Hopf fibration
A locally trivial fibration for
. This is one of the earliest examples of locally trivial fibrations, introduced by H. Hopf in [1]. These mappings induce trivial mappings in homology and cohomology; however, they are not homotopic to the null mapping, which follows from the fact that their Hopf invariant is non-trivial. The creation of the mappings requires the so-called Hopf construction.
Let be the join of two spaces
and
, which has natural coordinates
, where
,
,
. Here, for example,
, where
is the suspension of
. The Hopf construction
associates with a mapping
the mapping
given by
.
Suppose that mappings are defined for
by means of multiplications: in the complex numbers for
, in the quaternions for
, and in the Cayley numbers for
. Then
, and the Hopf mapping is defined as
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The Hopf mapping ,
, is a locally trivial fibration with fibre
. If
is a mapping of bidegree
, then the Hopf invariant of the mapping
is
. In particular, the Hopf invariant of the Hopf fibration is 1.
Sometimes the Hopf fibration is defined as the mapping given by the formula
,
. This mapping is a locally trivial fibration with fibre
. For
one obtains the classical Hopf fibration
.
References
[1] | H. Hopf, "Ueber die Abbildungen von Sphären niedriger Dimension" Fund. Math. , 25 (1935) pp. 427–440 |
[2] | D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) |
Hopf fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_fibration&oldid=11978