Eta Invariant of Spherical Space Form

Question

Is the eta invariant of spherical space form $\eta(S^3/\Gamma)$ always nonegative? Can we calculate it with the information of $\Gamma\in SO(4)$ explicitly?

In fact, i need a reference for the calculation of eta invariant. Can some one give me some advice or download the following paper for me? Thank you!

(1) Hitchin, N. J.(4-CAMB) Einstein metrics and the eta-invariant. (Italian summary) Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 95–105.

(2) G.W. Gibbons, C.N. Pope Index theorem boundary terms for gravitational instantons Nuclear Physics B Volume 157, Issue 3, 1 October 1979, Pages 377–386.

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Answer 1

This is not a full answer, but for the lens spaces (i.e., $\Gamma$ cyclic), the eta invariant is computed in the second paper of Atiyah, Patodi and Singer. In case you're outside the paywall, you can try Proposition 5.2 in this eprint.

Edit

There are formula for the general case in this Inventiones paper of Peter Gilkey's.

Answer 2

See Theorem 5.2 of this paper of C. Bär for an explicit formula.

The real projective spheres $\mathbb R\mathbb P^n$ with $n\equiv 3\pmod 4$ admits two spin structures. Bär shows in Corollary 5.4 that the eta invariant is negative for one of them.

Answer 3

The eta invariant of any locally symmetric space $\Gamma \backslash G/K$ is calculated by Moscovici and Stanton in Eta invariants of Dirac operators on locally symmetric manifolds. They related eta invariant with the conjugation class of $\Gamma$.