Skip to main content

Structure sets of triples of manifolds

Abstract

The structure set of a given manifold fits into a surgery exact sequence, which is the main tool for classification of manifolds. In the present paper, we describe relations between various structure sets and groups of obstructions which naturally arise for triples of manifolds. The main results are given by commutative braids and diagrams of exact sequences.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    A. Bak and Yu. V. Muranov, “Splitting along submanifolds and L-spectra,” J. Math. Sci., 123, 4169–4183 (2004).

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    W. Browder and G. R. Livesay, “Fixed point free involutions on homotopy spheres,” Bull. Amer. Math. Soc., 73, 242–245 (1967).

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    W. Browder and F. Quinn, “A surgery theory for G-manifolds and stratified spaces,” in: Manifolds, Univ. of Tokyo Press (1975), pp. 27–36.

  4. 4.

    S. E. Cappell and J. L. Shaneson, “Pseudo-free actions. I,” in: Algebraic Topology (Aarhus, 1978 ), Lect. Notes Math., Vol. 763, Springer, Berlin (1979), pp. 395–447.

    Chapter  Google Scholar 

  5. 5.

    M. M. Cohen, A Course in Simple-Homotopy Theory, Springer, New York (1973).

    MATH  Google Scholar 

  6. 6.

    I. Hambleton, “Projective surgery obstructions on closed manifolds,” in: Algebraic K-Theory, Proc. Conf., Oberwolfach 1980, Part II, Lect. Notes Math., Vol. 967, Springer, Berlin (1982), 101–131.

    Google Scholar 

  7. 7.

    I. Hambleton and A. F. Kharshiladze, “A spectral sequence in surgery theory,” Russ. Acad. Sci. Sb. Math., 77, 1–9 (1994).

    Article  MathSciNet  Google Scholar 

  8. 8.

    I. Hambleton and E. Pedersen, Topological Equivalences of Linear Representations for Cyclic Groups, preprint, MPI (1997).

  9. 9.

    I. Hambleton, A. Ranicki, and L. Taylor, “Round L-theory,” J. Pure Appl. Algebra, 47, 131–154 (1987).

    MATH  Article  MathSciNet  Google Scholar 

  10. 10.

    S. Lopez de Medrano, Involutions on Manifolds, Springer, Berlin (1971).

    MATH  Google Scholar 

  11. 11.

    W. Lück and A. A. Ranicki, “Surgery transfer,” in: Algebraic Topology and Transformation Groups, Proc. Conf., Göttingen/FRG 1987, Lect. Notes Math., Vol. 1361, Springer, Berlin (1988), pp. 167–246.

    Chapter  Google Scholar 

  12. 12.

    W. Lück and A. A. Ranicki, “Surgery obstructions of fibre bundles,” J. Pure Appl. Algebra, 81, No. 2, 139–189 (1992).

    MATH  Article  MathSciNet  Google Scholar 

  13. 13.

    J. Malešič, Yu. V. Muranov, and D. Repovš, “Splitting obstruction groups in codimension 2,” Mat. Zametki, 69, 52–73 (2001).

    Google Scholar 

  14. 14.

    Yu. V. Muranov, “Splitting obstruction groups and quadratic extension of antistructures,” Russ. Acad. Sci. Izv. Math., 59, No. 6, 1207–1232 (1995).

    MATH  MathSciNet  Google Scholar 

  15. 15.

    Yu. V. Muranov, “Splitting problem,” Proc. Steklov Inst. Math., 212, 115–137 (1996).

    MathSciNet  Google Scholar 

  16. 16.

    Yu. V. Muranov and R. Jimenez, “Transfer maps for triples of manifolds,” Mat. Zametki, to appear.

  17. 17.

    Yu. V. Muranov and A. F. Kharshiladze, “Browder-Livesay groups of Abelian 2-groups,” Math. USSR Sb., 70, 499–540 (1991).

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

    Yu. V. Muranov and D. Repovš, “Groups of obstructions to surgery and splitting for a manifold pair,” Russ. Acad. Sci. Sb. Math., 188, No. 3, 449–463 (1997).

    MATH  Google Scholar 

  19. 19.

    Yu. V. Muranov, D. Repovš, and R. Jimenez, “Surgery spectral sequence and stratified manifolds,” in: Preprint of the University of Ljubljana, Vol. 42, No. 935, 2004, pp. 1–33.

    Google Scholar 

  20. 20.

    Yu. V. Muranov, D. Repovš, and F. Spaggiari, “Surgery on triples of manifolds,” Sb. Math., 194, No. 8, 1251–1271 (2003).

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    A. A. Ranicki, Exact Sequences in the Algebraic Theory of Surgery, Math. Notes, Vol. 26, Princeton Univ. Press, Princeton (1981).

    Google Scholar 

  22. 22.

    A. A. Ranicki, “The total surgery obstruction,” in: Algebraic Topology (Aarhus, 1978 ), Lect. Notes Math., Vol. 763, Springer, Berlin (1979), pp. 275–316.

    Chapter  Google Scholar 

  23. 23.

    A. A. Ranicki, Algebraic L-Theory and Topological Manifolds, Cambridge Tracts Math., Cambridge Univ. Press (1992).

  24. 24.

    R. Switzer, Algebraic Topology—Homotopy and Homology, Grund. Math. Wiss., Bd. 212, Springer Berlin (1975).

    Google Scholar 

  25. 25.

    C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, London (1970). Second Edition: A. A. Ranicki, ed., AMS, Providence (1999).

    Google Scholar 

  26. 26.

    S. Weinberger, The Topological Classification of Stratified Spaces, The University of Chicago Press, Chicago (1994).

    MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 153–172, 2005.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Muranov, Y.V., Jimenez, R. Structure sets of triples of manifolds. J Math Sci 144, 4468–4483 (2007). https://doi.org/10.1007/s10958-007-0285-0

Download citation

Keywords

  • Manifold
  • Exact Sequence
  • Fundamental Group
  • Commutative Diagram
  • Spectrum Level