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Miscellany

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Abstract

At this point, it would be a good idea to briefly reread Chapter 2.6, as we now wish to prove higher-dimensional analogs of those theorems. In all of the theorems of that section the key step is the contradiction obtained when a continuous function tries to take a connected space to a non-connected space. This contradiction could be restated in terms of homology groups, since the group H0(K) detects connectivity. Only with the introduction of functions on topological spaces and their action on the homology groups can we get results comparing two different spaces.

Keywords

  • Vector Field
  • Homology Group
  • Euler Characteristic
  • Boundary Sphere
  • Antipodal Point

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“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”

Albert Einstein

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© 1993 Springer Science+Business Media New York

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Kinsey, L.C. (1993). Miscellany. In: Topology of Surfaces. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0899-0_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0899-0_10

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-6939-7

  • Online ISBN: 978-1-4612-0899-0

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