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: $$χ (\{f = α\} \cap \{g = δ\} \cap B_ε^n),\\ χ (\{f = α\} \cap \{g ≥ δ\} \cap B_ε^n) − χ (\{f = α\} \cap \{g ≤ δ\} \cap B_ε^n),$$ where (α, δ) is a regular value of (f, g) and 0 < |(α, δ)| « ε « 1.