A slice of Swiss cheese has one-dimensional holes;

a block of Swiss cheese has two-dimensional holes.

What is the dimension of a point? From classical geometry we have the definition “A point is that which has no parts” — *also sprach Euclid*. A point has dimension zero, a line has dimension one, a plane has dimension two, and so on.

**Missing the Point**

A point has dimension zero. But what if the point is not there? The real line is one-dimensional. It is a connected manifold: we can join any two points in by a path. This is not true of the punctured line : this manifold falls into two disconnected components, and ; the points and cannot be connected by a path within .

We define the unit -sphere in as . is the usual sphere in 3-space, is the unit circle in 2-space and is the “0-sphere”, a pair of points on the real line. Within the real line the 0-sphere can be continuously distorted, or shrunk, to a single point. The technical term for this shrinkability is homotopy: we say that is homotopic to a point in . Within the *punctured line*, , the 0-sphere cannot be shrunk in this way because of the gap or hole. We say that there is a zero-dimensional hole in .

In the plane , both the 0-sphere and the 1-sphere (a circle) can be continuously distorted, or shrunk, to a single point. In the punctured plane, , this is no longer true. The two points of the 0-sphere can be smoothly moved to a single point, but the 1-sphere cannot be shrunk to a point because it surrounds the missing point. We say that has a one-dimensional hole, even thought only a single point is removed. Obviously, this continues: has a 2-dimensional hole, since the unit 2-sphere cannot be shrunk to a point, whereas circles and point-pairs can.

**A Hole is not there**

We see that the removal of a single point from results in an -dimensional hole. The point may be zero-dimensional in , but the gap that it leaves in is an -dimensional hole. The crux is that, in the punctured space , the hole is “not there!” We have to study the properties of the manifold while remaining within . This is done by defining algebraic stuctures that enable us to identify and quantify holes of various dimensions in the manifold. Let’s start with a simple example.

Holes usually comprise more than a single point. We compare two manifolds, the plane annulus and the spherical shell . is like a sheet of paper with a hole punched in it. A loop within that surrounds the hole cannot be contracted to a point within ; there is a one-dimensonal hole in . By contrast, is more like a Swiss cheese. Loops in can all be shrunk to a point, but a sphere surrounding the hole cannot be so shrunk: it is not homotopic to a point. A thin slice of Swiss cheese has one-dimensional holes; a block of Swiss cheese has two-dimensional holes.

**A Little History**

Bernhard Riemann noticed that holes in a surface could be counted by checking how many cuts could be made while the surface remains in one piece. For a bounded surface like a disk or an annulus, each cut begins and ends on a boundary. Clearly, the disk falls into two pieces once a single cut is made (it has no holes) but the annulus remains connected after a cut from the inner to the outer boundary circle is made.

These ideas were greatly developed by Henri Poincaré in his paper *Analysis Situs* (1895) and five follow-up papers. He introduced the idea of homology, which generalizes Riemann’s ideas to higher dimensions. Poincaré called the number of holes of dimension the -th Betti number, after Italian mathematician Enrico Betti. The zeroth Betti number, , indicates the number of pieces. For a connected manifold, . The first Betti number is the number of one-dimensional holes. For the 2-sphere, ; for the torus, . The second Betti number is the number of cavities in the manifold. For both the 2-sphere and the spherical shell, . In general, gives the number of -dimensional holes in the manifold.Poincaré showed that the Betti numbers are topological invariants. They do not change under distortions of a shape as long as it is not cut or broken. Therefore, shapes that have different sets of Betti numbers cannot be homeomorphic or topologically equivalent. For example, the numbers for a sphere are while for a torus they are . Several other examples are shown in the table above. The invariance greatly helps topologists to categorize topological spaces.