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I am wondering what, precisely, the singular point of an orbit space is. Specifically, I am looking at quantum statistics and the orbit space $M^N/S_N,$ where $M^N$ is the classical configuration space of $N$ particles and $S_N$ is the permutation group and want to know what specifically makes the constant points of the group on the manifold singular as described in "Quantum Hall Systems: Braid groups, composite fermions, and fractional charge" by Lucjan Jacak.

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    $\begingroup$ They are singular because the "manifold" isn't a manifold at that point - just like a cone isn't a manifold at its tip. $\endgroup$
    – ACuriousMind ♦
    Commented Mar 20, 2015 at 17:34
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    $\begingroup$ In the example of a cone, the symmetry is rotation around the origin, and the orbits of the symmetry are then circles. At the tip of the cone, the orbit has degenerated to a single point. $\endgroup$
    – Surgical Commander
    Commented Mar 20, 2015 at 18:06

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