An intersection homology invariant for knots in a rational homology 3- sphere
To a homologically trivial knot \(K\) in a rational homology 3-sphere the authors associate a family \(\{\lambda_{n, d}, p_{n,d} (t)\}_{(n, d) \in \mathbb{N}^* \times \mathbb{N}}\) of computable homological invariants. They generalize the Casson invariant of knots. Some explicit calculations are presented. The case where \(K\) is a fibered knot is treated in detail. Invariants \(\lambda_{(n,d)}\) can be computed using the intersection Lefschetz number of the monodromy action of the moduli space of semistable holomorphic bundles of rank \(n\) and degree \(d\) and fixed determinant over a compact Riemann surface. Reasons to appeal to intersection homology are: (1) for some cases (\(n\) and \(d\) not relatively prime) this moduli space is typically singular and stratified spaces (in the sense of Goresky-MacPherson) appear instead of manifolds and (2) these invariants \(\lambda_{(n,d)}\) can be computed using intersection numbers. For the fibered case, the perversity used is the middle perversity; in the general case the situation is more complicated and stratum dependent perversities appear (instead of filtration dependent perversities).
moduli space of semistable holomorphic bundles
scientific article; zbMATH DE number 527442