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 Z2-equivariant Stiefel-Whitney numbers (see Section 1 for details). Then:THEOREM 0.1. Let. If M = ker*(η, G) ∩ ker*(SW, G), M = 0.
Topological spherical space form problem. III. Dimensional bounds and smoothing
