I am looking for any resources (book/online) that teaches and further elaborates on how the "cut and glue" technique works for $\Delta$-complexes.
To be precise, I am looking for techniques and at least some semi-rigorous theoretical framework explained nicely in a textbook style to solve questions like this in Hatcher:
Show that the $\Delta$-complex obtained from $\Delta^3$ by performing the edge identifications $[v_0,v_1]\sim [v_1,v_3]$ and $[v_0,v_2]\sim [v_2,v_3]$ deformation retracts onto a Klein bottle.
Hatcher doesn't explain it in his book as far as I read (does he?), and online solutions like Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle. unfortunately is hard to understand for one, and don't look very rigorous to me.
Thanks for any help! Explanations of how to do the question above will also be upvoted and accepted.