Equisingularity and The Euler Characteristic of a Milnor Fibre

We study the Euler characteristic of the Milnor fibre of a hypersurface singularity. This invariant is given in terms of the Euler characteristic of a fibre in between the original singularity and its Milnor fibre and in terms of the Euler characteristics associated to strata of the in-between fibre. From this we can deduce a result of Massey and Siersma regarding singularities with a one-dimensional critical locus. The result is also applied to the study of equisingularity. The famous Brian\c{c}on-Speder-Teissier result states that a family of isolated hypersurface singularities is equisingular if and only if its $\mu ^*$-sequence is constant. We show that if a similar sequence for a family of corank 1 complex analytic mappings from n-space to (n+1)-space is constant, then the image of the family of mappings is equisingular. For families of corank 1 maps from 3-space to 4-space we show that the converse is true also.