The Walker conjecture for chains in ℝd

A chain is a configuration in ℝd of segments of length ℓ1, ..., ℓn−1 consecutively joined to each other such that the resulting broken line connects two given points at a distance ℓn. For a fixed generic set of length parameters the space of all chains in ℝd is a closed smooth manifold of dimension (n − 2)(d − 1) − 1. In this paper we study cohomology algebras of spaces of chains. We give a complete classification of these spaces (up to equivariant diffeomorphism) in terms of linear inequalities of a special kind which are satisfied by the length parameters ℓ1, ..., ℓn. This result is analogous to the conjecture of K. Walker which concerns the special case d=