[Math] 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

The 3-simplex obviously deformation retracts onto the union of the surfaces obtained by $[v_0,v_1,v_3]$ and $[v_0,v_2,v_3]$. Note that the continuous image of a deformation retract, where the map identifies the points in the retract only, is still a deformation retract.