Join theorem for real analytic singularities

Abstract

Let $f_{1} : (\Bbb{R}^{n}, \mathbf{0}_{n}) \rightarrow (\Bbb{R}^{p}, \mathbf{0}_{p})$ and $f_{2} : (\Bbb{R}^{m}, \mathbf{0}_{m}) \rightarrow (\Bbb{R}^{p}, \mathbf{0}_{p})$ be analytic germs of independent variables, where $n, m \geq p \geq 2$. In this paper, we assume that $f_{1}, f_{2}$ and $f = f_{1} + f_{2}$ satisfy $a_{f}$-condition. Then we show that the tubular Milnor fiber of $f$ is homotopy equivalent to the join of tubular Milnor fibers of $f_1$ and $f_2$. If $p = 2$, the monodromy of the tubular Milnor fibration of $f$ is equal to the join of the monodromies of the tubular Milnor fibrations of $f_1$ and $f_2$ up to homotopy.