Abstract
The eta invariant of Atiyah et al is a spectral invariant measuring the spectral asymmetry of a self adjoint elliptic operator P. In general it is a geometric invariant. However, if dim(M)+order(P) is odd, it is rigid and is a homotopy invariant. There are many examples known where this invariant is non trivial if dim(M) is even and order(P)=1. However, it is not known if the invariant is non trivial if dim(M) is odd and order(P) is even. In this note, we discuss some of the functorial properties which this invariant has.
Research partially supported by the NSF and NSA; MOS number 58G25
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry I, Mat Proc Camb Philos Soc 77 (1975), 43–60; II Math Proc Camb Philos Soc 78 (1975); III Math Proc Camb Philos Soc 79 (1976), 71–99.
A. Bahri and P. Gilkey, The eta invariant, Pin c bordism, and equivariant Spin c bordism for cyclic 2 groups, Pacific journal of mathematics 128 (1987), 1–24.
R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. AMS 64 (1958), 87–89.
P.Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem. Publish or Perish Press (1984).
P.Gilkey, The geometry of spherical space form groups, (to appear).
P. Gilkey, The eta invariant and non singular bilinear products on R n, Canad. Math. Bull. 30 (1987), 147–154.
M.Karoubi, Algèbres de Clifford et K-théorie, Ann. Sci. Ec. Norm (1968), 161–270.
M. Karoubi, Sur la K-théorie équivariante, Springer Lecture Notes 136 (1970), 187–253.
R.T. Seeley, Complex powers of an elliptic operator, Proc. Sympos. Pure Math 10 (1967), 288–307.
S.Stolz, Exotic structures on 4 manifolds detected by spectral invariants (preprint, Notre Dame).
J.Wolf, Spaces of constant curvature, 5 th ed, Publish or Perish Press (1984).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this chapter
Cite this chapter
Gilkey, P.B. (1989). The eta invariant of even order operators. In: Carreras, F.J., Gil-Medrano, O., Naveira, A.M. (eds) Differential Geometry. Lecture Notes in Mathematics, vol 1410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0086423
Download citation
DOI: https://doi.org/10.1007/BFb0086423
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51885-3
Online ISBN: 978-3-540-46858-5
eBook Packages: Springer Book Archive