Let \(f:X\to Y\) be a finite, proper continuous map. We have the following definitions:
\(k\)th multiple point space:
\(D^k(f):=\text{closure}\{(x_1,\cdots,x_k)\in X^k \mid f(x_1)=\cdots=f(x_k)\), \(x_i \neq x_j\) for \(i \neq j\}\).
\(k\)th image multiple point space:
\(M_k(f):=\text{closure}\{y\in Y \mid \#f^{-1}\geq k\}\).
Let us consider \(g_i:X_i\to Y_i\), \(i=1,2\), two finite, proper continuous maps. In the paper under review the author determines (see Theorem 3.2) to which extent, assumed homotopy (homology) equivalence at the level of \(k\)th multiple point spaces \(D^k(g_i)\), \(k\geq 1\), induces similar equivalence at the level of \(k\)th image multiple point space \(M_k(g_i)\), \(k\geq 1\). If \(f:(\mathbb C^n,0)\to (\mathbb C^p,0)\), \(n<p\), is a finitely \(\mathcal A\)-determined map-germ, there exists a local topological stabilisation of \(f\), a deformation of \(f\), \(f_t\), such that \(f_t\) is topological stable , \(t\neq 0\) [W. L. Marar, Manuscr. Math. 80, No. 3, 273-281 (1993; Zbl 0793.32018)]. Let \(U_t=f_t^{-1}( B_{\varepsilon})\) conveniently chosen.
Disentanglement: For sufficiently small \(t\neq 0\), the map \(f_t:U_t \to B_{\varepsilon}\) is called the disentanglement map of \(f\). The \(k\)th image multiple point space of \(f_t\) is called the \(k\)th disentanglement of \(f\) and is denoted by \(\text{Dis}_k(f)\).
Real perturbation: The restriction of \(f_t\) to the real part of the source is called a real perturbation.
If \(f_{\mathbb C}=f:(\mathbb C^n,0)\to (\mathbb C^p,0)\) is the complexification of a finitely \(\mathcal A\)-determined map-germ and \(f_t:V_1 \subset \mathbb C^n \to V_2\subset \mathbb C^p\) is a stable perturbation giving the disentanglement map of \(f\), denote \(f_{\mathbb R}:= f|\mathbb R^n\) and define \(f_{\mathbb R,t}=f_{\mathbb C,t}|\mathbb R^n\) to be a real perturbation associated to \(f\) (no uniqueness). Then \(\text{Dis}_k(f_{\mathbb C})\) will be the \(k\)th complex disentanglement and \(\text{Dis}_k(f_{\mathbb R})= M_k(f_{\mathbb R,t})\) a \(k\)th real disentanglement.
The main result of this paper (Theorem 3.6), gives conditions on the inclusion maps \[ D^k(f_{\mathbb R,t})\to D^k(f_{\mathbb C,t}),\;k \geq 1 \] which imply the inclusions \(\text{Dis}_k(f_{\mathbb R})\to \text{Dis}_k(f_{\mathbb C})\), \(k \geq 1\) to be a \(\mathbb Z\)-homology equivalence, i.e. the real perturbation is good. In the case \(n=p+1\) and corank of \(f\) is 1 then under the same conditions he even obtains homotopy equivalences.