Background and Motivation for Homology Theory
Homology theory is a subject whose development requires a long chain of definitions, lemmas, and theorems before it arrives at any interesting results or applications. A newcomer to the subject who plunges into a formal, logical presentation of its ideas is likely to be somewhat puzzled because he will probably have difficulty seeing any motivation for the various definitions and theorems. It is the purpose of this chapter to present some explanation, which will help the reader to overcome this difficulty. We offer two different kinds of material for background and motivation. First, there is a summary of some of the most easily understood properties of homology theory, and a hint at how it can be applied to specific problems. Secondly, there is a brief outline of some of the problems and ideas which lead certain mathematicians of the nineteenth century to develop homology theory.
KeywordsHomology Group Homology Class Homotopy Type Free Abelian Group Homology Theory
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