Volume 16, issue 1 (2016)

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Intersection homology of linkage spaces in odd-dimensional Euclidean space

Dirk Schütz

Algebraic & Geometric Topology 16 (2016) 483–508
Abstract

We consider the moduli spaces ${\mathsc{ℳ}}_{d}\left(\ell \right)$ of a closed linkage with $n$ links and prescribed lengths $\ell \in {ℝ}^{n}$ in $d\phantom{\rule{0.3em}{0ex}}$–dimensional Euclidean 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 be used to distinguish the homeomorphism types of ${\mathsc{ℳ}}_{d}\left(\ell \right)$ for a large class of length vectors. These rings behave rather differently depending on whether $d$ is even or odd, with the even case having been treated in an earlier paper. The main difference in the odd case comes from an extra generator in the ring, which can be thought of as an Euler class of a stratified bundle.

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