Indefinite Stein fillings and $$\text {PIN}(2)$$-monopole Floer homology

We introduce techniques to study the topology of Stein fillings of a given contact three-manifold (Y,ξ) which are not negative definite. For example, given a spinc rational homology sphere (Y,s) with s self-conjugate such that the reduced monopole Floer homology group HM∙(Y,s) has dimension one, we show that any Stein filling which is not negative definite has b+2=1 or 2, and b−2 is determined in terms of the Froyshov invariant. The proof of this uses Pin(2)-monopole Floer homology. More generally, we prove that analogous statements hold under certain assumptions on the contact invariant of ξ and its interaction with Pin(2)-symmetry. We also discuss consequences for finiteness questions about Stein fillings.