On the Bauer--Furuta construction

Using the six-functor formalism for sheaves of spectra on topological spaces, we provide a novel construction of the Bauer--Furuta invariant, as well as its family version. This approach avoids the conventional arguments based on approximations by finite-dimensional subspaces, and we instead employ the Borel--Moore homology spectra relative to Fredholm maps between Banach spaces. A key observation here is that $C^1$-differentiable Fredholm maps between Banach manifolds are locally proper, thereby defining the shriek functors, whose dualizing objects may be described as the Thom spectra of the Atiyah--Singer families index. We also outline a possible candidate for the stable homotopy theory of genuine equivariant sheaves on topological spaces with Lie group actions. In this context, we investigate the proper pushforward functor, which accommodates the genuine equivariant Bauer--Furuta invariant.