Casson invariant of cyclic coverings via eta-invariant and Dedekind sums

AbstractLet Σ be a 3-dimensional oriented manifold and let K⊂Σ be a knot. We assume that Σ is an integer homology sphere and (Σ,K) has a plumbing representation. We denote the cyclic n-fold covering of Σ branched along K by Σ(K,n), and we assume that this manifold is integer homology sphere as well. If λ denotes the Casson invariant, then we show that λ(Σ(K,n))−n·λ(Σ) can be computed from homological information only. More precisely, we compute in terms of an eta-type-invariant associated with the isometric structure of the knot