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INIDEPENDENT UNIVERSITY OF MOSCOW

3. n AND THE HOPF BUNDLE

TOPOLOGY-2, FALL 2012

Problem 1. (a) Let X be a CW-complex. Prove that k (X ) = k (skk+1 (X )). (b) Prove that k (S n ) is trivial for
any k < n. ) = Z, (b) n (S group Z. (c) Describe the natural action of 1 on n .

Problem 2. Prove that (a) 1 (S

1 Sn

1 Sn

) is a direct sum of countably many copies of the

Let S 3 = {(z ; w) C2 | |z |2 + |w|2 = 1} C2 . The map p : S 3 CP 1 given by the formula p(z ; w) = [z : w] is called the Hopf bundle. Problem 3. Prove that the Hopf bundle is indeed a ber bundle with the ber S 1 . In particular, for any [z : w] CP 1 the set p-1 ([z : w]) is homeomorphic to a circle. Problem 4. (a) Prove that the sets A = {[z : w] CP 1 | |w=z | 1} and B = {[z : w] CP 1 | |w=z | 1} are homeomorphic to disks with the common boundary C = {[z : w] CP 1 | |w=z | = 1}. Prove that CP 1 is homeomorphic to S 2 . (b) Prove that p-1 (A) and p-1 (B ) are homeomorphic to solid tori S 1 в D2 and p-1 (C ) is homeomorphic to the torus S 1 в S 1 . Problem 5. (a) Prove that there exists a homeomorphism between the 2-disk D and the set Q = {(z ; w) S 3 | w [0; 1]} mapping @ D to p-1 ([1 : 0]) Q. (b) Prove that for any a = [1 : 0] the intersection Q p-1 (a) contains exactly one point. ab SU(2) is the group of matrices -b a where |a|2 + |b2 | = 1. Problem 6. (a) Prove that the standard (linear) action of SU(2) on C2 maps S 3 C2 to itself. (b) Prove that for any A SU(2) and x S 3 one has p(Ax) = Ap(x) where Ap(x) means the standard (pro jective) action of A on p(x) CP 1 . (c) Prove that the action of SU(2) on CP 1 is transitive: for any u; v CP 1 there is A SU(2) such that A(u) = v. (d) Prove using 6(c) and 5(b) that any two bers of the Hopf bundle are like two links of an anchor chain: for any u CP 1 there exists Du S 3 homeomorphic to the 2-disk D, such that the homeomorphism maps p-1 (u) Du to @ D and for any v = u the intersection Du p-1 (v) is one point. Let e0 def {[1 : 0 : 0]} CP 2 , e1 def {[x0 : 1 : 0]} CP 2 , e2 def {[x0 : x1 : 1]} CP 2 be a ne charts in CP 2 . = = = Problem 7. (a) De ne in CP 2 a cell space where e0 , e1 and e2 are cells of dimension 0, 2 and 4, respectively. Prove that e0 e1 is homeomorphic to S 2 . (b) Let 2 : D4 S 2 be the characteristic map of the cell e2 . Prove that 2 |@ D4 : S 3 S 2 is the Hopf bundle.