W-invariants and Neumann–Siebenmann invariants for Seifert homology 3-spheres

We give a reciprocity formula for w-invariants for homology 3-spheres defined by Fukumoto and Furuta, which enables us to compute these invariants recursively. Using this formula, we show that the w-invariant for a Seifert homology 3-sphere (together with a specific spin V 4-manifold bounded by it) equals its Neumann–Siebenmann invariant up to sign. This implies that the Neumann–Siebenmann invariant for a Seifert homology 3-sphere is a homology cobordism invariant if it has either at most six singular fibers or has a singular fiber of even multiplicity.