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We consider the moduli spaces $\mathcal{M}_d(\ell)$ of a closed linkage with$n$ links and prescribed lengths $\ell\in \mathbb{R}^n$ in $d$-dimensionalEuclidean space. For $d>3$ these spaces are no longer manifolds generically,but they have the structure of a pseudomanifold.  We use intersection homology to assign a ring to these spaces that can beused to distinguish the homeomorphism types of $\mathcal{M}_d(\ell)$ for alarge class of length vectors. These rings behave rather differently dependingon whether $d$ is even or odd, with the even case having been treated in anearlier paper. The main difference in the odd case comes from an extragenerator in the ring which can be thought of as an Euler class of a startifiedbundle.

Intersection homology of linkage spaces in odd dimensional Euclidean space