Involutions on compact 3-manifolds

Abstract

Let G be a finite group and X a closed fixed-point free G-manifold of odd dimension, that is G acts on X preserving the orientation. We have associated to (G,X) for each g E G, g ≠ 1, an invariant alpha, as follows. According to the free cobordism theory of Conner and Floyd (4), the disjoint union kX of k copies bounds a free G-manifold Y, for some k. alpha is defined by [equation]. When G has order two, [equation], we have a fixed-point free involution [equation] and it turns out that alpha coincides with the Browder-Livesay invariant ([beta)] of (T,X). In this thesis we develop the proof by F. Hirzebruch and K. Janich that alpha = beta, when H2m+2 (X,Q) = 0, where dim X = 4m + 3. We also compute the Browder-Livesay invariant of involutions derived from free actions of the generalized quaternion / groups [equation] on the spheres S4m-1. Furthermore, Lopez de Medrano constructs involutions on homology 3-spheres, as follows. Theorem (Medrano). For every i E Z , there is a fixed-point free in volution (T, sigma3) of a homology 3-sphere sigma3 such that beta(T, sigma3) = 8i. We work with the examples above and prove the following theorem. Theorem . If beta(T, sigma3)/8 is odd, where (T, sigma3) is one of Medrano's examples, then sigma3 cannot be h-cobordant to S3. Also, sigma3 does not imbed in R4. For this, we compute first the signature of a suitable 4-manifold with sigma3 as boundary, and compute the mu-invariant of sigma3.