10 Hopf Fibration and Its Applications 181h(f 0 )=h(f 1 ) it suffices to show that the deformationft:S^2 n−^1 ×I →
Sn connecting f 0 tof 1 and not passing through the points aandb can
be constructed. Then the submanifoldsft−^1 (a)andft−^1 (b)donotnon-
intersect under the homotopy; therefore the linking number is unaltered.
- The independence ofh(f) from the choice of regular valuesaandbin
Snis proved quite simply. Leta 1 andb 1 two other points inSn. There exists
a mapγ:Sn→Snthe sphere onto itself, homotopic to the identity (since
π 1 (Sn) = 0) and suchγ(a)=a 1 ,γ(b)=b 1. Then the mapsfandγfare
homotopic. Thereforeh(f)=h(γf).
10.2.2 Integral Representation of the Hopf Invariant
The Hopf invariant h(f) admits a remarkable integral representation
due to Whitehead [3]. This result has important applications in magneto-
hydrodynamics, field theory, condensed matter. The Hopf invariant act as the
topological conversation law.
First we formulate the Whitehead result for the classical Hopf fibration.
Proposition 4.Letw^2 be a normalized 2-form onS^2 , i.e.,
∫
S^2 w(^2) =1and
f:S^3 →S^2 is a smooth map. Consider the 2-formΩ^2 =f∗(w^2 )=dξ^1 ,where
ξ^1 is a 1-form. Then ∫
S^3
f∗(w^2 )∧ξ^1 =h(f). (10.7)
The multidimensional generalization of the Whitehead formula for the Hopf
fibrationS^2 n−^1 →Snis the following:
Proposition 5.Letωnbe a normed n-form onSn, i.e.,
∫
Snωn=1.Thenexists the n-formf∗ωnonS^2 n−^1 induced byfand exact, i.e.,f∗ωn=dξn−^1 ,
whereξn−^1 is an− 1 – form onS^2 n−^1 .Then
∫
f∗(ωn)∧ξn−^1 =h(f). (10.8)
The proof of (10.7) and (10.8) can be found in [4]. (See also [2].)10.3 Applications of Hopf Invariant
Two definitions of Hopf invariants via linking coefficients and integral formula
(10.7) intimately linked among themselves and admit different generalizations
and applications in many mathematical and physical problems. I discuss some
examples that find or that could find some applications in biology. (See also
the other chapters of this book.)