A Formula Similar to a Consequence of the Join Theorem

Let u:(Cm,0)→(C,0) and v:(Cn,0)→(C,0) be holomorphic function germs that have isolated critical points at the origin. It is known that μ(u+v)=μ(u)·μ(v) holds, where μ denotes the Milnor number. In this paper, we consider a formula similar to the above one. More precisely, for Morse functions f:M→R and g:N→R, let C(f+g) denote the fiber product of two copies of f+g. Two of the main results are as follows: Firstly, under a generic condition on the critical values of f and g, the equation χ(C(f+g))=χ(C(f))·χ(C(g)) holds. Here χ denotes the Euler characteristic. Secondly, under a certain condition, the converse of the first result is also true.