The author studies the rational homology cobordisms of rational homology 3-spheres. In particular it is shown that the \(\rho_{\alpha}\)- invariants of Atiyah-Patodi-Singer are, under some conditions, integral homology cobordism invariants of rational homology spheres.
Related to this problem is the question of when a rational homology sphere \(\Sigma\) bounds a rational homology ball. This can also be answered in some cases in terms of \(\rho_{\alpha}\)-invariants. These invariants can be used to answer questions concerning sliceness of knots. A. J. Casson and C. McA. Gordon [“Cobordism of classical knots”, in: A la recherche de la topologie perdue, 81-197 (1986; Zbl 0597.57001)] have constructed an invariant detecting when a two-bridge knot is not ribbon. This is just the \(\rho_{\alpha}\)-invariant for double branched cover \(\Sigma\) of \(S^ 3\) branched over \(K\subset S^ 3\) and character \(\alpha\) : \(H_ 1(\Sigma)\to U(1)\). For characters of prime power order, the above mentioned authors show that this is a slice invariant. Namely, if \(\rho_{\alpha}(\Sigma)=\sigma (K,\alpha)\neq \pm 1,\) then K is not ribbon and if \(\alpha\) is of prime power order then one can also conclude that this is not slice. If \(\Sigma\) is a spherical space form then R. Fintushel and R. J. Stern [Topology 26, 385-393 (1987; Zbl 0627.57011)] remove the condition that \(\alpha\) be a prime-power order.
In the paper under review, the condition that \(\Sigma\) be a spherical space form is replaced by a weaker condition that \(H^ 1(\Sigma,{\mathbb{L}}_{\alpha})=0\), where \({\mathbb{L}}_{\alpha}\) is the flat complex line bundle induced by the character \(\alpha\).
The technique used was to study the moduli spaces of solutions to perturbed self-duality and anti-self-duality equations in a V-manifold setting. In the last part applications to homology cobordisms and sliceness of knots questions are described.