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

Abstract

We introduce invariants of, in general, 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, Mg, we get an invariant of 4-dimensional Lefschetz fibrations over the 2-sphere. Moreover, the construction is applicable for the quantum representations of Mg derived from Chern-Simons field theory. We also see that our invariant is unstable with respect to fiber sum of Lefschetz fibrations.