Abstract
The Fundamental Theorem of Surfaces assures us that any connected compact surface is homeomorphic to one of the following closed surfaces: the two-dimensional sphere, a connected sum of tori, or a connected sum of real projective planes. We have seen that the homology groups of such closed surfaces are not isomorphic, and therefore, the surfaces under discussion cannot be homeomorphic. It is possible to arrive at this same result by computing another algebraic invariant of the polyhedra, the so-called fundamental group, which is clearly related to the first homology group. In what follows, we shall study such concepts in detail.
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Intuitively, the loop h is the loop f, but traveled in the opposite direction.
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© 2011 Springer Science+Business Media, LLC
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Ferrario, D.L., Piccinini, R.A. (2011). Homotopy Groups. In: Simplicial Structures in Topology. CMS Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7236-1_6
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DOI: https://doi.org/10.1007/978-1-4419-7236-1_6
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Online ISBN: 978-1-4419-7236-1
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