A generalized join theorem for real analytic singularities

Abstract

Let $f_{1} : (\mathbb{R}^{n}, \mathbf{0}_{n}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ and $f_{2} : (\mathbb{R}^{m}, \mathbf{0}_{m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ be real analytic map germs of independent variables, where $n, m \geq 2$. Then the pair $(f_{1}, f_{2})$ of $f_{1}$ and $f_{2}$ defines a real analytic map germ from $(\mathbb{R}^{n+m}, \mathbf{0}_{n+m})$ to $(\mathbb{R}^{4}, \mathbf{0}_{4})$. We assume that $f_{1}$ and $f_{2}$ satisfy the $a_{f}$-condition at $\mathbf{0}_{2}$. 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 $(\mathbb{C}^{2}, \mathbf{0}_{4})$ to $(\mathbb{C}, \mathbf{0}_{2})$. If we identify $\mathbb{C}$ with $\mathbb{R}^{2}$, then $g$ also defines a real analytic map germ from $(\mathbb{R}^{4}, \mathbf{0}_{4})$ to $(\mathbb{R}^{2}, \mathbf{0}_{2})$. Then the real analytic map germ $f : (\mathbb{R}^{n} \times \mathbb{R}^{m}, \mathbf{0}_{n+m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ is defined by the composition of $g$ and $(f_{1}, f_{2})$, i.e., $f(\mathbf{x}, \mathbf{y}) = (g \circ (f_{1}, f_{2}))(\mathbf{x}, \mathbf{y}) = g(f_{1}(\mathbf{x}), f_{2}(\mathbf{y}))$, where $(\mathbf{x}, \mathbf{y})$ is a point in a neighborhood of $\mathbf{0}_{n+m}$.

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 $f_{1}, f_{2}$ 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 $f_{1}, f_{2}$ and $g$.