Bauer--Furuta invariants and Galois symmetries

The Bauer-Furuta invariants of smooth 4-manifolds are investigated from a functorial point of view. This leads to a definition of equivariant Bauer-Furuta invariants for compact Lie group actions. These are studied in Galois covering situations. We show that the ordinary invariants of all quotients are determined by the equivariant invariants of the covering manifold. In the case where the Bauer-Furuta invariants can be identified with the Seiberg-Witten invariants, this implies relations between the invariants in Galois covering situations, and these can be illustrated through elliptic surfaces. It is also explained that the equivariant Bauer--Furuta invariants potentially contain more information than the ordinary invariants.