Surgery and excision for Furuta-Ohta invariants on homology \(S^1 \times S^3\) (Q6593665)

The Furuta-Ohta invariant is an invariant for manifolds with the homology \(S^1\times S^3\) defined via the Yang-Mills equations. It is an analogue of the Casson invariant of a homology \(3\)-sphere. An analogue in Seiberg-Witten theory was defined by \textit{T. Mrowka} et al. [J. Differ. Geom. 88, No. 2, 333--377 (2011; Zbl 1238.57028)]. Establishing the equivalence of these two invariants would among other things provide a new proof that there are higher-dimensional manifolds that are not trainable.\N\NIn this paper, Ma establishes torus surgery and excision formulae for the Furuta-Ohta invariant. This is one of the most technically challenging situations in which one considers gluing results in gauge theory. The invariant is already defined for \(b^2_+ = 0\) which is as low as possible and this makes it impossible to avoid all reducible by a transversality argument. Furthermore, the gluing takes place along a \(3\)-torus which requires one to analyze the behavior around reducible and central reducible limits. Gluing in a similar situation was analyzed by Morgan, Mrowka, and Ruberman in a long technical paper [\textit{J. W. Morgan} et al., The \(L^ 2\)-moduli space and a vanishing theorem for Donaldson polynomial invariants. Cambridge, MA: International Press (1994; Zbl 0830.58005)]. These results need to be refined in order to address the \(b^2_+\) situation.\N\NMa has written a careful, and technical argument addressing all of the points. He summarizes relevant parts of the literature very nicely, so this difficult paper may be read without referring back to other foundational papers. To get a sense of some of the techniques used, we will describe that the perturbations that Ma uses have two inputs. One input is a holonomy perturbation supported in the compact region of a \(4\)-manifold with a cylindrical end. The other input starts with a neighborhood of a link in the \(3\)-torus on the end and defines a holonomy perturbation down the end with exponential decay. These perturbations have to be carefully tuned to get the result.\N\NMa uses his surgery and excision formulas to perform several new computations of the Furuta-Ohta invariant.