On the possible forms of our space
What does the space we live in look like? Such a question appears
quite frequently in various journals and an answer is usually
provided by an architect, an ecologist or a geographer. What can
a mathematician add to this?
For instance, you can ask him about the geometric properties of
the surrounding space. Naturally, what we are interested in is
the space itself, and not the objects it contains. Therefore,
to make things easy, we will simply neglect the latter.
So, imagine you are standing in the middle of a huge and dusky
empty hangar. What geometric features of the space are you liable
to observe? It seems that the only reasonable observation is that
you can freely move around. If you focus your attention on one
particular point, e.g. on the tip
Fig. 1. Let
Now we have at our disposal a nice arithmetical model of the space.
The points correspond to number triples and the distance between
the points
By treating points as number triples we can describe even complicated subsets of our space by equations or inequalities, easily subject them to various kinds of transformations etc.
The above model of the space has one more advantage: it can be
easily carried over to "other dimensions". Indeed, why
not consider Does this really mean that we live in the three-dimensional Cartesian space ? Remember that our description originated in the hangar, i.e., it relies on the observation of a very small fragment of the space only. Thus it cannot be excluded that by hastily generalising the results of our local investigation to the entire space we have committed some important errors. You can easily imagine such errors in the two-dimensional case: a sphere with large radius does indeed look like a fragment of a plane when observed at a very close distance, although it certainly is not one. This is why the notion of a flat Earth once had so many fervent supporters.
What certain knowledge do we have, then? The point
This brings us to the notion of a The answer to this question requires some physical experiments. There is, however, something left for the mathematician. namely, he should provide a list of all possible worlds together with a description of their geometric properties. Only then will the physicist be able to decide which of the proposed models corresponds to the physical reality. Thus our problem consists in creating a complete list of three-dimensional manifolds.
How can we cope with this problem, if it's not even obvious that
there The reason for this difficulty lies in our habit to situate things in our nearest neighbourhood, i.e., in precisely. Therefore we must learn to think of spaces in a different way. For instance, we can imagine that the space has been divided into small polyhedral solids, which fill it completely. Rather then "seeing" the entire space in one glimpse, we may think of it as a collection of solids glued one to another by a pair of faces. Clearly, the gluing should be performed so that any point on the weld has a neighbourhood of the type. The solids can be tetrahedrons. If so, we say that the space has been triangulated. Moise's theorem of 1952 states that there is a triangulation for every manifold of dimension three. If you come down to dimension two, the analogue of our situation would be to use triangular block to construct a surface which resembles in the neighbourhood of each of its points. It is not hard to produce in this way an infinite series of cracknels (see Fig. 2) with an increasing number of holes.
Cracknels are
Fig. 2. All these are cracknels: with no holes (a
sphere), one hole (a torus), two holes and three holes. Let's return to dimension three. How can a compact and connected three-dimensional manifold be glued up of tetrahedrons? Obviously, the essence of the problem is to describe the gluing. This can be done quite easily due to a simple trick.
Assume for a while that we have been given such a manifold with
a triangulation. Then we can associate with it a system of points
and of segments between the points, which will be called the
Fig. 3
It can be easily proved by induction that every finite connected
graph contains a maximal tree, i.e., a subgraph that includes
all the vertices of the graph and has no loops (Fig. 4).
Fig. 4 Now break the manifold up into separate tetrahedral blocks and reassemble them gluing only those faces that correspond to the edges of the dual graph which belong to some fixed maximal tree. The result will be a polyhedron with triangular faces, which can be easily embedded into . Observe that it contains all the blocks of the original manifold; if you want to reconstruct the manifold, you just have to complete the gluing process and that consists in gluing pairs of triangular faces of our new solid. We have thus proved that every compact and connected three-dimensional manifold can be obtained from some polyhedron by an appropriate identification of its faces.
Consider, for example, the rectangular parallelepiped
Fig. 5. An impossible picture: by successively gluing
pairs of opposite faces of the rectangular parallelepiped we get
a three-dimensional torus.
By gluing pairs of opposite faces we get a manifold called the
If we change the polyhedron and the way its faces are identified, new examples of manifolds may arise. We know by now that all compact and connected manifolds can be produced by this procedure. But then, how many of them are there? Suppose two very patient students have been given a polyhedron each and asked to show all possible ways of gluing pairs of faces that yield a manifold. Assume moreover that the task has been completed. How can we be sure that the manifolds thus obtained are indeed different?
The problem of
This observation is, of course, completely trivial. All the art
lies in finding good functions
First of all, let's have a look at compact and connected manifolds
of dimension two. To make things simpler, we restrict our considerations
to two-sided surfaces, i.e. those that do not contain the
By copying the argument of the three-dimensional case you may
check for yourself that every such surface can be derived from
some polygon by appropriately gluing pairs of its sides. Some
not too sophisticated manipulation with the polygon further proves
that every two-sided surface is a cracknel
Let's start with
So, consider a function
with integer values given by the formula .
As for the sphere, it can be proved that the number
does not depend on the triangulation of the surface
We know already that .
Observe that the cracknel - cutting out two disjoint open (i.e. without borders) triangles of the triangulation,
- gluing their borders.
Fig. 6. From
Note also that the cracknel
Let's try to carry our experiments over to the three-dimensional
case. If the triangulation of a manifold
The beginning seems to be promising: it can be proved that also
in this case the number
does not depend on the triangulation of
The theorem has a simple and yet very ingenious proof. Once again
we'll make use of the cracknels. Observe that every one of them,
considered as a subset of ,
is the border of some three-dimensional figure which we will call
a
What is Euler-Poincaré's characteristic of the set
Now we note that each compact connected manifold
Thus we have placed two disjoint full cracknels within
Fig. 7
This decomposition will be of much help in the computation of
the characteristic of
Apply the formula to Heegaard's decomposition ,
where Summing up: Euler-Poincaré's characteristic is useless for distinguishing three-dimensional manifolds. Other, more efficient invariants are needed. Maybe you have some good ideas in this respect?
At the moment of this writing (i.e. December 1996), the classification
problem for three-dimensional manifolds, i.e. the problem of describing
all the possible forms of our space, remained open. |