Notions of simple type for Bauer--Furuta invariants

By extending the notion of simple type for the Seiberg–Witten invariant of a 4-manifold, we introduce notions of BF blowup simple type and BF homogeneous type for the Bauer–Furuta invariant and study their applications. Specifically, we show that the existence of an immersed 2-sphere with a certain condition guarantees BF blowup simple type. As an application, we determine the Bauer–Furuta invariant of a 4-manifold obtained by a logarithmic transformation along a torus in a fishtail neighborhood. We also give constraints on gluing decompositions of 4-manifolds by using BF homogeneous type. To prove these results, we also give gluing formulae and an immersed adjunction inequality for Bauer–Furuta invariants.