Intersection homology of linkage spaces (Q2797813)

Configuration spaces of closed linkages in Euclidean space modulo isometry group have occurred in many contexts in recent years. This is the moduli space NEWLINE\[NEWLINE{\mathcal M}_d(\ell )=\left.\left\{ (x_1,\ldots, x_n)\in (S^{d-1})^n\;|\;\sum_{i=1}^n\ell_ix_i =0\right\}\right/\mathrm{SO}(d)NEWLINE\]NEWLINE where \(\mathrm{SO}(d)\) acts diagonally on the product of spheres. The main question here is how the topology of \({\mathcal M}_d(\ell )\) depends on the length vector \(\ell\). A chamber is a maximal connected open subset of \(\mathbb R^n\) such that any two length vectors it contains admit homeomorphic moduli spaces. In the case \(d<n-1\) the topology of the moduli space does depend on the chamber by work of the author [Algebr. Geom. Topol. 13, No. 2, 1183--1224 (2013; Zbl 1269.58004)]. The main result of this paper shows that for a large class of length vectors the topology of the moduli space does recover the chamber of the length vector. Here is the main statement.NEWLINENEWLINEMain Theorem: let \(d\geq 4\) be even, \(\ell,\ell'\in \mathbb R^n\) be generic, \(d\)-normal length vectors. If \({\mathcal M}_d(\ell )\) and \({\mathcal M}_d(\ell' )\) are homeomorphic, then \(\ell\) and \(\ell'\) are in the same chamber up to a permutation.NEWLINENEWLINEA length vector is generic if it does not admit a collinear configuration (that is if \(\mathcal M_1(\ell)=\emptyset\)). The notion of \(d\)-normality is defined in \S2. For \(\ell\in \mathbb R^n\) a length vector, a subset \(J\subset\{1,\ldots, n\}\) is called \(\ell\)-short, if \(\sum_{j\in J}\ell_j<\sum_{i\not\in J}\ell_i\). It is called \(\ell\)-long if its complement is \(\ell\)-short. For a length vector to be not \(d\)-normal, one needs a long subset \(J\) with \(d-1\) elements such that \(J\) does not contain an element \(m\) with \(\ell_m\) maximal.NEWLINENEWLINEThe main theorem for \(d=2,3\) has been obtained earlier using cohomology. For linkages in \(3\)-space, the cohomology of the resulting moduli spaces has been calculated in [\textit{J.-C. Hausmann} and \textit{A. Knutson}, Ann. Inst. Fourier 48, No. 1, 281--321 (1998; Zbl 0903.14019)] with applications to the so-called Walker conjecture in [\textit{M. Farber} et al., J. Topol. Anal. 1, No. 1, 65--86 (2009; Zbl 1177.55008)]. Much less is known for linkages in higher dimensional Euclidean spaces, and this paper is a major and important step in this direction. Note that for \(d=2,3\) and \(\ell\) generic, the spaces \(\mathcal M_d(\ell )\) are closed manifolds, but for \(d\geq 4\) and \(n>d\) these spaces are pseudomanifolds (\S3). For \(d\geq 4\), cohomology turns out not to be enough to distinguish between the moduli spaces and the author resorts to intersection homology. By letting the perversity vary with the degree of the intersection homology group, the author uses the intersection pairing to assign a ring to each moduli space which behaves very similar to the cohomology ring in the case \(d=2\). For even \(d\geq 4\), he is able to describe this ring as an exterior algebra on explicit generators (Theorem 4.5). He then uses this calculation to prove his main theorem. The proof of Theorem 4.5 is remarkable and occupies a large part of this paper.