Local topology of images of finite complex analytic maps (Q1356380)

The author proves the following two interesting results: Theorem A: Let \(f: (\mathbb{C}^n,0) \to (\mathbb{C}^p,0)\) be a finitely \({\mathcal A}\)-determined map germ with \(0<n<p\), then the image of \(f_t|_{U_t}\) is homotopically equivalent to a wedge of spheres of different dimensions. The possible dimensions are \(p=(p-n-1) k-1\) for all \(2\leq k\leq p/(p-n)\). The mapping \(f_t\) is defined by \(f_t(x)= F(x,t)\), where \(F: (\mathbb{C}^n \times\mathbb{C}^r\), \(S\times 0) \to (\mathbb{C}^n \times \mathbb{C}^r\), \(0\times 0)\) is an \(r\)-parameter unfolding of \(f\) \((S\) is a finite set), and \(U_t= U\cap F^{-1} (\mathbb{C}^p \times\{t\})\), \(U= F^{-1} (B_\varepsilon)\), \(B_\varepsilon\) being a sufficiently small ball centred at zero in \(\mathbb{C}^p \times\mathbb{C}^r\). The author calls the image of \(f_t|_{U_t}\) a disentanglement of \(f\). Theorem B: Let \(f: X\to\mathbb{C}^p\) be a finite complex analytic map with \(X\) defined in \(C^n\) by \(r\) equations and \(n-r<p\). Suppose \(Y\) is defined in \(\mathbb{C}^p\) by \(s\) equations. Let \(m\) be the largest number of preimages of any point in \(f(X)\cap Y\). Then any complex link \({\mathcal L}\) of a stratum \(A\) of \(f(X)\cap Y\) has \[ \pi_i ({\mathcal L})=0 \quad \text{for} \quad i<(n-r-p+1) m+p-s- \dim_\mathbb{C} A-2. \]