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Homology and Cohomology Theories: An Axiomatic Approach with Consequences

In: Basic Topology 3

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  • Mahima Ranjan Adhikari

    (Institute for Mathematics, Bioinformatics, Information Technology and Computer Science (IMBIC))

Abstract

As the classical constructions of homology and cohomology groups are available in almost every textbook of algebraic topology, this chapter develops homology and cohomology theories with an emphasis on the axiomatic approach of Eilenberg and Steenrod with various interesting applications for accessible presentation. It starts with a brief discussion on simplicial homology theory with Euler-Poincaré theorem, and then it defines singular homology theory which is a natural generalization of simplicial homology theory. Their dual theories are also discussed. This chapter also gives a presentation of an approach formulating axiomatization of homology and cohomology theories which makes the subject algebraic topology elegant and provides a quick access to further study.

Suggested Citation

  • Mahima Ranjan Adhikari, 2022. "Homology and Cohomology Theories: An Axiomatic Approach with Consequences," Springer Books, in: Basic Topology 3, chapter 0, pages 163-234, Springer.
    • Handle: RePEc:spr:sprchp:978-981-16-6550-9_3
    DOI: 10.1007/978-981-16-6550-9_3
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