Indefinite Stein fillings and \(\text{PIN}(2)\)-monopole Floer homology (Q2310820)

The paper under review applies Pin(2)-equivariant monopole Floer homology to study Stein fillings of a contact \(3\)-manifold \((Y,\xi)\). While many classification results have been proved under the assumption of negative definite intersection forms, the paper under review considers the case that the intersection form of the Stein filling is not negative definite. Under the further assumptions that the \(\mathrm{spin}^c\)-structure \({\mathfrak s}_\xi\) is induced by a spin structure and that the reduced Floer homology satisfies \(HM(Y,{\mathfrak s}_\xi)=F_2\), the author proves that the (by assumption not negative definite) intersection form of a Stein filling is even, and he computes the Betti numbers \(b_2^+\) and \(b_2^-\) (the latter in terms of Frøyshov's invariant). As a consequence, for a rational homology sphere satisfying the above conditions, the set of Euler characteristics of Stein fillings is finite.