Hatcher 2.1.1

Hatcher, Algebraic Topology, Chapter 2, Section 1

1. What familiar space is the quotient \Delta-complex of a 2-simplex [v_0,v_1,v_2] obtained by identifying the edges [v_0,v_1] and [v_1,v_2], preserving the ordering of the vertices?

Proof. It’s a Möbius band.

This can be seen by considering the 2-simplex, cutting along the perpendicular bisector connecting [v_0,v_2] to v_1, and rearranging. I (crudely) illustrate the process here. \square

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