Topological Equivalence | Research & Encyclopedia Articles

Topological Equivalence

Two objects are topologically equivalent if one object can be continuously deformed to the other. In one or two dimensions, this is something we can visualize: to continuously deform a surface means to stretch it, bend it, shrink it, expand it, etc.--anything that we can do without actually tearing the surface or gluing parts of it together. Intuitively, we imagine that the surface is made of infinitely flexible rubber, and any shape that we can transform this surface into without tearing or gluing the rubber is topologically equivalent to it.

In two dimensions, there are many examples of surfaces that are topologically equivalent, and also many examples of surfaces that are topologically distinct (i.e., surfaces that cannot be continuously deformed to each other). To begin with, consider an ordinary piece of paper, without the edges (that is, the edges of the paper are not considered to be part of the surface). Now crumple it up. The crumpled paper has ridges and valleys that were not originally there, but it is topologically equivalent to the flat paper. The flat paper (and hence the crumpled paper) is also topologically equivalent to an entire (i.e. infinite) plane, because we can stretch the paper--in theory--as much as we like. It is also equivalent to the interior of a circle, rectangle, hexagon, etc., but not to a washer (i.e. the area between two concentric circles), because we cannot change a circle into a washer without punching a hole in it. A cone, however, is equivalent to a plane, even though it sits in three-dimensional space, because we can flatten the cone down into the plane and get a circle. (To see this, imagine standing an empty ice cream cone on a table, and then pushing each point of the cone down onto the table. Remember, this ice cream cone is made of infinitely stretchable rubber!) On the other hand, if we take a flat piece of paper and roll it up into a cylinder, we obtain a topologically distinct surface, because parts of the paper that were not originally attached to each other (the opposite edges of the paper) are attached now. Thus a cylinder is distinct from a plane. A cylinder is, in fact, topologically equivalent to a washer.

There are other objects, such as a hollow sphere, that are "locally" two-dimensional, because even though the surface itself does not lie in any two-dimensional plane, each point has a neighborhood that is topologically equivalent to a plane. Among such surfaces, the sphere is topologically distinct from the plane (which is, obviously, globally two-dimensional as well as locally), and a torus (that is, the "shell" of a doughnut) is topologically distinct from both the sphere and the plane. We can obtain an infinite collection of mutually topologically distinct surfaces by adding more and more holes, since adding a hole automatically changes the topological "type" of a surface: a two-holed torus (imagine doughnut dough with two holes punched out, instead of one), a three-holed torus (like the shell of a pretzel), a four-holed torus, etc. In fact, one of the basic ways that topology distinguishes among surfaces is by counting their holes. A standard joke about topologists is that they are mathematicians who can't tell the difference between a coffee cup and a doughnut.

Topological equivalence can also be defined in dimensions other than two. In one dimension, the intuition is the same as in two--for example, a circle is topologically distinct from a line segment. In three dimensions or higher, however, we cannot visualize objects other than subsets of our own three-dimensional environment--for instance, what does a three-dimensional sphere look like, and how can we tell what it can or cannot be deformed into? (Remember that the "usual" hollow sphere is two-dimensional, even though it sits in three-dimensional space.) For three or more dimensions, we need a more rigorous notion of topological equivalence.

To come up with a definition, we need to see how to express mathematically the concept of deforming a surface. If surface A is being deformed to surface B, that means that we have a function f that takes each point on A to a point on B, and moreover this map is what is called a homeomorphism: it is continuous (intuitively, this says that points that are near each other on A will be fairly near each other on B--this is the condition that allows for stretching and shrinking, but not tearing) and one-to-one (no two points on A are sent to the same point on B--intuitively, this is the "no gluing" condition), and moreover its inverse map from B to A is also continuous. However, a deformation involves more than just this function f. Imagine that while A is being deformed to B, we freeze the process several times. We then see a collection of surfaces that are "in between" A and B. Suppose there are nine of them. First A is deformed to a surface A₁ that is one-tenth of the way towards B, then A₁ is deformed to a surface A₂ that is two tenths of the way, and so on, until A₉ is deformed to B. So this gives us a sequence of homeomorphisms that are "in between" the identity map on A (i.e. the map that leaves A unchanged--think of this as the zero-th stage of the process) and the map f:

The identity map takes A to A f_1/10 takes A to A₁ f_2/10 takes A to A₂ (in two steps, "via" the surface A₁) f_3/10 takes A to A₃ (in three steps, via A₁ and A₂) ... f_9/10 takes A to A₉ 9 (in nine steps, via A₁ through A₉) f takes A to B (in ten steps, via A₁ through A₉).

Of course, there is nothing special about the number nine. We could freeze the procedure 90 times, 1000 times, 10 million times, etc.; we can freeze the procedure at any specific fraction of the way between the identity map and f and obtain an intermediate surface and an intermediate function. Thus a deformation is actually a class of homeomorphisms f_t that vary continuously with t, where t ranges between 0 and 1, f₀ is the identity map on A and f₁ is the function f from A to B.

The definition we just constructed works just as well in twelve dimensions as in two. Thus the general, n-dimensional definition of topological equivalence is: A is topologically equivalent to B if there exist a family of homeomorphisms f_t varying continuously with t in [0,1] such that f₀ is the identity map on A and f₁ takes A to B.