Picard—Lefschetz—Pham Theory and Singularity Theory

Abstract

Let p : E → T be a locally trivial fibre bundle with locally contractible base. The corresponding homological bundle is a fibre bundle with the same base, whose fibre over a point x is the group H_*(p^-1(x)). This bundle admits a natural local trivialization (i.e. a flat connection): the cycles in the fibres can be continuously transported in the neighbouring fibres, and the homology classes of the transported cycles do not depend on the choice of this transportation. The formal definition of this trivialization is as follows. (We denote the fibre p^-1(x) by F.) Consider an arbitrary contractible domain U in T. The bundle p over this domain is equivalent to the trivial one. Then, by the Künneth formula, for any point x∈U,

$$ {H_*}\left( {{\mathcal{F}_x}} \right) = {H_*}\left( {{p^{ - 1}}\left( U \right)} \right) $$

.