The paper under review studies the moduli space \[ \mathcal{M}_d (\ell) = \left\{ (x_1,\dots, x_n) \in (S^{d-1})^n \mid \sum_{i=1}^n \ell_i x_i =0 \right\}/\mathrm{SO}(d) \] of closed \(n\)-gon linkages in \(\mathbb{R}^d\), where \(\ell \in \mathbb{R}^n\) is a given length vector, \(\ell_i >0\) for all \(i\). For \(d\geq 4\), these spaces acquire singularities and have the structure of a pseudomanifold. By thinking of points \(x\in \mathcal{M}_d (\ell)\) as represented by \(d\times n\)-matrices, one obtains a stratification of the moduli space by the rank of these matrices. In previous work, the author has shown that in the case of even \(d\geq 4\), intersection homology can be used to distinguish moduli spaces for many \(\ell\). In the present paper, he uses this method for odd \(d\geq 5\).
Upon removal of the hyperplanes \(\{ \sum_{j\in J} x_j = \sum_{j\not\in J} x_j \},\) \(J \subset \{ 1, \dots, n \},\) the quadrant space \(\mathbb{R}^n_{>0}\) is a union of finitely many components, called chambers. If \(\ell\) and \(\ell'\) are in the same chamber, then \(\mathcal{M}_d (\ell)\) and \(\mathcal{M}_d (\ell')\) are homeomorphic, a result of Hausmann. For length vectors \(\ell\), notions of genericity and \(d\)-normality can be defined. The main result of the paper under review is that if \(d\geq 2\) and \(\ell, \ell' \in \mathbb{R}^n\) are generic, \(d\)-normal length vectors such that \(\mathcal{M}_d (\ell)\) and \(\mathcal{M}_d (\ell')\) are homeomorphic, then \(\ell\) and \(\ell'\) lie in the same chamber (up to permutation). This is shown by describing the intersection homology ring of the moduli spaces, formed by considering many perversities at once. The main difference to the even-dimensional case is that in the odd-dimensional case, a Euler class coming from the intersection homology of the Thom space of a stratified disk-bundle over the moduli space enters. This stratified bundle is obtained by taking the orbit space of the diagonal action of \(\mathrm{SO}(d-1)\) on \(\mathcal{C}_d (\ell) \times D^{d-1},\) where \(\mathcal{C}_d (\ell)\) is essentially the moduli space before quotienting by the \(\mathrm{SO}(d)\)-action.