Homological type of spaces of configurations of structurally stable type in \({\mathbb{C}}^ 2\) (Q1081117)

The author considers the set \(W_ n\) of ordered n-tuplets \(z_ 1,...,z_ n\) of points in the complex plane \({\mathbb{C}}^ 2\) that are in general position (no three points collinear), and the orbit space \(\tilde W_ n=W^ 0_ n/GL_ 2{\mathbb{C}}\) of \(W_ n\) modulo the action of the group \({\mathbb{C}}^ 2\cdot GL_ 2{\mathbb{C}}\) of affine maps. (Here, \(W^ 0_ n\subseteq W_ n\) is the subset defined by \(z_ n=0.)\) For \(n\geq 3\) he shows that the first homology group of \(\tilde W_ n\) is free abelian with \(\left( \begin{matrix} n\\ 3\end{matrix} \right)-1\) generators. To this end, he constructs a homeomorphism between \(\tilde W_ n\) and the space \(GW^ 0_ n\) of all planes \({\mathbb{C}}^ 2\cong E\leq {\mathbb{C}}^ n\) such that orthogonal projection \({\mathbb{C}}^ n\to E\) maps the standard basis of \({\mathbb{C}}^ n\) to an n-tuplet \(z_ 1,...,z_ n\) in general position satisfying \(z_ n=0\).