Refined Kirby calculus for three-manifolds of first homology groups of odd prime orders

Every integral homology 3-sphere is presented by a framed link with framing ±1 and without linking numbers. Restricting such presentations, Habiro arranged Kirby calculus so that it preserves framings and linkings and moreover showed that his calculus suffices to relate all links with the same results. This paper provides an extension of his result for manifolds of first homology groups of odd prime orders. After defining our set of links, we establish Habiro calculus over it, and show that, for many orders, it works on those manifolds. We further give the existence of the Casson–Walker invariant for them.