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Homology

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Abstract

The object of topology is the classification and description of the shape of a space up to topological equivalence. We have in Theorem 4.14 a technique for classifying the surfaces, but this is, as you may have noticed, rather arduous. The euler characteristic can be used to shorten the process, but for some cases a lengthy procedure is still necessary. Neither of these options provide a clear accounting for the ways in which the surfaces vary, e.g., which enclose cavities, which are non-orientable, etc., and neither can be completely generalized to higher-dimensional manifolds. Ideally, one would like some sort of algebraic invariant or computable quantity that would codify a lot of information: how many connected pieces a space has, how the gluing directions work, whether the surface is orientable or not, etc. The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have X = 0. See Figure 6.1.

Keywords

  • Simplicial Complex
  • Homology Group
  • Euler Characteristic
  • Prove Theorem
  • Betti Number

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“Mathematicians are like Frenchmen; whatever you say to them they translate into their own language and forthwith it is something completely different.”

Goethe

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© 1993 Springer Science+Business Media New York

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Kinsey, L.C. (1993). Homology. In: Topology of Surfaces. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0899-0_6

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  • DOI: https://doi.org/10.1007/978-1-4612-0899-0_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6939-7

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