An approach to Lagrangian specialisation through MacPherson's graph construction

Let $f: M \to N$ be a holomorphic map between two complex manifolds. Assume $f$ is flat and sans \'{e}clatement en codimension 0 (no blowup in codimension 0). We study the theory of Lagrangian specialisation for such $f$, and prove a Gonz\'{a}lez-Sprinberg type formula for the local Euler obstruction relative to $f$. With the help of this formula and MacPherson's graph construction for the vector bundle map $f^*T^*N \to T^*M$, we find the Lagrangian cycle of the Milnor number constructible function $\mu$. As an application, we study the Chern class transformation of $\mu$ when $f$ has finite contact type.