2 - An axiomatic approach to homology theory

Summary

The following note by Eilenberg and Steenrod is the first announcement of their axiomatic approach to homology theory. Of course, this work has influenced later developments very strongly. It should perhaps be pointed out that in 1945, when this note appeared, we had essentially no examples of what are now called ‘generalised homology theories’; nor did we have the techniques, such as spectral sequences, which are now used for dealing with them. The only prerequisite is a minimal acquaintance with homology groups (see §1 of the introduction).

Introduction.—The present paper provides a brief outline of an axiomatic approach to the concept, homology group. It is intended that a full development should appear in book form.

The usual approach to homology theory is by way of the somewhat complicated idea of a complex. In order to arrive at a purely topological concept, the student of the subject is required to wade patiently through a large amount of analytic geometry. Many of the ideas used in the constructions, such as orientation, chain and algebraic boundary, seem artificial. The motivation for their use appears only in retrospect.

Since, in the case of homology groups, the definition by construction is so unwieldy, it is to be expected that an axiomatic approach or definition by properties should result in greater logical simplicity and in a broadened point of view. Naturally enough, the definition by construction is not eliminated by the axiomatic approach. It constitutes an existence proof or proof of consistency.