Mathematical Notes

, Volume 71, Issue 1–2, pp 245–261 | Cite as

The Eta-Invariant and Pontryagin Duality in K-Theory

  • A. Yu. Savin
  • B. Yu. Sternin


The topological significance of the spectral Atiyah--Patodi--Singer \(\eta \)-invariant is investigated. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. Pontryagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented.

eta-invariant K-theory Pontryagin duality linking coefficients Atiyah--Patodi--Singer theory modulo n index 


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  1. 1.
    P. B. Gilkey, “The eta invariant of even-order operators,” Lecture Notes in Math., 1410 (1989), 202–211.Google Scholar
  2. 2.
    M. Atiyah, V. Patodi, and I. Singer, “Spectral asymmetry and Riemannian geometry, I,” Math. Proc. Cambridge Phil. Soc., 77 (1975), 43–69, “II,” 78 (1976), 405–432, “III,” 79 (1976), 315–330.Google Scholar
  3. 3.
    P. B. Gilkey, “The eta invariant for even dimensional Pinc manifolds,” Advances in Math., 58 (1985), 243–284.Google Scholar
  4. 4.
    A. Bahri and P. Gilkey, “The eta invariant, Pinc bordism, and equivariant Spinc bordism for cyclic 2-groups,” Pacific J. Math., 128 (1987), no. 1, 1–24.Google Scholar
  5. 5.
    L. S. Pontrjagin, “Ñber den algebraischen Inhalt topologischer Dualitatssatze,” Math. Ann., 105 (1931), no. 2, 165–205.Google Scholar
  6. 6.
    L. S. Pontryagin, “Topological duality theorems,” Uspekhi Mat. Nauk [Russian Math. Surveys], 2 (1947), no. 2, 21–44.Google Scholar
  7. 7.
    H. Seifert and W. Threllfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934.Google Scholar
  8. 8.
    A. Fomenko and D. Fuks, Course of Homotopy topology [in Russian], Nauka, Moscow, 1989.Google Scholar
  9. 9.
    J. Milnor, “Torsion et type simple d'homotopie,” in: Essays Topol. Relat. Top. Mem. dédiés à Georges de Rham, 1970, pp. 12–17.Google Scholar
  10. 10.
    G. Brumfiel and J. Morgan, “Quadratic functions, the index modulo 8, and a ℤ/4-Hirzebruch formula,” Topology, 12 (1973), 105–122.Google Scholar
  11. 11.
    M. Kervaire and J. Milnor, “Groups of homotopy spheres, I,” Ann. Math., 77 (1963), no. 3, 504–537.Google Scholar
  12. 12.
    G. Moore and E. Witten, “Self-duality, Ramond-Ramond fields and K-theory,” in: E-print hepth/9912279, 1999.Google Scholar
  13. 13.
    A. Yu. Savin and B. Yu. Sternin, “Elliptic operators in even subspaces,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 190 (1999), no. 8, 125–160.Google Scholar
  14. 14.
    A. Yu. Savin and B. Yu. Sternin, Elliptic Operators in Odd Subspaces, Preprint no. 242 99/11 (Juni 1999), Univ. Potsdam, Institut für Mathematik, Potsdam, 1999, in: E-print math.DG/9907039.Google Scholar
  15. 15.
    A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Operators in Subspaces, Preprint no. 242 00/04 (Februar 2000), Univ. Potsdam, Institut für Mathematik, Potsdam, 2000.Google Scholar
  16. 16.
    A. Savin, B.-W. Schulze, and B. Sternin, Elliptic Operators in Subspaces and the Eta Invariant, Preprint no. 242 99/14 (Juni 1999), Univ. Potsdam, Institut für Mathematik, Potsdam, 1999, in: E-print math.DG/9907047.Google Scholar
  17. 17.
    M. Karoubi, K-Theory. An Introduction, vol. 226, Grundlehren Math. Wiss, Springer-Verlag, Berlin, 1978.Google Scholar
  18. 18.
    M. F. Atiyah and I. M. Singer, “The index of elliptic operators, I,” Ann. of Math., 87 (1968), 484–530.Google Scholar
  19. 19.
    M. S. Birman and M. Z. Solomyak, “On the subspaces admitting a pseudodifferential projection,” Vestnik LGU (1982), no. 1, 18–25.Google Scholar
  20. 20.
    R. Bott and L. Tu, Differential Forms in Algebraic Topology, vol. 82, Graduate Texts in Math., Springer-Verlag, Berlin-Heidelberg-New York, 1982.Google Scholar
  21. 21.
    P. Baum and R. G. Douglas, “Toeplitz operators and Poincaré duality,” in: Toeplitz Centennial. Proceedings of Toeplitz Memory Conf. (Tel Aviv, 1981). Operator Theory Adv. Appl., 4 (1982), 137–166.Google Scholar
  22. 22.
    M. Karoubi, “Algèbres de Clifford et K-théorie,” Ann. Sci. École Norm. Sup., 4 (1968), no. 1, 161–270.Google Scholar
  23. 23.
    A. S. Mishchenko, Vector Bundles and Their Applications [in Russian], Nauka, Moscow, 1984.Google Scholar
  24. 24.
    A. Connes, D. Sullivan, and N. Teleman, “Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes,” Topology, 33 (1994), no. 4, 663–681.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • A. Yu. Savin
    • 1
  • B. Yu. Sternin
    • 1
  1. 1.M. V. Lomonosov Moscow State UniversityRussia

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