Every point P on a regular torus X can be described by means of two angles θ and ∅, as shown in Fig. 43.16. That is, we can associate coordinates (θ, ∅) with P. For each of the mappings f of the torus X onto itself given below, describe the induced map f_*n of H_n(X) into H_n(X) for n = 0, 1, and 2, by finding the images of the generators for H_n(X) described in Example 42.12. Interpret these group homomorphisms geometrically as we did in Example 43.9. 

a. f: X → X given by f((θ, ∅)) = (2θ, ∅)

b. f :X → X given by f((θ, ∅)) = (0. 2∅)

c. f: X → X given by f((0, ∅)  = (2θ, 2∅)

Data from Figure 43.16

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