FIBRING THE COMPLEMENT OF THE FENN-ROLFSEN LINK

Define a link map of two spheres to be a map defined by FIBRING THE COMPLEMENT OF THE FENN-ROLFSEN LINK


. Introduction
f : Si U Si -) R' (or S'n ) of disjoint spheres of dimension p and q into a Euclidean space or sphere of the same dimension m, such that the images of the two componente are disjoint .
Let f1 = f1 Sp and f2 = f1 S2 .By an abuse of notation the suffixes 1 and 2 will often be dropped when no confusion can arise.
In this paper only low dimensions will be considered : in particular the classical case p = q = 1, m = 3 and the next interesting case p = q = 2, m = 4.
For higher dimensions see [K] or [F, H] .Two link maps are said to be link-homotopic if they are homotopic through link maps .
Given a link map f as above there is a map The homotopy class of cp written a(f), may be regarded as lying in irp+ .(S' -1 ) if p, q < m -2 and is a link homotopy invariant .
In the classical case a(f) is an integer and corresponds to the linking number .Moreover ca(f) in this dimension classifies links up to homotopy.
As examples consider the Hopf link and the whitehead link illustrated in figure 1 .

Figure 1
Hopf link Whitehead link The Hopf link has linking number f1 and so is not null homotopic (homotopic to a pair of points) .
However the Whitehead link is nullhomotopic as can easily be seen.Actually it will be shown that the Whitehead link is nullhomotopic in a rather stronger sense which depends on its symmetry.
For the case of link maps f : S2 U SZ -+ R4 the a-invariant lies in 7r4 S3 = Zz .It is known that a is not a complete invariant in this case, see [K] .

. Constructing .linked 2-spheres from linked circles
A link map f : SI U Sz -+ R3 is said to be doubly null homotopic if each component is null homotopic in the complement of the other .For linked circles with unknotted components being doubly nullhomotopic is the same as having linking number zero although it is not often easy to see.
Figure 2 illustrates the fact that the Whitehead link is doubly null homotopic .
So f = flo U feo whilst fll and f21 are constant maps.Moreover flt U feo and fl o U f2t are link-homotopies which shrink one component while fixing the other .
Diagramatically we can think of f as in figure 3. The correspondence f -> f is well defined up to link homotopy because of the asphericity of knots .
If f is the Whitehead link, f is the Fenn-Rolfsen link.This case is illustrated in figure 4. Sections are drawn according to approximate choices of the coordínate t .
to Figure 4.Sections of the Fenn-Rolfsen link In figure 4 critical occurences happen at times to, ti, t2,'t3 .They are described in the following list: to : minima of fi and f2 ti : crossing point of fi t2 : crossing point of f2 t3 : maxima of fi and f2 Between tl and t2 is an isotopy of the Whitehead link.

The double Hopf map
Let h : S3 ___> S2 denote the Hopf map .If x E S2 then h-' x is an unknotted circle and for two distinct points x,y E SZ h -l x, h-1 y is a Hopf link.As x varies over S2 the circles h-'x form the Hopf fibration of S3.
Identify ES' with S4 and ES 2 with S3 .Under the identification of ES2 with S3 let the suspension points líe in h-'x+ and h-l x_.The composition of Eh with h gives a map S4 ~-q _+ S2 .

SZ t3
Consider now the inverse images q-' x as x varíes, over S2 .If x :~xf then g -l x := Tx in a torus.If x+ qÉ x_, which can be assumed by slightly changing the identification of ES 2 with S3 if necessary, g-'x+ = T+ and g-'x_ = Tare pinched tori.That is a torus with one meridian identified to a point .The map T is of degree 1 and u is a 3-duality so clearly the class of the composite [cp] = a cannot be zero.

Figure 5 Figure 6 .
Figure 5It is interesting to note that if x+ = x_, i.e. the suspension points both lie in the same fibre of S3 , then T+ = T_ ¡Sí a Montesinos twin, that is a pair of spheres meeting in two points .See[Mo]  and the discussion in [Mal .Theorem .With the notation abone the pinched tori Tf C S4 may be identified with the image of the Fenn-Rolfsen link .Proof.Generic cross sections of the tori Tt look as follows : The torus T+ is indirected by a heavier line than T-

Figure 6
Figure 6 is shown to represent the same two pinched tori as figure 4 by cancelling a saddle with a minimum on T+ and a saddle with a maximum on T-as illustrated in figure 7 on the component T+ .This is achieved by pushing over a 3-ball (shown shaded) .

Figure 7 .
Figure 7. Cancelling a saddle with a minimum This process applied to T+ and T-res.ults in the cross sections of figure 8 which can be readly seen to be the same as figure 4.