A Formula for the Euler Characteristic of the Fiber Product of Morse Functions

Let η:X→R be a Morse function on a connected closed manifold X. We denote by C(η) the fiber product of two copies of η. For Morse functions f:M→R and g:N→R, we define the function f∗g:M×N→R by (f∗g)(p,q):=f(p)·g(q). The purpose of this paper is twofold: Firstly, we study the sufficient condition for which χ(C(f∗g))=χ(C(f))·χ(C(g)) holds, where χ denotes the Euler characteristic. Secondly, for the case that f is the well-known Morse function on CPn, we determine χ(C(f∗f)).