An integer valued SU(3) Casson invariant (Q1598395)

For \(n>2\), trying to generalize the \(SU(2)\) Casson invariant of a homology 3-sphere \(X\) to \(SU(n)\), a signed count of conjugacy classes of irreducible \(SU(n)\) representations of \(\pi_1(X)\) depends on the previously necessary perturbation of the space of representations (to make it generic and hence finite). In a previous paper [J. Differ. Geom. 50, No. 1, 147-206 (1998; Zbl 0934.57013)], the first two authors defined an \(SU(3)\) Casson invariant \(\lambda_{SU(3)}\) for homology 3-spheres \(X\) using the perturbation approach of Taubes (Taubes used a gauge-theoretic interpretation of Casson's \(SU(2)\) invariant as a signed count of gauge orbits of perturbed flat \(SU(2)\) connections on \(X\)). For a generic perturbation \(h\), the count of gauge orbits of irreducible, \(h\)-perturbed flat \(SU(3)\) connections depends on the perturbation \(h\), so in order to define a topological invariant one needs a correction term. In the present paper, an improved version of such a correction term is given, and the behaviour of the resulting invariant is studied under orientation reversal and connected sum. In contrast to \(\lambda_{SU(3)}\) the new invariant is integer-valued, and as other advantages the authors mention that it is easier to compute and therefore seems more likely to satisfy a surgery formula; also, in contrast to \(\lambda_{SU(3)}\) it may be related to finite-type invariants.