Using six-functor sheaf theory, the Bauer-Furuta invariant is defined as the proper pushforward f_* f^!(1), with f^!(1) computed as the Thom spectrum of the family index.
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On the Bauer--Furuta construction
Using six-functor sheaf theory, the Bauer-Furuta invariant is defined as the proper pushforward f_* f^!(1), with f^!(1) computed as the Thom spectrum of the family index.