I have been stuck on this question for a long time:
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.
What I tried: I am aware that the way to do this type of question is by repeated cutting and gluing, until the diagram resembles the fundamental polygon of the Klein bottle. Unfortunately, I am not sure how to go about this.
I drew a basic picture of $\Delta^3$ as such. How do I go about making the first cut to identify $[v_0,v_1]\sim [v_1,v_3]$?
P.S.: I am aware of this question (Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle.), but I don't really understand his solution, besides in the comments someone said it is wrong.
Thanks for any help. I am looking for the simplest possible proof, if possible just using diagrams of "cutting and gluing", a proof-without-words of sorts. Not sure if such a proof is possible though.