__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.

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 .

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