  We obtain the Hausdorff dimension, $h=2-2s$, for particles with fractional
spins in the interval, $0\leq s \leq 0.5$, such that the manifold is
characterized by a topological invariant given by, ${\cal W}=h+2s-2p$. This
object is related to fractal properties of the path swept out by fractional
spin particles, the spin of these particles, and the genus (number of anyons)
of the manifold. We prove that the anyonic propagator can be put into a path
integral representation which gives us a continuous family of Lagrangians in a
convenient gauge. The formulas for, $h$ and ${\cal W}$, were obtained taking
into account the anyon model as a particle-flux system and by a qualitative
inference of the topology.
