Introduction

There are three different ways in which mathematicians have reached the subject now called algebraic topology:

1. Classification, properties and invariants. One looks for ways of distinguishing topological spaces.
2. Obstructions. What are the difficulties in getting certain types of maps between spaces.
3. Categorical. What is the structure of the category of topological spaces or some subcategories of it?

As one realises during the study of algebraic topology all these are different ways of stating the same question--and the answer is to apply the methods of algebraic topology.

Classification. The method of “divide and conquer” can be applied to break the study of “all” topological spaces into a number of different classes. For example we can see that the real line is not homeomorphic to -{0} because the former is connected while the latter is not. Similarly, the real line is not homeomorphic to the circle S¹ = {(x,y)|x² + y¹ = 1}, which is a subspace of ², because the latter is compact but the former is not. One way of stating the goal of algebraic topology is to say that we want to find enough such properties to classify all spaces upto homeomorphism. However, this proves too hard and we must restrict ourselves to some class of “reasonable” spaces (as shown by Steenrod one such class is the collection of “compactly-generated spaces”); further, we must weaken the equivalence of spaces to “homotopy equivalence” instead of homeomorphism. Even this problem is too hard based on “computable” invariants and so the problem is usually reduced to the study of “weak homotopy equivalence” of these spaces or the study of a still smaller class of spaces called CW-complexes.

Obstructions. A different way of approaching the problem is through the study of continuous maps. Instead of asking whether is homeomorphic to S¹ we can ask whether the map t(cos(t),sin(t)) has an inverse. Such questions arise naturally in other contexts also. For example, to each line in the plane we can associate an angle. Can we find a parametric family of lines which is (continuously) parametrised by the angle of the line? Can we find such a parametric family that is at unit distance from the origin? As another example consider equations of the form x³ + ax + b = 0 with a and b real; we know that such an equation always has a solution. Can we find a (continuous) function f(a,b) that gives a solution? The study of such problems is called obstruction theory. It too leads to algebraic topology in a natural way. For example, we can show that there is no surjective contiuous map from to -{0} because the latter is disconnected. Similarly, there is no surjective continuous map from S¹ to since the former is compact. Thus the invariants of algebraic topology can be used to solve problems in obstruction theory as well. In fact, and perhaps surprisingly, the problem of classification can be reduced to problems in obstruction theory.

Category Theory. The relation between classification and obstruction can be better understood if we look at the totality (carefully avoiding Russellian paradoxes) of spaces and maps. The algebraic mechanism for this is called a category. A category is defined by its objects and morphisms. Morphisms can be composed and composition is associative. For each object we have an identity morphism from the object to itself which acts as a both sided identity for composition. As an important example, we can take the category of topological spaces; its objects are topological spaces and morphisms are continuous maps between topological spaces. The problem of classification can thus be re-stated as the study of this category; can one find an algebraic description of this category that does not refer to the underlying notions of topological spaces and continuous maps. As an example of such a description consider the category of all vector spaces over a field (say ), with morphisms given by linear maps. The theorem of basis for vector spaces and the matrix representation of morphisms says that this category can be “represented” as the category whose objects are non-negative integers and morphisms between n and m are n × m matrices.

Different approaches. There are essentially three approaches to the study of algebraic topology corresponding to the three categories that arise there. First of all there is the Steenrod category of compactly generated spaces. Next there is Whitehead’s category of CW-complexes and finally there is Kan’s category of simplicial complexes; this list in increasing order of “algebraic”-ness. Ultimately, the result is going to say that Kan’s category will capture all the computable algebraic topological invariants of the other two categories. In the interest of “concrete”-ness it seems (to me!) to be better to start with this category as it restricts the types of spaces that we will study and the kinds of maps allowed between them. Thus the study will be much simpler than the study of “all” topological spaces and “all” continuous maps; as we shall see we can (essentially) construct these spaces out of building blocks as also construct maps out of simple incidence and collapsing maps. Of course, the problem of studying more general spaces and more general maps will remain. This will be resolved by proving triangulability and the simplicial approximation theorem. A final word of advice. Since are taking a very algebraic approach it is possible that what we will do will look “abstract”. The solution is to think geometrically and write algebraically. It is easier to verify (line by line) a well-written algebraic computation but it is easier to contruct/re-construct one if we can visualise the geometric meaning of each step.

Assignment.

1. Classify the letters of any alphabet (of your choice) upto homeomorphism. Can you prove the above classification? Between which classes are their injective or surjective continuous maps?
2. Give a topological classification of plane conics. Prove this classification.
3. Let S³ be the 3-sphere (unit vectors in 4-space). Show that it is possible to choose a continuous map f : S³ S³ such that f(v) is orthogonal to v. Is this possible for S²? What about S¹?
4. Let P be the category whose objects are polynomials f(x) over a field k. A morphism h : f g is a polynomial h(x) such that g(h(x)) = f(x). Relate this category to some subcategory of the category of rings.