


Chapters 7 and 8 : Fibration
The
mathematician Heinz Hopf describes his "fibration". Using complex
numbers he builds beautiful arrangements of circles in space.



1. Heinz Hopf and topology


Topology is the
science that studies deformations. For example, the cup and the tyre
here on the right are of course two different objects but one can pass
from one to the other by a continuous deformation that does not
introduce any tear: the mathematician says that the cup and the tyre
are homeomorphic
(same form). And a topologist, this is somebody who can’t
distinguish their cup of coffee from their donut!!
There too, the theory took a very long time to
achieve the status of an autonomous discipline, with its own problems
and its original methods, often of a qualitative nature. Even though he
had prestigious predecessors (such as Euler,
Riemann, Listing or Tait), Henri Poincaré is often
considered as being the one who laid the solid foundations of topology
(that he called analysis
situs).
Our presenter, Heinz
Hopf (18941971), is one of his most remarkable followers, in
the first half of the twentieth century.








2. The sphere S^{3}
in C^{2}
We saw that the sphere S^{3}
of unit radius in 4 dimensional space is the set of points at distance
one from the origin. If one takes four real coordinates x_{1},y_{1},x_{2},y_{2}
in this space, the equation of this sphere is :
x_{1}^{2}
+ y_{1}^{2} + x_{2}^{2}
+ y_{2}^{2}= 1.
But one can think of (x_{1},y_{1})
as a complex number z_{1}
= x_{1}+i y_{1} and
of (x_{2},y_{2})
as a complex number z_{2} = x_{2}+i
y_{2}, and the sphere S^{3}
can then be thought as the set of pairs of complex numbers (z_{1},z_{2})
such that
z_{1}^{2}
+ z_{2}^{2} = 1.


In other words, the sphere S^{3}
can be regarded as the unit sphere in the plane of complex dimension 2.
By analogy, but only by analogy, one can thus draw the sphere S^{3}
as a circle in a plane, but it is important not to forget that this
plane is complex, that each one of its coordinates z_{1}
and z_{2} is
a complex number. The axis z_{2}=0
for example is a complex line, therefore a real plane, and it meets the
sphere S^{3}
at the set of points (z_{1},0)
such that z_{1}^{2}
= 1, in other words, on the circle S^{1}.
The same thing is true for the axis z_{1}=0
but also for every line passing through the origin whose equation is of
the
form z_{2}= a.z_{1},
where a is a complex number.
Thus each complex number a
defines a complex line z_{2}= a.z_{1}
that meets the sphere S^{3}
on a circle. There is thus a circle in S^{3}
for each complex number a.
Moreover, although the axis z_{1}=0
is not an equation of this form, one can regard it as corresponding to a
being infinite (isn’t the vertical axis a line of infinite
slope?).
The sphere S^{3}
is therefore filled with circles, one for each point of S^{2},
that is, for each complex number a
(that we allow to be infinite). No two of these circles meet for
different values of a.
It is this decomposition of the 3 dimensional sphere into circles that
one calls the Hopf
fibration.


Click
the image to see a film. 


Recall that if X
and Y
are two sets, a map
f
from X to
Y,
often
denoted f
: X → Y, is a rule which allows us to
associate to each point x
of X
a point f(x)
in Y.
For example, we can consider the Hopf map f
: S^{3} → S^{2}
which associates to a point (z_{1},z_{2})
the point z_{2}/z_{1}.
This deserves two explanations:
First, a point of S^{3}
is a point of the plane of complex dimension 2 and it can be described
by its complex coordinates (z_{1},z_{2}).
Second, we saw, by stereographic projection, that
if one adds to the plane a point at infinity, one obtains a sphere S^{2}.
And of course, the complex number z_{2}/z_{1}
is well defined only when z_{1}
is nonzero and if it isn’t, we choose z_{2}/z_{1}
to be the point at infinity, so that z_{2}/z_{1 }does
indeed define a point of S^{2}.

For each point a of S^{2},
the set of points of S^{3}
whose image by f is the point a (that is, the
preimage of a),
which we call the fiber
over a,
is a circle in S^{3}.
What is the connection with the preceding explanation: quite simply
that all the points of a line z_{2}=
a.z_{1} are such
that z_{2}/z_{1}
is constant (obviously, because it equals a !).




The film first invites us to closely observe this
"fibration". For each a,
we have a circle in S^{3}.
How do we visualize this? By stereographic projection of course! One
projects the sphere S^{3}
onto the 3 dimensional tangent space of the pole opposite the point of
projection. This projection is a circle in space, which you can admire
(remember the lizards!). Of course, it can happen that the circle of S^{3}
passes through the North Pole so that its stereographic projection is a
line (that is, a circle that is missing a point... that has gone to
infinity!).




Several sequences illustrate the fibration :
First, we see only one Hopf circle, associated
with the value of a.
This point a moves on the sphere S^{2} (remember,
the complex plane plus a point at infinity) and we see that the circle
moves in space and become a line from time to time, when a passes
through the point at infinity.
Then, we see two Hopf circles, associated with two
values of a,
that are both on the move. At the bottom of the screen, you see the two
points a moving
and
simultaneously, so too do the two circles. Incidentally, notice that
the two circles are linked, like two links of a chain. One cannot
separate them without breaking them.
Then, we see three Hopf circles for three values
of a in
choreographed movement ... The circles separate, approach...

Click
the image to see a film. 

Finally, we see many Hopf circles at the same
time. The values of a
are chosen at random and the corresponding circles appear little by
little. We can thus "see" that space is filled by the circles and that
these circles do not meet each other. But also, we can now understand
the origin of the word "fibration": all these circles are arranged like
fibers of a fabric: locally, they are well organized like a packet of
spaghetti. This concept of fibration, whose prototype is the Hopf map,
became a central concept in topology and mathematical physics. Some
fibrations are much more complicated, on spaces of much higher
dimension, but it is certainly instructive to have a clear view of this
historical example!
To think of the real plane as a complex line is
useful, but to think of the space of real dimension 4 as a plane of
complex dimension 2 is even more useful!



4. The fibration ... continued
See in the film: Chapter 8 : Fibration,
continued.



To better understand the Hopf fibration f
: S^{3} → S^{2},
consider a line of latitude p
in S^{2}
and then its “preimage” p
by f that is, the set of
points of S^{3}
for which the image by f is in
p.
Since the preimage of each point of S^{2}
(each fiber) is a Hopf circle and since a line of latitude is also a
circle, the preimage of p
is composed of a family of circles that depends on a parameter
pertaining to the circle p.
So it’s a surface in S^{3}
for which the film shows the stereographic projection in the 3
dimensional space, as usual.
When a line of latitude is very close to a
pole of S^{2 }
and is thus a very small circle, the preimage of p
is a small tube, in the vicinity of the fiber above this pole. When the
line of latitude grows gradually, becomes the equator, then decreases
again to finally approach the opposite pole, the tube grows bigger
gradually then decreases again and ends up being a very fine tube.
These tubes are tori in S^{3}
but we only observe them through their projections in 3 dimensional
space, so they do not appear very fine when they pass close to the
North Pole of the sphere S^{3}.

Click
on the image to see a film. 

Strictly speaking, a torus is the surface of
revolution in space obtained by turning a circle around an axis that is
in its plane. A point of the torus has two angular coordinates: one to
describe the position on the circle and another one to describe the
angle through which the circle is turned.
Notice the analogy with longitude and latitude.
Beings who lived on a torus (and not on a sphere, like our Earth) would
have also invented the ideas of meridian lines, parallels, longitude
and latitude.


In fact, the topologists often call a "torus" a
surface
which is "homeomorphic" to a torus of revolution, like a coffee cup
for example! When they want to speak about a torus obtained by turning
a circle, they make it clear by saying torus of revolution.
On a torus of revolution, one clearly sees two
families of
circles: the meridian lines (in blue) and the parallels (in red).
Distinguishing between meridian lines and parallels is now a bit more
difficult. On a sphere it was easy: all the meridian lines pass through
the poles, but there are no poles on a torus of revolution! One then
agrees (but this is a convention only) to name the blue circles
"meridians" because they lie on planes that contain the symmetry axis
of the torus of revolution, and to name the red circles "parallels"
because they lie on planes that are perpendicular to this axis.
It is a little
marvel of geometry that it is possible to trace many other circles on a
torus of revolution... This chapter explains how to construct them.



Recall the formula which expresses the Hopf
projection. In terms of the complex coordinates, it sends (z_{1},z_{2})
to the point a
considered as a point of S^{2}.
Fixing a line of latitude p
in S^{2},
is the same as fixing the modulus of a complex number, so the preimage
of a line of latitude is described by an equation of the form
z_{2}/z_{1}
= constant.
For example, let us choose 1 for this constant so
that z_{1}
and z_{2}
have the same modulus. But don’t forget that
z_{1}^{2}
+ z_{2}^{2} = 1,
so the modulus of z_{1}
and of z_{2}
are both equal to √2/2. Therefore, the preimage of the line
of latitude consists of the (z_{1},z_{2})
where z_{1} and z_{2}
are chosen arbitrarily on the circle centered at the origin with radius
√2/2. Thus we see that the preimage of the line of latitude
is a surface that is parameterized by two angles : so it’s a
torus, as we saw in the film. If we fix z_{1},
we obtain a circle in S^{3},
and if we fix z_{2} we obtain
another circle, but for a torus of revolution in dimension 4, it is
impossible to say which is a meridian and which is a parallel.
When we stereographically project this torus into
3 dimensional space from the North Pole, with coordinates (0,1), it
isn’t difficult to verify that the projection of the torus is
not only homeomorphic to a torus but it is actually a torus of
revolution. Revolution about which axis? Quite simply about the
stereographic projection of the Hopf circle that passes through the
North Pole; this projection is certainly a line! So we can see how to
interpret a torus of revolution as the preimage of a line of latitude
by the Hopf map.
Here is a consequence of this interpretation: for
each point of the chosen line of latitude, the corresponding Hopf
circle is obviously contained in the torus of revolution. We have just
found other circles on the torus of revolution....
Here are some formulas. Consider the torus of
revolution in space that is obtained by projecting
z_{1}
= √2/2 ; z_{2} =
√2/2
from the North Pole (0,1).
Let's then consider the maps that send (z_{1},z_{2})
to (ω.z_{1},z_{2})
where ω
describes the circle of complex numbers with module 1. As the modules
of z_{1} and z_{2} do not
change, these maps preserve the sphere S^{3}.
Points of the form (0,z_{2})
do not change either.
What we are looking at is rotations in the space of dimension 4 around
the complex line z_{1}=0.
As this line passes through the projection pole (0,1),
its stereographic projection is not a circle, but a straight
line. Therefore, these maps that depend on the parameter ω define
rotations of our space around a line. These rotations also preserve the
torus of revolution that we are looking at, so that the line z_{1}=0 is the axis
of symmetry of the torus
!


As a consequence, the parallel
that passes through (z_{1},z_{2})
is the set of points of the form (ω.z_{1},
z_{2}) where ω belongs to
the circle of complex numbers of modulus 1. The meridian
passing through (z_{1},z_{2})
is the set of points of the form (z_{1},
ω.z_{2}).
The Hopf circle
passing through (z_{1},z_{2}) is the set of
points of the form (ω.z_{1},
ω.z_{2}) (note that if we
multiply z_{1}
and z_{2}
by ω, we don’t change z_{2}/z_{1} so all these
points have the same image under f;
they are in the same fiber). We don’t stop here either;
through each point (z_{1},z_{2})
we can also consider the "symmetric" circle of points of the form (ω.z_{1},
ω^{1}.z_{2})
which gives us a fourth circle traced on the torus of revolution.
We have just shown that through each point of a torus
of revolution one can draw four circles: a meridian, a parallel, a Hopf
circle and the symmetric circle of a Hopf circle.




This fact has been known for a long time. These
circles are usually called Villarceau
circles, in honor of a mathematician of the nineteenth century. But, as
the reader will have already realized, it is quite rare in mathematics
that a theorem is due to the person whose name it carries, so long and
complex is the process of creationassimilation. Indeed, a staircase at
the museum of the cathedral of Strasbourg, dating to the XVI Century
shows that sculptors didn’t have to wait for Villarceau in
order to cut circles on tori!


The second part of this chapter depicts the
Villarceau circles in a way that is independent of the Hopf fibration.
Starting with a torus of revolution, one slices it by a bitangent plane
and observes that the section consists of two circles.
How do we prove this? One can write equations and
calculate... It is possible (see here) but well, it’s not
very enlightening. But algebraic geometry makes it possible to prove it
in a masterful way almost without calculation, provided we use concepts
such as "cyclic points". These are points that are not only infinite
but also imaginary! You see, imagination is infinite! For a proof of
Villarceau’s theorem with this kind of ideas, see the article.




Given a surface in 3 dimensional space, we can
regard it as a surface in S^{3},
by adding a point at infinity. Since S^{3}
is the unit sphere in 4 dimensional space, one can turn it by
fourdimensional rotations before once more stereographically
projecting it
into 3 dimensional space! One obtains another surface which resembles
the first but which is different! If one starts from a torus of
revolution, the surfaces thus obtained are called Dupin
cyclides and they were extensively studied in the nineteenth
century. Since stereographic projection transforms the circles which do
not pass through the pole into circles, the existence of four families
of circles on the tori of revolution implies that there are also four
families of circles on the cyclides...
A torus in the space of dimension 3 can thus be
thought of as a surface in S^{3}
that rotates in the space of dimension 4. If we observe this through
stereographic projection, we see a film where the Dupin cyclide changes
shape, and at a certain moment, when the surface passes through the
projection pole, becomes infinitely large, and then returns to its
initial shape. However, you can see that meridians have become
parallels, and vice versa, and that the torus has been turned inside
out!

Click
on the image to see a film. 

The geometry of the circles in space is
magnificent. It sometimes bears the name of anallagmatic
geometry. It
is subject on which much that could be said and shown!
5. Hopf and homotopy
To
finish this page, here are some brief comments on Hopf’s
motivations, about which we unfortunately don’t speak in the
film.
In
topology we often consider maps between topological spaces X
and Y.
We won’t give the definition here, but you can think for
example that X and Y
are two spheres of dimension n
and p.
Of course, until now we only discussed spheres of dimension 0,1,2 and 3
but you must have guessed that the story wouldn’t
stop
there... Of course, there isn’t much interest in studying
completely arbitrary maps, and we focus on continuous maps,
that is, those for which the point f(x)
doesn’t change much when the change in x
is sufficiently small. For example the map that associates to a real
number x the number +1 if x
is not zero and 1 if x is zero
isn’t continuous since it "jumps" when we pass 0. But the map
that associates to a number x
its square x^{2}
is continuous; if we change a number only a little, its square is only
changed a little. One of the fundamental problems of topology consists
of understanding continuous maps between topological spaces, such as
the spheres.
In fact, topology is less demanding; it seeks to
understand the homotopies.
Another complicated word that means something simple! Suppose that we
are given two continuous maps f_{0}
and f_{1}
from the sphere S^{n}
to the sphere S^{p}.
We say that f_{0}
and f_{1}
are homotopic if we can deform the first one into
the second. In other words, this means there is a family of maps f_{t}
that depend on a parameter t,
which is a number between 0 and 1, and that connects f_{0}
and f_{1}.
Even more precisely, this means we can associate to each x
of S^{n}
and to each number t between 0
and 1 a point f_{t}(x)
in a way that defines a continuous map of x
and of t such that for t=0 we
have f_{0}
and for t=1 we have f_{1}.


For example, a map f :
S^{1}→
S^{2}
is nothing other than a closed curve plotted on the 2 dimensional
sphere. For example, the map f_{0}
might be the one that sends all the points x
of S^{1}
to the North Pole: this is what one calls a constant map.
As for the map f_{1},
this could be for example the map that sends the circle S^{1}
to the equator of S^{2}.
To say that these two maps are homotopic, means that we can
progressively deform the equator until it becomes the North Pole. This
is what you can see in the image on the right. In fact, it tuns out
that this is always possible; every pair of maps S^{1}
in S^{2}
are always homotopic. The topologist says that all the curves
plotted on the sphere S^{2}
are homotopic to constant curves, or that S^{2}
is simply connected.
It shouldn’t be difficult either to convince yourself that
the same thing is true for the spheres S^{p},
of all dimensions greater than or equal to two. (Check out this page too)




A map between S^{1 }and
S^{1}
amounts transforming each point of a circle to another point on the
circle: it is to some extent a
curve plotted on a circle. Such a map has a degree : this is
simply the number of complete revolutions that it makes. For example,
the constant map does not turn at all: its degree is 0. The identity
map that sends every point to itself, makes of course one revolution;
its degree is 1. The map that sends any complex number of module 1 to
its square doubles the argument. So as one travels once around the
circle, the square makes two revolutions: its degree is 2. When a map
is deformed, its degree is not changed, so there are maps of S^{1}
to S^{1}
that
are not deformable to constant maps... It is a little more
difficult to see than two maps of the same degree are deformable to
each other.



But what about the maps between S^{2}
and S^{2}?
It is similar to the case of S^{1}
to S^{1};
one can also define a degree, even if it is not a question any more of
counting a "number of revolutions": it is now a question of counting
how often the image of f
"covers" the sphere and this isn’t easy to define. The
simplest example is the identity: the map which sends any point to
itself: its degree is 1. One may well suspect that it is not possible
to deform the identity of the sphere S^{2}
to make it constant, without tearing the sphere. But it is still
necessary to prove it!
The surprise came in 1931, when Heinz Hopf showed
that certain maps of S^{3}
to S^{2}
could not be deformed continuously to constant maps. Of course, his
example is the Hopf fibration which we have just met. Little by little
it became an extremely important object in mathematics, but also in
physics.
It is the property that each pair of fibers is
interlinked which underlies the fact that it is impossible to deform
the Hopf map f : S^{3}→
S^{2}
to a constant map. It would require a lengthy explanation to give a
convincing justification! See this book for a complete but difficult
exposition or even Hopf’s original article for a proof and many more
details.


What does one know about the maps between S^{n}
and S^{p}
with arbitrary values of n
and p?
Many things are known, but we are far from knowing everything: the
"homotopy classes of the maps between spheres" remain largely a
mystery!
The "Hopf fibration" is only one of the
contributions of Heinz Hopf. He had a profound impact on the
mathematics of the twentieth century.


