The Dimension of a Point that isn’t there

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 {\mathbb{R}} is one-dimensional. It is a connected manifold: we can join any two points in {\mathbb{R}} by a path. This is not true of the punctured line {\mathbb{R}\setminus \{0\}}: this manifold falls into two disconnected components, {\{x\in\mathbb{R}:x<0\}} and {\{x\in\mathbb{R}:x>0\}}; the points {-1} and {+1} cannot be connected by a path within {\mathbb{R}\setminus \{0\}}.

We define the unit {n}-sphere in {\mathbb{R}^{n+1}} as {\mathbb{S}^n = \{x  :|x|=1\}}. {\mathbb{S}^2} is the usual sphere in 3-space, {\mathbb{S}^1} is the unit circle in 2-space and {\mathbb{S}^0 = \{-1,+1\}} is the “0-sphere”, a pair of points on the real line. Within the real line {\mathbb{R}} the 0-sphere can be continuously distorted, or shrunk, to a single point. The technical term for this shrinkability is homotopy: we say that {\mathbb{S}^0  } is homotopic to a point in {\mathbb{R}}. Within the punctured line, {\mathbb{R}\setminus\{0\}}, 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 {\mathbb{R}\setminus\{0\}}.

In the plane {\mathbb{R}^2}, both the 0-sphere {\{-1,+1\}} and the 1-sphere (a circle) can be continuously distorted, or shrunk, to a single point. In the punctured plane, {\mathbb{R}^2\setminus\{0\}}, 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 {\mathbb{R}^2\setminus\{0\}} has a one-dimensional hole, even thought only a single point is removed. Obviously, this continues: {\mathbb{R}^3\setminus\{0\}} 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 {\mathbb{R}^n} results in an {(n-1)}-dimensional hole. The point may be zero-dimensional in {\mathbb{R}^n}, but the gap that it leaves in {\mathbb{R}^n\setminus\{0\}} is an {(n-1)}-dimensional hole. The crux is that, in the punctured space {\mathrm{X}=\mathbb{R}^n\setminus\{0\}}, the hole is “not there!” We have to study the properties of the manifold {\mathrm{X}} while remaining within {\mathrm{X}}. 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 {\mathbb{A}^2 = \{x\in\mathbb{R}^2: a \le |x| \le b \}} and the spherical shell {\mathbb{A}^3 = \{x\in\mathbb{R}^3:  a \le |x| \le b \}}. {\mathbb{A}^2} is like a sheet of paper with a hole punched in it. A loop within {\mathbb{A}^2} that surrounds the hole cannot be contracted to a point within {\mathbb{A}^2}; there is a one-dimensonal hole in {\mathbb{A}^2}. By contrast, {\mathbb{A}^3} is more like a Swiss cheese. Loops in {\mathbb{A}^3} 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

Enrico Betti (1823–1892).

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 {\mathbb{A}^2} 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 {p} the {p}-th Betti number, after Italian mathematician Enrico Betti. The zeroth Betti number, {b_0}, indicates the number of pieces. For a connected manifold, {b_0 = 1}. The first Betti number is the number of one-dimensional holes. For the 2-sphere, {b_1 = 0}; for the torus, {b_1 = 2}. The second Betti number is the number of cavities in the manifold. For both the 2-sphere and the spherical shell, {b_2 = 1}. In general, {b_p} gives the number of {p}-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 {(b_0, b_1, b_2)} for a sphere are {(1, 0, 1)} while for a torus they are {(1, 2, 0)}. Several other examples are shown in the table above. The invariance greatly helps topologists to categorize topological spaces.

 


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