Show that the $\Delta$-complex obtained from $\Delta^3$ by performing edge identifications deformation retracts onto a Klein bottle.

Question

I am going through some exercises in Hatcher's Algebraic Topology. You have a $\Delta$-complex obtained from $\Delta^3$ (a tetrahedron) and perform edge identifications $[v_0,v_1]\sim[v_1,v_3]$ and $[v_0,v_2]\sim[v_2,v_3]$. How can you show that this deformation retracts onto a Klein bottle?

Answer 1

Flatten the tetrahedron and draw it in the plane (triangle with a vertex inside and edges going out to the vertices of the triangle); i.e. smoosh the 3-cell. If you smoosh and cut it up a little, you're looking at the standard "rectangle-with-sides-identified" picture of the Klein bottle.