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invariant? They are more general than motions or isometries. Preserving distance or
curvature of curves or surfaces is not important.
As mentioned in Chapter 5, topology is sometimes called “rubber-sheet geometry”
because a
topologist
does not distinguish between two subsets in
R
n
if, upon pretend-
ing that they are made of rubber, he/she can stretch one into the other without tearing
or puncturing anything in the process. Compare the difference between the sphere and
an ellipsoid and the sphere and the torus. The difference between the former is clearly
minor. The ellipsoid is just a slightly elongated or deformed sphere. On the other hand,
note that if one cuts along any circle on a sphere, then one will divide the sphere into
two parts. The allowable deformations of spaces that we have in mind should preserve
this property. Since cutting the torus along a meridian does not separate it, a torus
cannot be a deformed sphere, and we shall consider the sphere and torus to be topo-
logically distinct. As another example, consider the sphere and the disk. An obvious dif-
ference between the two is that the latter has a boundary whereas the former does not,
but there is also a more subtle difference. Our intuition should tell us that there is no
way to flatten out a sphere without first puncturing it. To put it another way, a sphere
has a “hole” but a disk does not. A torus can also be thought of as having two “holes,”
but its “holes” are quite different from the one in a sphere.
There is one important observation that needs to be made in the context of rubber-
sheet geometry. When we talk about stretching an object, the stretching is not con-
fined to take place in a fixed
R
n
but could take place in any higher-dimensional
R
n+k
for some k. For example, a nontrivial knot in
R
3
like the trefoil knot cannot be
deformed into the standard circle
within R
3
(otherwise it would not be knotted), but
it can be deformed into the circle by a deformation that takes place in
R
4
. The point
is that the space in which an object happens to be imbedded is unimportant. In fact,
it would be better to forget that it is there at all, because one is only interested in the
intrinsic properties of the object itself. In analogy with the concept of Platonic forms
one should think of equivalent spaces as being merely two different representations
of some
single
ideal object. (Of course, what we are talking about now concerns
invariant properties of
objects
. We could also ask about invariants for
maps
. The
classification of knots has to do with invariants of imbeddings of circles in
R
3
. This
is a whole other story that is postponed to later.)
The transformations that capture the spirit of rubber-sheet geometry are homeo-
morphisms and the basic problem that the topologist is trying to solve is determin-
ing when two spaces are homeomorphic. It would be nice if there were a simple
algorithm that would accomplish this. The search for such an algorithm takes us out
of general or point set topology and into the domain of algebraic topology and the
special case of combinatorial topology. In
algebraic topology
the goal is to associate to
each space certain algebraic invariants such that two spaces will be homeomorphic
if and only if they have the same invariants. A good simple example of such an invari-
ant is the Euler characteristic that was discussed earlier. However, note that our point
of view has changed. Instead of starting with a polyhedron with a given Euler char-
acteristic and discovering other polyhedra with the same Euler characteristic, we are
thinking of the Euler characteristic as a number that is the same for all homeomor-
phic spaces. Returning to the general case, if we had methods for computing our alge-
braic invariants, then studying a space would reduce to studying its invariants. In this
way topological or geometric questions would reduce to problems in algebra (hence
the term “algebraic” topology) for which a great deal of theory has already been devel-
