intro to topology

Problem 1 (20 marks] Consider the map p: S2 SO(3) defined through 2.c? - 1 2.cy 2.cz p(x, y, z) = 2.xy 2y2 - 1 2yz 2.cz 2yz 222 - 1 (i) Prove that the map p is continuous. (ii) Show that p(S) SO(3). Consider now a map o : S3 + SO(3) defined through wa + x2 - y2 - 2 2(xy - wz) 2(wy + x2) (w, x, y, z) = 2xy + wz) w2 - 12 + y2 - 22 2yz - wx) 2(x2 - wy) 2(y2 + wx) w2 - 22 - y2 + z2 (i) Prove that the map o is continuous. (ii) Show that o(S) CSO(3). (iii) Verify that every point (w, x, y, z) R4 satisfying the condition wa + 2a + y2 + 22 = 1 can be expressed as (cos 0, x' sin 0, y sin 0, z sin )) with 0