A surgery formula for the μ̄ -invariant

We introduce a relative version, μ̄′ , of the μ̄ -invariant of Neumann and Siebenmann for graph 3-manifolds, and use it to prove a surgery formula for the μ̄ -invariant. We further identify μ̄′ with the linking number of certain Montesinos links, and relate it to the Floer homology in the special case of Seifert manifolds.