# Configuration space of identical particles - fractional statistics

In Khare's book of fractional statistics and quantum theory, when discussing why we need fractional statistics he arrives at the configuration space

for a system of two identical particles in $$d$$ spatial dimensions.
Q1: I do not see how the second equality is justified. ($$P_{d-1}$$ is the $$d-1$$ dimensional real projective space, and $$\langle 0,\infty \rangle$$=$$(0,\infty)-\{0\}$$).

Later on, he writes this:

Q2: He seems to imply that $$(\mathbb R^2 - \{0\})/\mathbb Z_2 = \mathbb R P_{1}$$. But actually $$\mathbb R P_{1} = (\mathbb R^2 - \{0\})/\sim$$ with $$x\sim y$$ iff they lie on the same line. Maybe there is a physical reason to make this identification $$(\mathbb R^2 - \{0\})/\mathbb Z_2 = \mathbb R P_{1}$$?

Denote

$$M_d := (\mathbb{R}^d -\{0\})/\{x \sim - x\} \ .$$

To answer Q2: for the equation (2.14) to hold, it is not necessary that $$M_d = \mathbb{R}P_{d-1}$$. It's enough if these two spaces are homotopy equivalent, which they are. Indeed, consider an equivalence class $$[x] \in M_d$$. $$[x]$$ may be written as $$e^\lambda [\hat{x}]$$, where $$\lambda \in \mathbb{R}$$ and $$\hat{x}$$ is a unit vector in $$\mathbb{R}^d$$. Then consider the following map:

$$F:[0,1] \times M_d \mapsto M_d \ , \quad (t,e^\lambda [\hat{x}]) \mapsto e^{(1-t)\lambda} [\hat{x}] \ .$$

Then $$F(0,\cdot) = \text{id}_{M_d}(\cdot)$$, but $$F(1,\cdot)$$ maps $$M_d$$ on the projective space.