We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

中文翻译：

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Image Milnor 数和出色的展开

我们展示了图像 Milnor 数 µI(f) 的三个基本属性 $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0 )$ 具有孤立的不稳定性。首先，我们展示了图像 Milnor 数的守恒，从中可以推断出族的上半连续性和拓扑不变性。其次，我们证明了弱 Mond 猜想，即当且仅当 f 稳定时 µI(f) = 0。最后，我们证明了休斯顿的一个猜想，即任何家庭 $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ 与 $\mu_I (\,f_t)$ 常数在 Gaffney 的意义上是极好的。出于技术原因，在最后两个属性中，我们只考虑 corank 1 的情况。