Notions of simple type for Bauer--Furuta invariants

TL;DR

This paper extends the concept of simple type to Bauer--Furuta invariants, introduces new notions, and explores their implications for 4-manifold topology, including invariants computation and gluing constraints.

Contribution

It defines BF blowup simple type and BF homogeneous type for Bauer--Furuta invariants and applies these notions to compute invariants and derive constraints in 4-manifold topology.

Findings

Existence of an immersed 2-sphere guarantees BF blowup simple type.

Determination of Bauer--Furuta invariants for manifolds after logarithmic transformations.

Constraints on 4-manifold decompositions using BF homogeneous type.

Abstract

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.