Topology of toric gravitational instantons

For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton (M,g) with toric symmetry, we express the signature of (M,g) directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.