In detail, there are projective representations of SO(2,1) which don't arise from linear representations of SO(2,1), or of its double cover, Spin(2,1). These representations are called **anyons**.

The topological reason behind the phenomenon is this: the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is **Z** (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. On the other hand, for *n* > 2, for SO(n,1) and Poincaré(2,1), it's only **Z**_{2} (cyclic of order 2); meaning that the spin group is simply connected.

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