# Exchange statistics from topology of configuration space

I'm trying to understand Leinaas and Myrheim's famous 1976 argument for exchange symmetry of the wavefunction.

If we consider the configuration space $$\mathcal{M}_2$$ for two identical particles moving in $$d$$ spatial dimensions we have

$$\mathcal{M}_2 = \mathbb{R}^d \times r(d,2)$$

where the first factor parametrizes the center of mass coordinate and the second factor $$r(d,2) =(\mathbb{R}^d / \mathbb{Z}_2)$$ corresponds to relative motion.

Leinaas and Myrheim's argument can be summarized as follows. If one artificially removes the origin from the orbifold space $$r(d,2)$$ then one of the resulting factors is the real projective space of dimension $$d-1$$,

$$r(d, 2) - \{0\} = (0,\infty) \times \mathbb{RP}^{d-1}$$

It is known that the first homotopy group of real projective space is $$\pi_1(\mathbb{RP}^{n}) = \mathbb{Z}_2$$ for $$n > 1$$ while $$\pi_1(\mathbb{RP}^{n}) = \mathbb{Z}$$ for $$n=1$$.

The physical dimension $$d = 3$$ (corresponding to $$n>1)$$ thus allows for the possibility of double-valued wavefunctions on the configuration space $$\mathcal{M}_2$$ which lift to single-valued wave functions on $$\mathbb{R}^d \times \mathbb{R}^d$$ satisfying exchange antisymmetry.

My concern is that the above argument depended on artificially removing the origin from the relative space $$r(d,2)$$ which physically corresponds to disallowing coincident particle positions. This seems to be contradicted by bosonic statistics which allow for coincident particle positions.