The answer might be a little late, but I'm also going through Nakahara at the moment:)

So I think, you might have missed a point of defining the negative elements of $\mathbb{Z}$; see Example 4.13 for this (in Nakahara 2004 Ed.). So the idea is that positive elements correspond to the mappings where the orientation of the surface does not change. Obviously, the minimum angle you need to map the zero point to by the texture having the same orientation is the point $2\pi$; any other angle multiple to this also won't change the orientation. Those multiples correspond to positive elements of $\mathbb{Z}$. Whereas the negative elements correspond to multiples of $\pi$ since they change the orientation. This answers your second question: the texture, that relates to $-1$, is the one with $f(0)=\pi$ instead of $2\pi$.

Hope that clarified some things (and I hope my understanding is correct; please correct me if I'm wrong).

The answer might be a little late, but I'm also going through Nakahara at the moment:)

So I think, you might have missed a point of defining the negative elements of $\mathbb{Z}$; see Example 4.13 for this (in Nakahara 2004 Ed.). So the idea is that positive elements correspond to the mappings where the orientation of the surface does not change. Obviously, the minimum angle you need to map the zero point to by the texture having the same orientation is the point $2\pi$; any other angle multiple to this also won't change the orientation. Those multiples correspond to positive elements of $\mathbb{Z}$. Whereas the negative elements correspond to multiples of $\pi$ since they change the orientation. This answers your second question: the texture, that relates to $-1$, is the one with $f(0)=\pi$ instead of $2\pi$.

Hope that clarified some things (and I hope my understanding is correct; please correct me if I'm wrong).

EDIT:

See the answer in another OP's post here, as mentioned in the comment. As the OP has mentioned there, the answer to the second question is $\Omega = \Omega(x,-y,z)$.

The answer might be a little late, but I'm also going through Nakahara at the moment:)

So I think, you might have missed a point of defining the negative elements of $\mathbb{Z}$; see Example 4.13 for this. So the idea is that positive elements correspond to the mappings where the orientation of the surface does not change. Obviously, the minimum angle you need to map the zero point to by the texture having the same orientation is the point $2\pi$; any other angle multiple to this also won't change the orientation. Those multiples are positive elements of $\mathbb{Z}$. Whereas the negative elements are multiples of $\pi$ since they change the orientation. This answers your second question: the texture, that relates to $-1$, is the one with $f(0)=\pi$ instead of $2\pi$.

Hope that clarified some things (and I hope my understanding is correct; please correct me if I'm wrong).

The answer might be a little late, but I'm also going through Nakahara at the moment:)

So I think, you might have missed a point of defining the negative elements of $\mathbb{Z}$; see Example 4.13 for this (in Nakahara 2004 Ed.). So the idea is that positive elements correspond to the mappings where the orientation of the surface does not change. Obviously, the minimum angle you need to map the zero point to by the texture having the same orientation is the point $2\pi$; any other angle multiple to this also won't change the orientation. Those multiples correspond to positive elements of $\mathbb{Z}$. Whereas the negative elements correspond to multiples of $\pi$ since they change the orientation. This answers your second question: the texture, that relates to $-1$, is the one with $f(0)=\pi$ instead of $2\pi$.

Hope that clarified some things (and I hope my understanding is correct; please correct me if I'm wrong).