998), 469-485.
  32. Milnor J.W., Stasheff J.D., Characteristic classes, Ann. of Math. Stud., Vol. 76, Princeton University Press, Princeton, NJ, 1974.
  33. Montgomery D., Yang C.T., Differentiable actions on homotopy seven spheres, Trans. Amer. Math. Soc. 122 (1966), 480-498.
  34. Montgomery D., Yang C.T., Differentiable actions on homotopy seven spheres. II, in Proc. Conf. on Transformation Groups (New Orleans, La., 1967), Springer, New York, 1968, 125-134.
  35. Reiser P., Generalized surgery on Riemannian manifolds of positive Ricci curvature, Ph.D. Thesis, Karlsruher Institut für Technologie, 2024, available at https://doi.org/10.5445/IR/1000150280.
  36. Reiser P., Generalized surgery on Riemannian manifolds of positive Ricci curvature, Trans. Amer. Math. Soc. 376 (2023), 3397-3418, arXiv:2103.05517.
  37. Reiser P., Metrics of positive Ricci curvature on simply-connected manifolds of dimension $6k$, J. Topol. 17 (2024), e70007, 50 pages, arXiv:2210.15610.
  38. Schultz R., The nonexistence of free $S^{1}$ actions on some homotopy spheres, Proc. Amer. Math. Soc. 27 (1971), 595-597.
  39. Thomas A., Almost complex structures on complex projective spaces, Trans. Amer. Math. Soc. 193 (1974), 123-132.
  40. Wall C.T.C., Classification problems in differential topology. V. On certain $6$-manifolds, Invent. Math. 1 (1966), 355-374.
  41. Wall C.T.C., Corrigendum to ''Classification problems in differential topology. V. On certain $6$-manifolds'', Invent. Math. 2 (1967), 306.
  42. Wang X., On the classification of certain 1-connected 7-manifolds and related problems, Q. J. Math. 73 (2022), 711-727, arXiv:1810.08474.
  43. Wraith D., Exotic spheres with positive Ricci curvature, J. Differential Geom. 45 (1997), 638-649.
  44. Xu F., Free circle actions on certain simply connected 7-manifolds, Topology Appl. 375 (2025), 109548, 26 pages, arXiv:2409.04938.
  45. Zhubr A.V., Classification of simply connected six-dimensional spin manifolds, Math. USSR-Izv. 9 (1975), 793-812.
  46. Zhubr A.V., Closed simply connected 6-manifolds: the proofs of classification theorems, St. Petersburg Math. J. 12 (2001), 605-680.

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