The topology of spaces of polygons (Q2838117)

It is known that essentially any closed smooth manifold can be realized as the configuration space of a mechanism. In this paper the authors consider the inverse problem for the spaces of polygons of arbitrary dimension \(d\geq 2\). Let \(\ell = (l_1,\ldots, l_n)\) be a tuple of positive numbers (the \textit{length vector}). For every \(d\geq 2\) one considers the space NEWLINE\[NEWLINEE_d(\ell ) = \{(u_1,\ldots, u_n)\in (S^{d-1})^n\;|\;\sum_{k=1}^n l_ku_k= 0\}NEWLINE\]NEWLINE This is the space of closed \(n\)-gons in \(\mathbb R^d\) up to translations. A length vector \(\ell\) is called generic if there is no subset \(J\subset\{1,\ldots, n\}\) so that \(\sum_{j\in J}l_j = \sum_{j\not\in J}l_j\). For generic \(\ell\), \(E_d(\ell)\) is a closed smooth manifold of dimension \((n-1)(d-1)-1\).NEWLINENEWLINEThe main theorem of this paper gives three equivalent conditions for when the manifolds \(E_d(\ell)\) and \(E_d(\ell^\prime )\) are \(O(d)\)-equivariantly diffeomorphic. The authors show that it is enough for the cohomology rings mod two to be isomorphic. Equivalently it is enough for the rings \(H^{(d-1)*}(E_d(\ell);{\mathbb Z}_2)\) and \(H^{(d-1)*}(E_d(\ell^\prime);{\mathbb Z}_2)\) to be isomorphic. Finally they give the strikingly simple and purely combinatorial characterization which states that these spaces of polygons are diffeomorphic if and only if the length vectors \(\ell\) and \(\ell^\prime\), after a permutation of entries if need be, lie in the ``same chamber''. Two generic vectors \((l_1,\ldots, l_n)\) and \((l^\prime_1,\ldots, l^\prime_n)\) are in the same chamber if \(\sum_{j\in J}l_j > \sum_{j\not\in J}l_j\) implies that \(\sum_{j\in J}l_j^\prime > \sum_{j\not\in J}l_j^\prime\), and vice-versa.NEWLINENEWLINEThe proof of these results invokes techniques of Morse theory and exploits the result of \textit{J. Gubeladze} [J. Pure Appl. Algebra 129, No. 1, 35--65 (1998; Zbl 0931.20053)] on the isomorphism problem for monoidal rings. The main computation here being Proposition 4.2 which for any ordered \(\ell\) (generic or not) gives the explicit description of the graded ring \(H^{(d-1)*}(E_d(\ell^\prime);{\mathbb Z}_2)\) in terms of generators and relations. In order to make this computation, the authors develop a ``lacunary principle'' for Morse-Bott functions (\S2) which is of independent interest. This principle gives a decomposition of \(H_*(M,{\mathbb Z})\) for a smooth compact manifold \(M\), possibly with boundary, from the data of a Morse-Bott function whose connected critical submanifolds \(C\subset M\) have torsion free homology and are trivial in all degrees not divisible by a given \(k\geq 2\).NEWLINENEWLINEThe authors close by giving an example of two length vectors \(\ell\) and \(\ell'\) such that the corresponding spaces \(E_d(\ell)\) and \(E_d(\ell')\) have the same mod-two Betti numbers but do not lie in different orbits of chambers under the permutation action. This shows that the product structure of the cohomology ring is necessary to compare the polygon spaces.