work with 1 B , 2 ,3 and 4

I think the idea is clear

Algebraic Topology State the Seifert and Vankampen Theorem for a space X given as a union of two sets. b) Let U = S'xs! - int(disk) where S' is a unit circle, and let V = K - int(disk) where K is a Klein bottle. Hence au and the aV are both circles. Let f:au - V be a function defined by f(t) = enio where n is a positive integer. Compute (AI) where M = UUV where aU is identified with a via f. +2) a) Let K be a Klein bottle. Show using Seifert and Vankampen Theorem that +1(K) Ja, 6 | bab-I = 2-?). b) Show N = (a, 62 ) is an abelian normal subgroup isomorphic to Zx Z. c) Let p: F + K be a covering space with p. (#1(F)) = N. Show F is a surface and find F. Justify your answer. d) What is the group of automorphisms A(F,p)? a) Let X be a space such that s (X) - Zz. Let f: x + SX S1 = T be a continuous map, and let p: R2-T be the universal covering space. Show f lifts to a map g: X Rso that pog=f. b) Show 9 is homotopic to a constant map. c) Show f is homotopic to a constant map. 4) Identify the following surface as a certain connected sum of either tori or projective planes with holes: o Algebraic Topology State the Seifert and Vankampen Theorem for a space X given as a union of two sets. b) Let U = S'xs! - int(disk) where S' is a unit circle, and let V = K - int(disk) where K is a Klein bottle. Hence au and the aV are both circles. Let f:au - V be a function defined by f(t) = enio where n is a positive integer. Compute (AI) where M = UUV where aU is identified with a via f. +2) a) Let K be a Klein bottle. Show using Seifert and Vankampen Theorem that +1(K) Ja, 6 | bab-I = 2-?). b) Show N = (a, 62 ) is an abelian normal subgroup isomorphic to Zx Z. c) Let p: F + K be a covering space with p. (#1(F)) = N. Show F is a surface and find F. Justify your answer. d) What is the group of automorphisms A(F,p)? a) Let X be a space such that s (X) - Zz. Let f: x + SX S1 = T be a continuous map, and let p: R2-T be the universal covering space. Show f lifts to a map g: X Rso that pog=f. b) Show 9 is homotopic to a constant map. c) Show f is homotopic to a constant map. 4) Identify the following surface as a certain connected sum of either tori or projective planes with holes: o