10
Hopf Fibration and Its Applications
M. Monastyrsky
Summary.In this chapter we deal with Hopf fibration – one of the key exam-
ples in the topology of manifolds – and vividly illustrate the power and diversity of
applications of topology. We especially point out less familiar applications of Hopf
fibrations. For the sake of volume limit we omit the proofs and refer to the appro-
priate literature.
10.1 Classical Hopf Fibration
Hopf in his celebrated paper “ ̈uber die Abbildungen der 3-Sphare auf die
Kugelfleche,” Math. Ann. 104 , 637–665, (1931) studied the space of homo-
topically nontrivial mappings of spheres:
S^3 →S^2. (10.1)In modern language it is a groupπ 3 (S^2 ).
Later, more general mappings
S^2 n−^1 →Sn (10.2)came to be calledgeneral Hopf fibrations.
10.1.1 Constructing the Hopf Fibrations
We first show that
π 3 (S^2 )=Z. (10.3)
The proof follows immediately from the exactness of the sequence of
homotopy groups
πi(S^1 )→πi(S^3 )→πi(S^2 )→πi− 1 (S^1 ). (10.4)