Home Free cyclic group actions on highly-connected 2n-manifolds
Article
Licensed
Unlicensed Requires Authentication

Free cyclic group actions on highly-connected 2n-manifolds

  • Yang Su EMAIL logo and Jianqiang Yang
Published/Copyright: November 18, 2020
Forum Mathematicum
From the journal Forum Mathematicum Volume 33 Issue 2

Abstract

In this paper we study smooth orientation-preserving free actions of the cyclic group ℤ/m on a class of (n-1)-connected 2⁢n-manifolds, ♯⁡♯⁡g⁢(Sn×Sn)Σ, where Σ is a homotopy 2⁢n-sphere. When n=2, we obtain a classification up to topological conjugation. When n=3, we obtain a classification up to smooth conjugation. When n≥4, we obtain a classification up to smooth conjugation when the prime factors of m are larger than a constant C⁢(n).

Keywords: Highly-connected manifolds; free group actions; cobordism; surgery
MSC 2010: 57R65; 57R19; 57S17; 57S25

Communicated by Frederick R. Cohen

Acknowledgements

We would like to thank Matthias Kreck for clarifying the relation between the L- and tilde L-groups in Lemma 3.5. We would like to thank Ian Hambleton and Diarmuid Crowley for helpful communications.

References

[1] J. F. Adams, On the groups J⁢(X). IV, Topology 5 (1966), 21–71. 10.1016/0040-9383(66)90004-8Search in Google Scholar

[2] A. Bak, K-Theory of Forms, Ann. of Math. Stud. 98, Princeton University, Princeton, 1981. Search in Google Scholar

[3] H. J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, De Gruyter Exp. Math. 2, Walter de Gruyter, Berlin, 1991. 10.1515/9783110854480Search in Google Scholar

[4] H. J. Baues and B. Bleile, Poincaré duality complexes in dimension four, Algebr. Geom. Topol. 8 (2008), no. 4, 2355–2389. 10.2140/agt.2008.8.2355Search in Google Scholar

[5] D. Crowley and I. Hambleton, Finite group actions on Kervaire manifolds, Adv. Math. 283 (2015), 88–129. 10.1016/j.aim.2015.06.010Search in Google Scholar

[6] S. R. Galatius and O. Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds. I, J. Amer. Math. Soc. 31 (2018), no. 1, 215–264. 10.1090/jams/884Search in Google Scholar

[7] I. Hambleton, Free-Involutions on Highly-Connected Manifolds, ProQuest LLC, Ann Arbor, 1973; Thesis (Ph.D.)–Yale University. Search in Google Scholar

[8] I. Hambleton, Free involutions on 6-manifolds, Michigan Math. J. 22 (1975), no. 2, 141–149. 10.1307/mmj/1029001471Search in Google Scholar

[9] I. Hambleton and M. Kreck, Cancellation, elliptic surfaces and the topology of certain four-manifolds, J. Reine Angew. Math. 444 (1993), 79–100. 10.1515/crll.1993.444.79Search in Google Scholar

[10] I. Hambleton and M. Kreck, Recognizing products of surfaces and simply connected 4-manifolds, Proc. Amer. Math. Soc. 143 (2015), no. 5, 2253–2262. 10.1090/S0002-9939-2014-12425-7Search in Google Scholar

[11] I. Hambleton, M. Kreck and P. Teichner, Nonorientable 4-manifolds with fundamental group of order 2, Trans. Amer. Math. Soc. 344 (1994), no. 2, 649–665. 10.1090/S0002-9947-1994-1234481-2Search in Google Scholar

[12] I. Hambleton and L. R. Taylor, A guide to the calculation of the surgery obstruction groups for finite groups, Surveys on Surgery Theory. Vol. 1, Ann. of Math. Stud. 145, Princeton University, Princeton (2000), 225–274. 10.1515/9781400865192-013Search in Google Scholar

[13] M. A. Kervaire, On the Pontryagin classes of certain SO⁢(n)-bundles over manifolds, Amer. J. Math. 80 (1958), 632–638. 10.2307/2372775Search in Google Scholar

[14] M. A. Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161–169. 10.1215/ijm/1255455861Search in Google Scholar

[15] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. 10.1142/9789812836878_0002Search in Google Scholar

[16] R. C. Kirby and L. R. Taylor, A calculation of Pin+ bordism groups, Comment. Math. Helv. 65 (1990), no. 3, 434–447. 10.1007/BF02566617Search in Google Scholar

[17] M. Kreck, Isotopy classes of diffeomorphisms of (k-1)-connected almost-parallelizable 2⁢k-manifolds, Algebraic Topology, Lecture Notes in Math. 763, Springer, Berlin (1979), 643–663. 10.1007/BFb0088108Search in Google Scholar

[18] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no. 3, 707–754. 10.2307/121071Search in Google Scholar

[19] T. Lance, Free cyclic actions on manifolds, Comment. Math. Helv. 50 (1975), 59–80. 10.1007/BF02565734Search in Google Scholar

[20] B. Mazur, Stable equivalence of differentiable manifolds, Bull. Amer. Math. Soc. 67 (1961), 377–384. 10.1090/S0002-9904-1961-10626-5Search in Google Scholar

[21] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358–426. 10.1090/S0002-9904-1966-11484-2Search in Google Scholar

[22] J. Milnor, Classification of (n-1)-connected 2⁢n-dimensional manifolds and the discovery of exotic spheres, Surveys on Surgery Theory, Vol. 1, Ann. of Math. Stud. 145, Princeton University, Princeton (2000), 25–30. 10.1515/9781400865192-004Search in Google Scholar

[23] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Pure Appl. Math. 121, Academic Press, Orlando, 1986. Search in Google Scholar

[24] I. Reiner, Integral representations of cyclic groups of prime order, Proc. Amer. Math. Soc. 8 (1957), 142–146. 10.1090/S0002-9939-1957-0083493-6Search in Google Scholar

[25] J.-P. Serre, Cohomologie modulo 2 des complexes d’Eilenberg–MacLane, Comment. Math. Helv. 27 (1953), 198–232. 10.1007/BF02564562Search in Google Scholar

[26] P. Teichner, On the signature of four-manifolds with universal covering spin, Math. Ann. 295 (1993), no. 4, 745–759. 10.1007/BF01444915Search in Google Scholar

[27] C. M. Thatcher, On free ℤp actions on products of spheres, Geom. Dedicata 148 (2010), 391–415. 10.1007/s10711-010-9506-7Search in Google Scholar

[28] C. T. C. Wall, Classification of (n-1)-connected 2⁢n-manifolds, Ann. of Math. (2) 75 (1962), 163–189. 10.2307/1970425Search in Google Scholar

[29] C. T. C. Wall, Surgery of non-simply-connected manifolds, Ann. of Math. (2) 84 (1966), 217–276. 10.2307/1970519Search in Google Scholar

[30] C. T. C. Wall, Classification of Hermitian Forms. VI. Group rings, Ann. of Math. (2) 103 (1976), no. 1, 1–80. 10.2307/1971019Search in Google Scholar

Received: 2020-09-16
Published Online: 2020-11-18
Published in Print: 2021-03-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
https://doi.org/10.1515/forum-2020-0262
Keywords for this article
Highly-connected manifolds; free group actions; cobordism; surgery

Articles in the same Issue

  1. Frontmatter
  2. Lower bounds for regular genus and gem-complexity of PL 4-manifolds with boundary
  3. Free cyclic group actions on highly-connected 2n-manifolds
  4. The distinction problems for Sp4 and SO3,3
  5. Affine cones over cubic surfaces are flexible in codimension one
  6. Permutations of zero-sumsets in a finite vector space
  7. On the finiteness of solutions for polynomial-factorial Diophantine equations
  8. Galois action on Fuchsian surface groups and their solenoids
  9. On a Lévy process pinned at random time
  10. Borsuk–Ulam theorem for filtered spaces
  11. A non-commutative differential module approach to Alexander modules
  12. Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behavior
  13. Quantum modularity of partial theta series with periodic coefficients
  14. Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian
  15. Zeros of GL2 𝐿-functions on the critical line
  16. Weighted boundedness of multilinear Calderón commutators
  17. Two characterizations of central BMO space via the commutators of Hardy operators
  18. Weyl 𝑛-algebras and the Swiss cheese operad
  19. Syzygies in equivariant cohomology in positive characteristic
  20. Epsilon factors of symplectic type characters in the wild case
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 27.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2020-0262/html
Scroll to top button