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?
flatten the tetrahedron and draw it in the plane (triangle with a vertex inside and edges going out to the vertices of the triangle). if you cut it up a little, you're looking at the standard "rectangle-with-sides-identified" picture of the klein bottle. sorry for the terrible picture, mspaint hasnt changed since 3.x as far as i can tell...