Abstract
In this chapter, we discuss simplicial homology theory. We explain, with examples, the main motivation behind various homology theories. Simplicial Homology involves the ideas of simplicial complexes and triangulations for topological spaces, which are introduced at appropriate places. We discuss concrete examples of triangulations for surfaces. Further the chain complex, the boundary operator and the simplicial homology groups are defined. We compute simplicial homology for several examples explicitly. At the end of the chapter, we see the usefulness of homology groups in distinguishing topological spaces.
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References
[1] M. Greenberg and J. Harper, Algebraic Topology: A First Course(Mathematics Lecture Notes Series), (Westview Pr (Short Disc); Revised edition, 1982),
[2] A. Hatcher, Algebraic Topology, (Cambridge University Press, Cambridge, 2002).
[3] J. R. Munkres, Elements of algebraic topology, (Addison-Wesley Publishing Company, Menlo Park, CA, 1984).
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Kulkarni, D. (2017). Homology. In: Bhattacharjee, S., Mj, M., Bandyopadhyay, A. (eds) Topology and Condensed Matter Physics. Texts and Readings in Physical Sciences, vol 19. Springer, Singapore. https://doi.org/10.1007/978-981-10-6841-6_4
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DOI: https://doi.org/10.1007/978-981-10-6841-6_4
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