Invariants of Homogeneous Spaces

We derive a formula for the η-invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extensions. As an example, we give some computations for spheres. Quotients M = G/H of compact Lie groups provide many important examples of Riemannian manifolds with non-negative sectional curvature. The primary characteristic classes and numbers of these spaces have been computed by Borel and Hirzebruch in [5]. The η-invariant has been introduced by Atiyah, Patodi and Singer in [1] as a boundary contribution in an index theorem for manifolds with boundary. It can be used to construct certain secondary invariants of compact manifolds M that were originally defined using zerobordisms. For example, the Eells-Kuiper and Kreck-Stolz invariants distinguish homeomorphic homogeneous manifolds that are not diffeomorphic. These invariants can be expressed in terms of η-invariants and Chern-Simons numbers, see e.g. [6], [15]. Although the diffeomorphism type of many homogeneous manifolds G/H is well-known, in some cases the explicit values of certain η-invariants are needed to complete the diffeomorphism classification. It is therefore worthwhile to have a formula for the η-invariants of equivariant Dirac operators on homogeneous spaces. First steps in this direction have been made in [8], [9], [10]. There, we computed the equivariant η-invariant of a different operator, called “reductive” or “cubic” Dirac operator, and explained how to recover the η-invariant of the classical Dirac operator. However, one complicated local term remained, called “Bott localisation defect” below. The central result of the present article is a formula for this defect term that is similar to the formula for the equivariant η-invariant of the reductive Dirac operator itself. Thus we get a tractable formula for equivariant η-invariants of homogeneous spaces that is useful for explicit computations. This formula has already been applied in a joint paper [11] with N. Kitchloo and K. Shankar to calculate the Eells-Kuiper invariant of the Berger space SO(5)/SO(3), and to determine its diffeomorphism type. Our formula can be summarised as follows; details and notation will be explained later in the article. Theorem (see Theorem 2.33 below). Let G ⊃ H be compact Lie groups, and let Dκ be the equivariant Dirac operator on M = G/H twisted by the local bundle V κM associated to an irreducible h-representation with highest weight κ. Then η(Dκ) is a sum of the following terms: 2000 Mathematics Subject Classification. 58J28; 53C30. 1