Bilinear-form invariants of Lefschetz fibrations over the 2-sphere

Abstract

We introduce invariants of Hurwitz equivalence classes with respect to arbitrary group $G$. The invariants are constructed from any right $G$-modules $M$ and any $G$-invariant bilinear function on $M$, and are of bilinear forms. For instance, when $G$ is the mapping class group of the closed surface, ${\mathcal{M}}_g$, we get an invariant of 4-dimensional Lefschetz fibrations over the 2-sphere. Moreover, the construction is applicable for the quantum representations of ${\mathcal{M}}_g$ derived from Chern-Simons field theory. We compute the associated invariants in some cases, and find infinitely many Lefschetz fibrations which have the same Seiberg-Witten invariant and are homeomorphic but not mutually isomorphic as fibrations.

23 pages. Several figures. I revised some sentences and typomiss in the previous preprint. Moreover, I found some gap of the main theorem in Subsection 4.3 in the previous one; so I deleted the subsection

References in corpus (1)

• Lefschetz fibrations on knot surgery $4$-manifolds via Stallings twist