On the Euler characteristics of real Milnor fibres of partially parallelizable maps of (Rⁿ, 0) to (R², 0)

Abstract

We consider a real analytic map-germ (f,g):(Rⁿ,0) → (R²,0) such that the fibres of f are simultaneously parallelizable. We call such a map a partially parallelizable map. We establish degree formulas for the following quantities:
χ ({f = α} ∩ {g = δ} ∩ B_εⁿ),
χ ({f = α} ∩ {g ≥ δ} ∩ B_εⁿ) - χ ({f = α} ∩ {g ≤ δ} ∩ B_εⁿ),
where (α, δ) is a regular value of (f,g) and 0 < |(α, δ)| « ε « 1.