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Join theorem for real analytic singularities. (English) Zbl 1487.32158

Summary: Let \(f_1 : (\mathbb{R}^n, \mathbf{0}_n) \rightarrow (\mathbb{R}^p, \mathbf{0}_p)\) and \(f_2 : (\mathbb{R}^m, \mathbf{0}_m) \rightarrow (\mathbb{R}^p, \mathbf{0}_p)\) 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.

MSC:

32S05 Local complex singularities
32S55 Milnor fibration; relations with knot theory
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References:

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