A generalized join theorem for real analytic singularities

Abstract

Let 𝑓₁ : (ℝ^𝑛, 𝟎_𝑛) → (ℝ², 𝟎₂) and 𝑓₂ : (ℝ^𝑚, 𝟎_𝑚) → (ℝ², 𝟎₂) be real analytic map germs of independent variables, where 𝑛, 𝑚 ≥ 2. Then the pair (𝑓₁, 𝑓₂) of 𝑓₁ and 𝑓₂ defines a real analytic map germ from (ℝ^𝑛+𝑚, 𝟎_𝑛+𝑚) to (ℝ⁴, 𝟎₄). We assume that 𝑓₁ and 𝑓₂ satisfy the 𝑎_𝑓-condition at 𝟎₂. Let 𝑔 be a strongly non-degenerate mixed polynomial of 2 complex variables which is locally tame along vanishing coordinate subspaces. A mixed polynomial 𝑔 defines a real analytic map germ from (ℂ², 𝟎₄) to (ℂ, 𝟎₂). If we identify ℂ with ℝ², then 𝑔 also defines a real analytic map germ from (ℝ⁴, 𝟎₄) to (ℝ², 𝟎₂). Then the real analytic map germ 𝑓 : (ℝ^𝑛 ×ℝ^𝑚, 𝟎_𝑛+𝑚) → (ℝ², 𝟎₂) is defined by the composition of 𝑔 and (𝑓₁, 𝑓₂), i.e., 𝑓(𝐱, 𝐲) = (𝑔 ∘ (𝑓₁, 𝑓₂))(𝐱, 𝐲) = 𝑔(𝑓₁(𝐱), 𝑓₂(𝐲)), where (𝐱, 𝐲) is a point in a neighborhood of 𝟎_𝑛+𝑚.

In this paper, we first show the existence of the Milnor fibration of 𝑓. We next show a generalized join theorem for real analytic singularities. By this theorem, the homotopy type of the Milnor fiber of 𝑓 is determined by those of 𝑓₁, 𝑓₂ and 𝑔. For complex singularities, this theorem was proved by A. Némethi. As an application, we show that the zeta function of the monodromy of 𝑓 is also determined by those of 𝑓₁, 𝑓₂ and 𝑔.