Abstract
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. Second, there is a brief outline of some of the problems and ideas which led certain mathematicians of the nineteenth century to develop homology theory.
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References
History of Algebraic Topology
E. Betti, Sopra gli spazi di un numero qualunque di dimensioni, Ann. Mat. Pura Appl. 4 (1871), 140–158.
M. Bollinger, Geschichtliche Entwicklung des Homologiebegriffs, Arch. Hist. Exact Sci. 9 (1972), 94–166.
J. Dieudonné, Les débuts de Ia topologie algébrique, Expositiones Math. 3 (1985), 347–357.
J. Dieudonné, A History of Algebraic and Differential Topology: 1900–1960. Birkhäuser Boston, Inc., New York, 1989.
S. Lefschetz, The Early Development of Algebraic Topology, Bol. da Soc. Brasileira Mat. 1 (1970), 1–48.
J. C. Maxwell, A Treatise on Electricity and Magnetism. Oxford University Press, Oxford, 1873 (1st ed.); 1881 (2nd ed.); 1904 (3rd ed.).
H. Poincaré, Oeuvres, Gauthier-Villars, Paris, 1953, Vol. VI, pp. 183–499.
Jean-Claude Pont, La Topologie Algébrique, des Origines à Poincaré, Presses Universitaires de France, Paris, 1974.
Jean-Claude Pont, Petit Enfance de la Topologie Algébrique, L’Enseignement Math. 20 (1974), 111–126 (a brief summary of the preceding book).
G. F. B. Riemann, Fragment aus der Analysis Situs, in Gesammelte Mathematische Werke (2nd ed.), Dover, New York, 1953, pp. 479–483.
Motivation and Background for Homology Theory
D. W. Blackett, Elementary Topology, A Combinatorial and Algebraic Approach, Academic Press, New York, 1967, p. 219.
M. Frechet and K. Fan, Initiation to Combinatorial Topology, Prindle, Weber, and Schmidt, Boston, 1967, pp. 113–119.
J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, Mass., 1961, pp. 218–222.
H. Seifert and W. Threlfall, A Textbook of Topology, Academic Press, New York, 1980, Chapter 1.
A. H. Wallace, An Introduction to Algebraic Topology, Pergamon Press, Elmsford, N.Y., 1957. pp. 92–95.
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© 1991 Springer Science+Business Media, LLC
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Massey, W.S. (1991). Background and Motivation for Homology Theory. In: A Basic Course in Algebraic Topology. Graduate Texts in Mathematics, vol 127. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9063-4_6
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DOI: https://doi.org/10.1007/978-1-4939-9063-4_6
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