Basic Algebraic Topology

Abstract

A grand theme in any mathematical discipline is the classification of its objects: When are two such objects “essentially the same”?

In linear algebra, for example, the objects of study are the finite-dimensional vector spaces. One can agree that two such vector spaces are “essentially the same” if they are isomorphic as linear spaces, and one learns in any introduction to the subject that two finite-dimensional vector spaces (over the same field) are isomorphic if and only if their dimensions coincide. For example, R³ and the vector space of all real polynomials of degree at most two are isomorphic because they are both three-dimensional, but R³ and R² are not isomorphic because their respective dimensions are different.

The dimension of a finite-dimensional vector space is what’s called a numerical invariant: a number assigned to each such vector space, which can be used to tell different spaces apart.

It would be nice if a classification of topological spaces could be accomplished with equal simplicity, but this is too much to expect. There is a notion of dimension for topological spaces (we have only encountered zerodimensional spaces in this book; see Exercise 3.4.10), and there are other numerical invariants for (at least certain) topological spaces. In general, however, mere numbers are far too unstructured to classify objects as diverse as topological spaces.

In algebraic topology, one therefore often does not use numbers, but algebraic objects, mostly groups, as invariants. To each topological space, particular groups are assigned in such a way that, if the spaces are “essentially the same”, then so are the associated groups.