Algebraic Topology · Problem 2.1.2

Homology — Allen Hatcher

Mathematics — Topology · simplicial-homology-delta-complexes

Problem

Show that the Δ\Delta-complex obtained from Δ3\Delta^3 by performing the order-preserving edge identifications [v0,v1]∼[v1,v3][v_0, v_1] \sim [v_1, v_3] and [v0,v2]∼[v2,v3][v_0, v_2] \sim [v_2, v_3] deformation retracts onto a Klein bottle. Also, find other pairs of identifications of edges that produce Δ\Delta-complexes deformation retracting onto a torus, a 2-sphere, and RP2\mathbb{RP}^2.

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