What does Polchinki mean when he says identifying under involution? I am reading chapter 7 of Polchinki's String Theory vol. 1 textbook and I am trying to understand why one obtains the cylinder (annulus) from the torus by identifying points under involution.
To be more specific, he starts this chapter by saying that one can describe the torus as the complex plane with the identification
$$w \cong w + 2\pi \cong w + 2 \pi \tau, \quad \quad (1)$$
Where $w= \sigma^1 + i \sigma^2$. Then, he states that the cylinder can be obtained from the torus with imaginary $\tau = it$ by identifying under the involution
$$w'= - \bar{w}. \quad \quad (2)$$
He also says that (2) implies that the lines $\sigma^1 = 0,\pi$ are fixed by this reflection.
Here are some related questions that can help me clarify what I don't understand:
(1) Why is the line $\sigma^1 = \pi$ fixed? For me, equation (2) implies that $w=\pi$ is identified with $w = -\pi$, consequently, this point is not a fixed point.
(2) And finally, why does this extra identification describes a cylinder with fundamental region $0 \leq Re\;w \leq \pi$, $w \cong w + 2 \pi it$ (I can't see why the width of the cylinder becomes $\pi$, given that in equation (1) we have $w \cong w + 2\pi$)?
 A: I'm not familiar with this material, so I don't know what the author is talking about physics-wise, but in terms of mathematical intuition (and you may already understand this, but just so that we're on the same page), a torus described in these terms
$$w \cong w + 2\pi \cong w + 2 \pi \tau$$

is just a tiling of the plane with a parallelogram, with points within it identified to (considered the same as) corresponding points in each copy. In particular, the edges are identified - much like the edges of the screen in the Pacman game (if you exit at one end, you reappear at the other).

(screengrab from here)
Now imagine tiling Pacman screens; exiting at one end would just look like entering another screen. The positions of each copy of Pacman, of each ghost, etc., are identified as being the same (e.g. the red ghost at $w$ is also at $w + 2\pi$, at $w + 2 \pi \tau$, at $w + 2 \pi + 2 \pi \tau$, etc...)

Anything drawn in one tile appears in the same way in all tiles, and everything animates the same way (every tile is the same movie). I'm using this fact in the image below - for a torus, any collection of points in one tile that visually forms some shape or "constellation" can be copied over to their identified points in every other tile, maintaining the same local offset and visual appearance. Keeping this in mind when doing the $w'= - \bar{w}$ identification should help you see where each $-\bar{w}$ lands in the context of the source fundamental poligon.
I've been able to obtain a copy of the book, so from what I've seen (Chapter 5.1., Fig. 5.1.), in general $\tau \neq it$, but is angled (the fundamental polygon is in general a parallelogram; if I'm not mistaken, $\tau = it$ is a special case where it's a rectangle).
I'm going to assume $\tau = it$, that is, a rectangular fundamental polygon. Now, $w'= - \bar{w}$ is a reflection across the imaginary axis. If you take a number of representative points in the highlighted square, $w'= - \bar{w}$ sends them to the adjacent square (horizontally connected dots in the leftmost image below). Since these points are identified on top of the underlying toroidal topology, each $w'$ in the left square also corresponds to $w' + 2\pi$ (I'm ignoring the other direction). This lands them back in source square (second image), revealing which points are identified to the (grayed out) source points within the original fundamental polygon (the grayed out features are the dots/lines from the previous picture).
Remember the Pacman tiling: the entirety of the surface of the torus is just the fundamental polygon (here, the highlighted square); every other tile is a copy of it).

This corresponds to squishing the torus to a flat surface, producing a ring of width $\pi$.
So with this picture in mind, I think I can tackle your questions:

(1) Why is the line $\sigma^1 = \pi$ fixed? For me, equation (2) implies that $w = \pi$ is identified with $w = -\pi$, consequently, this point is not a fixed point.

The lines $\sigma^1 = 0$ and $\sigma^1 = \pi$ are the vertical lines that form the "creases" (the boundary edges) of the ring. In the image above, on the torus on the right, $\sigma^1 = \pi$ is the line parallel to the red arrow that goes through the point where the black arrow turns. Not sure what exactly is the technical meaning of "fixed line", but intuitively, the qualitative difference is that, in the Pacman analogy, you cannot go through that edge; instead, you are reflected back (or appear as such on the ring).

(2) And finally, why does this extra identification describes a cylinder with fundamental region $0 \leq Re\;w \leq \pi$, $w \cong w + 2 \pi it$ (I can't see why the width of the cylinder becomes $\pi$, given that in equation (1) we have $w \cong w + 2\pi$)?

I think you can see it from the image: the interplay of the two identifications results in a mirror symmetry within the original fundamental polygon with the axis at $\sigma^1 = \pi$ (and at any $\sigma^1 = n\pi$). It's essentially the accordion fold with creases at $n\pi$; when you fold a piece of paper in this way and punch a hole through it, you get the same identification of points (at least, in the horizontal direction):

