I'm reading Kosniowski's A first course in algebraic topology. In chapter 25 he discusses the fundamental group of some quotient spaces using Van Kampen theorem. I'm having troubles understanding how the boundaries of the following quotient spaces in figure 25.7 become the spaces in figure 25.9 after the identifications are made.
Since he doesn't explain any process to visualize the transformations under the identifications i'm guessing he's doing it by eye.
I've been able to handle simpler cases like the boundary of the torus becoming $\mathbb{S}^1 \lor \mathbb{S}^1$ after the identification, but not these cases.
I do not know the purpose of the triangles inside the hexagons. What happens if you identify the oriented edges denoted by $a_1,a_2, a_3$?
Let us first look at the vertices which we denote by $v_0, v_1, \dots, v_5$ beginning at the top and going clockwise (that is, $v_i$ corresponds to $2i$ o'clock am).
In the first picture the $a_1$-egdes begin at the vertices $v_0, v_2, v_4, v_5$ and end at the vertices $v_0, v_1, v_3$. Hence $v_0, v_2, v_4, v_5$ are mapped to a common point $w$ in the quotient space and $v_0, v_1, v_3$ are mapped to a common point $w'$ in the quotient space. Since $v_0$ occurs in both groups of points , we have $w = w'$. Hence the quotient space is obtained by attaching two loops denoted by $a_1, a_2$ to $w$.
Im am sure you can do the other two cases by yourself.