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Geometriae Dedicata

, Volume 200, Issue 1, pp 265–285 | Cite as

On the cardinality of the manifold set

  • Diarmuid CrowleyEmail author
  • Tibor Macko
Original Paper
  • 54 Downloads

Abstract

We study the cardinality of the set of homeomorphism classes of manifolds homotopy equivalent to a given manifold M and compare it to the cardinality of the structure set of M, as defined in surgery theory.

Keywords

Manifold set Structure set Rigidity Surgery Divisibility 

Mathematics Subject Classification (2010)

Primary: 57R65 57R67 

Notes

Acknowledgements

We would like to thank Jim Davis, Wolfgang Lück and Shmuel Weinberger for helpful and stimulating conversations.

References

  1. 1.
    Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 2(72), 20–104 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adams, J.F.: On the groups \(J(X)\). IV. Topology 5, 21–71 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brookman, J., Davis, J.F., Khan, Q.: Manifolds homotopy equivalent to \(P^{n} \# P^{n}\). Math. Ann. 338(4), 947–962 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bass, H., Heller, A., Swan, R.G.: The Whitehead group of a polynomial extension. Inst. Hautes Études Sci. Publ. Math. 22, 61–79 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bott, R.: The stable homotopy of the classical groups. Proc. Natl. Acad. Sci. USA 43, 933–935 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crowley, D., Hambleton, I.: Finite group actions on Kervaire manifolds. Adv. Math. 283, 88–129 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crowley, D., Sixt, J.: Stably diffeomorphic manifolds and \(l_{2q+1}(\mathbb{Z}[\pi ])\). Forum Math. 23(3), 483–538 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, S., Weinberger, S.: On invariants of Hirzebruch and Cheeger–Gromov. Geom. Topol. 7, 311–319 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jahren, B., Kwasik, S.: Free involutions on \(S ^{1} \times S ^{n}\). Math. Ann. 351(2), 281–303 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Khan, Q.: Free transformations of \(S ^{1} \times S ^{n }\) of square-free odd period. Indiana Univ. Math J. 66(5), 1453–1482 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kreck, M., Lück, W.: Topological rigidity for non-aspherical manifolds. Pure Appl. Math. Q. 5(3, Special Issue: In honor of Friedrich Hirzebruch. Part 2), 873–914 (2009)Google Scholar
  12. 12.
    Kervaire, M.A., Milnor, J.W.: Groups of homotopy spheres. I. Ann. Math. 2(77), 504–537 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kreck, M.: Surgery and duality. Ann. Math. (2) 149(3), 707–754 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kirby, R.C., Siebenmann, L.C.: Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press, Princeton. With notes by John Milnor and Michael Atiyah, p. 88. Annals of Mathematics Studies, No (1977)Google Scholar
  15. 15.
    Lück, W., Reich, H.: The Baum–Connes and the Farrell–Jones conjectures in \(K\)- and \(L\)-theory. In: Friedlander, E.M., Grayson, D.R. (eds.) Handbook of \(K\)-Theory, Vols. 1, 2, pp. 703–842. Springer, Berlin (2005)Google Scholar
  16. 16.
    Lück, W: A basic introduction to surgery theory. In: Topology of High-Dimensional Manifolds, No. 1, 2 (Trieste, 2001), volume 9 of ICTP Lecture Notes, pp. 1–224. The Abdus Salam International Centre for Theoretical Physics, Trieste (2002)Google Scholar
  17. 17.
    Madsen, I.B., Milgram, R.J.: The Classifying Spaces for Surgery and Cobordism of Manifolds, Volume 92 of Annals of Mathematics Studies. Princeton University Press, Princeton (1979)Google Scholar
  18. 18.
    Milnor, J., Spanier, E.: Two remarks on fiber homotopy type. Pac. J. Math. 10, 585–590 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Madsen, I., Taylor, L.R., Williams, B.: Tangential homotopy equivalences. Comment. Math. Helv. 55(3), 445–484 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Novikov, S.P.: On manifolds with free abelian fundamental group and their application. Izv. Akad. Nauk SSSR Ser. Mat. 30, 207–246 (1966)MathSciNetGoogle Scholar
  21. 21.
    Ranicki, A.A.: Algebraic \(L\)-Theory and Topological Manifolds. Cambridge Tracts in Mathematics, vol. 102. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  22. 22.
    Ranicki, A.: A composition formula for manifold structures. Pure Appl. Math. Q. 5(2, part 1), 701–727 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shaneson, J.L.: Wall’s surgery obstruction groups for \(G\times Z\). Ann. Math. 2(90), 296–334 (1969)CrossRefzbMATHGoogle Scholar
  24. 24.
    Wall, C.T.C.: Classification problems in differential topology. IV. Thickenings. Topology 5, 73–94 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wall, C.T.C.: Surgery on Compact Manifolds, Volume 69 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (1999). Edited and with a foreword by A. A. RanickiGoogle Scholar
  26. 26.
    Weinberger, S.: On smooth surgery. Commun. Pure Appl. Math. 43(5), 695–696 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Whitehead, G.W.: Elements of Homotopy Theory. Graduate Texts in Mathematics, vol. 61. Springer, New York (1978)CrossRefzbMATHGoogle Scholar
  28. 28.
    Weinberger, S., Xie, Z., Yu, G.: Additivity of Higher Rho Invariants and Nonrigidity of Topological Manifolds. arXiv:1608.03661 (2017)
  29. 29.
    Weinberger, S., Guoliang, Y.: Finite part of operator \(K\)-theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds. Geom. Topol. 19(5), 2767–2799 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

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