Stable homotopy invariants for 4-manifolds
The Bauer-Furuta invariant can be considered as a stable homotopy
refinement of the Seiberg-Witten invariant for a closed 4-manifold.
the idea to define a relative invariant for a 4-manifold with boundary.
For an example, we can explicitly compute the Bauer-Furuta invariants for
or without boundary) possessing a metric of positive scalar curvature. The
technique is to deform the Seiberg-Witten map into its linearization.