Question

Let S denote the 2-dimensional sphere. Define the complex projective line CP as the quotient space C {0}/~, where is the equivalence relation on {0} that I ~y if r = Ay for some 1 € C. Prove that S and CPI are homeomorphic.
Let S² denote the 2-dimensional sphere. Define the complex projective line ¹ as the quotient space $\mathbb{C}$ ² \ {0} / ∼ , where ∼ is the equivalence relation on $\mathbb{C}$ ² \ {0} that x ∼ y if x = λy for some λ∈C. Prove that S² and ¹ are homeomorphic.

Answer 1

solutions let s? be a 2-sphere CP'-Complex projective une for 240 in C. (ex ) 1209/xnxx where e-Complex then cpl cambe written as. cu{o} plane and soos ix the print at infinity because all ix the space of lines through orizin in C2. we establisha a homeomorphism from onto Cipt = cu soos Cuscola op1 Define fi S² N(0,0,1) we do a stereographic projection. we project by a line geining N(0,0,1) and a print of the sphere (a, b) which intersects the complex plane . C. at some pointp. we make the function. f. od - P. which tarkes (a, b, c) to the punt P. on a sphere S² Equation of line sinning (0,0,1) and a print Q (933, C) ix ziventy. n-o a-o Y-O b-o z-1 e-1 olsay. ax=2a, y=73 and Zz ne-at1 putkuz z=o. we get ace-4) +120. 3 x = 1/2 .., 23 qelya Pe 2-0. The map. fi s2 Culo} = CP' is defined by. (a,b,c) liceo eo) if (4,5,6)*(0,0,1) (0,0,1) 400 and the inverse map. gsft; cusas Equation of rine joining (Bito) and the print (0,0,1) = a (say) S? in givenby. X-0 00 y-o zh t-o and Z=-7 +1. y=nt 7-16 Now the print (2,7, 2) ES?

thus. are y + z =) & Xsvt avtr + (1-2) 21 (srety + 1-2x + 7 = 1 x (sretre) -2720. ( alo'et'+1)-2)=0. 020, or a=2 srttel 29 svetre ya 9 2 sefel of A20, will give the printa(0,0,1) ral. and if Az 2 26 then. a= sretrel sett and z= 1- 2 2 sret) sretren i thus the inverse map of = f1: Cusos S2 ix given by svetres (3,4) (23 at sret sretel svetel, oo (0;0,1) then. f and its inverse go ft. are continuous because the projection map in continuousas it în done by via a line from a fixed punt so. slight change of pink in 82 will produce slight chemze of print in cupos, and vice versa. f in a homeomorphism. (proved, So. 82 and CP' are homeomorphic