A Furuta-like inequality for spin orbifolds and the minimal genus problem

We generalize Furutaʹs 10/8ths inequality involving the index of the Dirac operator on a smooth spin 4-manifold to the setting of spin orbifolds. These spin orbifolds are obtained by coning off a tubular neighborhood of an embedded sphere representing a characteristic homology class in a smooth 4-manifold. This inequality can be used to obtain minimal genus bounds for characteristic classes in the original manifold. For example, for X with positive definite intersection form of rank n⩾2, we show a smoothly embedded characteristic sphere ξ satisfies ξ·ξ⩽9n−16.