WHAT IS ALGEBRAIC TOPOLOGY?
THE BEGINNINGS OF ALGEBRAIC TOPOLOGY
Algebraic topology is a twentieth century field of mathematics that can trace its origins and connections back to the ancient beginnings of mathematics. For example, if you want to determine the number of possible regular solids, you use something called the Euler characteristic which was originally invented to study a problem in graph theory called the Seven Bridges of Konigsberg. Can you cross the seven bridges without retracing your steps? No and the Euler characteristic tells you so. Later, Gauss defined the so-called linking number, a precise invariant which tells you whether two circles are linked. It is called an invariant because it remains the same even if we continuously deform the geometric object. Gauss also found a relationship between the total curvature of a surface and the Euler characteristic. All of these ideas are bound together by the central idea that continuous geometric phenomena can be understood by the use of discrete invariants.
One of the strengths of algebraic topology has always been its wide degree of applicability to other fields. Nowadays that includes fields like physics, differential geometry, algebraic geometry, and number theory. As an example of this applicability, here is a simple topological proof that every nonconstant polynomial p(z) has a complex zero. Consider a circle of radius R and center at the origin of the complex plane. The polynomial transforms this into a closed curve in the complex plane. If this image curve ever passes through the origin, we have our zero. Well, suppose the radius R is very large. Then the highest power of p(z) dominates and hence p(z) transforms the circle into a curve which winds around the origin the same number of times as the degree of p(z). This is called the winding number of the curve around the origin. It is always an integer and it is defined for every closed curve which does not pass through the origin. If we deform the curve, the winding number has to vary continuously but, since it is constrained to be an integer, it cannot change and must be a constant unless the curve is deformed through the origin. Now deform the image curve by shrinking the radius R to zero and suppose that the image curve never passes through the origin, that is to say that we never get the zero of the polynomial. The image curve gets very small, hence must have winding number 0 around the origin unless it is shrinking to the origin. If the image curve is shrinking to the origin, the origin is a zero. If not, the winding number is 0 which means that the polynomial must have degree 0, in other words, it is a constant.
The winding number of a curve illustrates two important principles of algebraic topology. First, it assigns to a geometric odject, the closed curve, a discrete invariant, the winding number which is an integer. Second, when we deform the geometric object, the winding number does not change, hence, it is called an invariant of deformation or, synomynously, an invariant of homotopy.
THE TWENTIETH CENTURY VIEWPOINT
Modern algebraic topology is the study of the global properties of spaces by means of algebra. Poincare' was the first to link the study of spaces to the study of algebra by means of his fundamental group. This is a generalization of the concept of winding number which applies to any space.
To get an idea of what algebraic topology is about, think about the fact that we live on the surface of a sphere but locally this is difficult to distinguish from living on a flat plane. One way of telling that we live on a sphere is to measure the sum of the three angles of a triangle. For a small triangle, it is slightly more than 180 degrees. For a large triangle, it is much more. This tells us that we live on a surface with what is called positive curvature. But, since we can use small triangles, this is a local property, not a global one. It properly belongs to the field known as differential geometry. Algebraic topology is concerned with the whole surface and points to the obvious fact that the surface of a sphere is a finite area with no boundary and the flat plane does not have this property. It expresses this fact by assigning invariant groups to these and other spaces. Usually, these groups are something called homotopy groups or another kind called homology groups. The groups are invariant in the sense that they do not change if the space is continuously deformed. The sphere is assigned an infinite group which is a measure of the fact that the sphere has a hole in it and the plane is assigned the zero group because it does not. The fact that these groups are different tells us that the spaces are fundamentally globally different. No doubt about it. Algebraic topology includes but is not confined to the study of spaces of dimensions only two or three. It includes, for example, the contemplation of the shape of the three dimensional universe itself or even the contemplation of the shape of the four dimensional space-time.
The concept of continuous deformation can be illustrated by the following examples. Consider a coffee cup (with a handle) and a donut. If they are both made of some pliable substance like modeling clay, they can be deformed continuously (without ripping them apart into pieces) one into the other. This means that they have the same homotopy and homology groups, that is, the homotopy groups and the homology groups are invariant. On the other hand, a donut cannot be continuously deformed into a sphere. This means that their homotopy and homology groups can be different and they are.
WHAT ARE HOMOTOPY GROUPS?
N-dimensional spheres, or more generally, n-dimensional manifolds (the torus is an example of a two dimensional manifold which is not a sphere) play a central role. One of the ways in which spheres interact with each other is via continuous functions with domain one sphere of dimension k and range another sphere of dimension n. Such functions are called homotopic if there is a continuous deformation of one into the other. If we regard homotopic functions are being equivalent, we can make them into the algebraic object called a group, more precisely a homotopy group and, in the case we mentioned above, it is called the k-th homotopy group of the n-dimensional sphere. Precise information about these groups have been obtained. One of the most important properties of any group is the number of times a member of it must be added to itself before it becomes trivial (represented by the constant function in the case of homotopy groups). This is called the order of the member of the group. For example, if one takes the homotopy groups of the 3-dimensional sphere, one can find members of these groups with order 4 but not 8, with order 3 but not 9, with order 5 but not 25, etc. If one takes the homotopy groups of the 5-dimensional sphere, one can find members with order 8 but not 16, with order 9 but not 27, with order 25 but not 125, etc. This is true no matter how large the dimension of the domain sphere is. These facts were discovered in the late 1970s by three people who are now or have been at the University of Rochester. It is called the Cohen-Moore-Neisendorfer Theorem.
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