Configuration spaces of weighted graphs in high dimensional Euclidean spaces (Q1610959)

Here a weighted graph \(G\) is a finite graph with a positive weight associated to each edge of \(G\). An ordered set of points in \(\mathbb{R}^n\) representing the vertices of \(G\) is called a realisation of \(G\) if, for each edge of \(G\), the distance of the corresponding points is equal to the weight of the edge. The configuration space of \(G\) is the set of all possible realisations of \(G\) modulo the group of proper isometries of the space \(\mathbb{R}^{n}\) (with the natural topology). The main result reads as follows. Let \(G\) be a connected weighted graph with \(n+1\) vertices and \(e\) edges which admits not only degenerated realisations in \(\mathbb{R}^{n}\). Then the configuration space of \(G\) is homeomorphic to the sphere of dimension \(n(n+1)/2-e\).