Hatcher, Algebraic Topology, Chapter 0
1. Construct an explicit deformation retraction of the torus with one point deleted
onto a graph consisting of two circles intersecting in a point, namely, longitude and
meridian circles of the torus.
Proof. Let be the torus and let be the space in question. By considering the square gluing diagram of the torus sans a point, a diagrammatic representation for can be given as shown in Figure 1 below.
Note that the gray part of Figure 1 represents the fact that the torus is filled in and that one can visualize the deformation retract in question by grabbing hold of the hole in and stretching it out so that the diagram in Figure 1 is hollow (i.e., not filled in, i.e. all white, etc.). Thus, the deformation retract in question amounts to the projection of the interior of a square (the square representing the gluing diagram of ) onto its boundary, the formula for which can be attained by presupposing that the gluing diagram for is placed in as where . Basic arithmetic shows that, for , the family given by
“does the trick.” Indeed, note that , that is an element of , and that due to the fact that on . Finally, note that continuity of the family is given due to the fact that is the composition of continuous functions of for all .