10 Hopf Fibration and Its Applications 183∫B 1u 1 ∧u 2 =−∫
B 2u 2 ∧u 1 =k(l 1 ,l 2 ). (10.9)Let us write the linking coefficientk(l 1 ,l 2 ) using the first definition via
intersection number. In fact it is contained in the Whitehead theorem (Propo-
sition 4). We associateliwith a closed 2-formviwith support outsideli.The
formviis determined by the equality
∫
zvi= Ind(z,li).Herezis an arbitrary 2-cycle in the complement toli.
3-formsu 1 ∧v 2 andv 1 ∧u 2 are defined on the whole sphereS^3 and
∫
S^3u 1 ∧v 2 =∫
S^3k(l 1 ,l 2 ). (10.10)The proof of (10.10) is actually equivalent to the proof of (10.7).
The tuple ofk(li,lj) is one of the numerical characteristics of the link
l=l 1 ,...,ln. It is also natural to introduce the linking number for the whole
oflby the formula
̄k(l) = max
1 ≤i<j≤n
|k(li,lj)|.
Iflis isotopically unlinked, then ̄k(l) = 0. However for the links in
Fig. 10.2a Whitehead link and 10.2b Borromean rings ̄k(l) = 0, but they re-
main linked. So we should introduce high-order linking numbers. If ̄k(l)=0,
wherel=(l 1 ,l 2 ), then there exists a 1-formu 12 on the complement to l- and
2-formsv 12 ,v′ 12 with compact support onS^3 such that
du 12 =u 1 ∧u 2 ,dv 12 =v 1 ∧u 2 ,dv′ 12 =u 1 ∧v 2.Assume that ̄k(l)=0forl=(l 1 ,l 2 ,l 3 ). In the addition to the above,
we define 1-formsu 3 andu 23 and 2-forms with compact supportv 3 ,v 23 ,v′ 23
and verify by differentiation that the 2-form ̃u 123 =u 12 ∧u 3 +u 1 ∧u 23 is
closed. ̃u 123 is defined on the complement ofl=(l 1 ,l 2 ,l 3 ). We also check by
differentiation that the 3-forms
v ̃ 123 =v 12 ∧u 3 +v 1 ∧u 23 , ̃u′ 123 =u 12 ∧v 1 +u 1 ∧v 23are closed.v 12 andv 23 can be picked so that the latter 3-forms are defined on
the whole sphereS^3.
The cohomology classes inH^2 (S^3 \l)andH^3 (S^3 ) determined by ̃u 123 , ̃v 123 ,
and ̃v′ 123 are called the Massey products of cohomology classesu 1 ,u 2 ,u 3 ,v 1 ,u 2 ,
u 3 ,u 1 ,u 2 ,v 3 , denoted by〈clu 1 ,clu 2 ,clu 3 〉,...,respectively. They do not de-
pend on the choice ofui,vj,inclui,clvi.