Geodesic
The Eta Invariant and Equivariant SpinC Bordism for Spherical Space form Groups
Canadian journal of mathematics, Tome 40 (1988) no. 2, pp. 392-428

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A finite group G is a spherical space form group if it admits a fixed point free representation τ:G → U(k) for some k; for the remainder of this paper, we assume G is such a group. The eta invariant of Atiyah et al [2] defines Q/Z valued invariants of equivariant bordism. In [6], we showed the eta invariant completely detects the reduced equivariant unitary bordism groups and completely detects all but the 2-primary part of the reduced equivariant SpinC bordism groups . The coefficient ring is without torsion; all the torsion in is of order 2. The prime 2 plays a distinguished role in the discussion of equivariant SpinC bordism and is quite different from at the prime 2. Let ker*(η, G) denote the kernel of all eta invariants and let ker*(SW, G) denote the kernel of the Z 2-equivariant Stiefel-Whitney numbers (see Section 1 for details). Then:THEOREM 0.1. Let. If M = ker*(η, G) ∩ ker*(SW, G), M = 0. 
Gilkey, Peter B. The Eta Invariant and Equivariant SpinC Bordism for Spherical Space form Groups. Canadian journal of mathematics, Tome 40 (1988) no. 2, pp. 392-428. doi: 10.4153/CJM-1988-016-8
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