Frohman, Charles; Nicas, Andrew An intersection homology invariant for knots in a rational homology 3- sphere. (English) Zbl 0822.57008 Topology 33, No. 1, 123-158 (1994). 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). Reviewer: M.Saralegi-Aranguren (Madrid) Cited in 5 Documents MSC: 57M25 Knots and links in the \(3\)-sphere (MSC2010) 55N33 Intersection homology and cohomology in algebraic topology 14D20 Algebraic moduli problems, moduli of vector bundles Keywords:homologically trivial knot; rational homology 3-sphere; Casson invariant; fibered knot; intersection Lefschetz number; moduli space of semistable holomorphic bundles; stratum dependent perversities PDFBibTeX XMLCite \textit{C. Frohman} and \textit{A. Nicas}, Topology 33, No. 1, 123--158 (1994; Zbl 0822.57008) Full Text: DOI