Abstract
The main goal of this chapter is to provide an introduction to simplicial and singular homology. We begin by stating the structure theorem for finitely abelian groups. Using this result, we define the Betti numbers. We state the fundamental fact that if two complexes have the same geometric realization, then these complexes and their geometric realization have the same homology groups. We also show how the Euler–Poincaré characteristic of a complex is given by the alternating sum of the Betti numbers of its homology groups. This shows that the Euler–Poincaré characteristic is an invariant of all the complexes corresponding to the same polytope. We determine the homology groups of surfaces with or without borders.
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Notes
- 1.
This means that Im f = Ker g, that f is injective, and that g is surjective.
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Gallier, J., Xu, D. (2013). Homology Groups. In: A Guide to the Classification Theorem for Compact Surfaces. Geometry and Computing, vol 9. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34364-3_5
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DOI: https://doi.org/10.1007/978-3-642-34364-3_5
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