= Set n > 1. Set an equivalence relation ~ on the space W C2 {0} as follows: (y, v) (y', v') there exists some integer j E Z such that (y', v') = (e2triky, eztikv). Let X = W/ ~ denote the quotient space. Let q:W + X be the quotient map. Let wo = (1, 1) E W. Set Xo = q(wo) E X. = 9 a. Prove that q must be an n-fold covering map, where n-fold is equiva- lent to q-1(x) containing n elements for each x E X. b. Prove that 11(X, Xo) has n elements where all of these elements are powers of a single element[8]. = c. Prove that the image B = 9(53) of the unit sphere S3 in C2 under 9 is a deformation retract of X. = Set n > 1. Set an equivalence relation ~ on the space W C2 {0} as follows: (y, v) (y', v') there exists some integer j E Z such that (y', v') = (e2triky, eztikv). Let X = W/ ~ denote the quotient space. Let q:W + X be the quotient map. Let wo = (1, 1) E W. Set Xo = q(wo) E X. = 9 a. Prove that q must be an n-fold covering map, where n-fold is equiva- lent to q-1(x) containing n elements for each x E X. b. Prove that 11(X, Xo) has n elements where all of these elements are powers of a single element[8]. = c. Prove that the image B = 9(53) of the unit sphere S3 in C2 under 9 is a deformation retract of X