180 M. Monastyrsky
In the nonorientable case Ind(P◦Q) is defined as the residue module 2 of
the number of intersection points.
Proposition 2.Letf be a map of two-dimensional diskD^2 →R^3 that co-
incides withγ 1 on the boundary∂D^2 =S^1 and is in general position onγ 2.
ThenInd(f(D^2 )∩γ 2 )is equal to{γ 1 ,γ 2 }.
Outline of the proof. The closed curvesγ 1 (t)andγ 2 (t) define a two-
dimensional oriented surfaceγ 1 ×γ−2:(t 1 ,t 2 )=(r 1 (t 1 ),r 2 (t 2 )) inR^6.
Letγ 1 andγ 2 be disjoint. Then the map (the so-called Gauss map)
φ(t 1 ,t 2 )=r 1 (t 1 )−r 2 (t 2 )
|r 1 (t 1 )−r 2 (t 2 )|is defined, with the degree given by integral (10.5). Therefore degφis an in-
teger. degφremaining unaltered under nonintersecting deformation of curves
γ 1 andγ 2 , i.e., the linking coefficient{γ 1 ,γ 2 }is invariant under homotopies
ofγi. Since the intersection number in (10.6) depends linearly on pointsxiit
is suffices to calculate Ind (10.6) in two cases (a)γ 1 andγ 2 are unlinked and
(b)γ 1 andγ 2 are two orthogonally linked circles (γ 1 is in the (x, y) plane and
γ 2 in the (y, z) plane).
10.2 Hopf Invariant
10.2.1 Definition of Hopf Invariant
Letf :S^3 →S^2 be a Hopf fibration. Consider two regular pointsaandb
on the sphereS^2 and take their preimagesl^1 a=f−^1 (a)andl^1 b=f−^1 (b). The
manifoldsl^1 aandl^1 bare two closed curves inS^3. Consider the linking number
{l^1 a,l^1 b}.
Definition 3.The Hopf invarianth(f)is the linking number{l^1 a,lb^1 }.
Proposition 3.h(f)is the homotopic invariant offand is independent of
the choice of pointsaandb.
Proposition 3 is valid in a considerably more general situation of 2n−
1-dimensional Hopf fibrationS^2 n−^1 →S^2. The Hopf invarianth(f) is de-
fined for a 2n−1-dimensional Hopf fibration similar to the fiber bundle
S^3 →S^2. It should only be noticed that inverse of preimages of two points
in 2n−1-dimensional sphere are (n−1)-dimensional closed submanifolds
M 1 n−^1 andM 2 n−^1. Outline of proof of Proposition 3.1. We show thath(f)
is unaltered under a homotopy of f.Letf 0 and f 1 be mutually homo-
topic mapsS^2 n−^1 →Snandf−tthe connecting homotopy. To prove that