Abstract
We have been studying groups in the past three chapters in order to lay the groundwork for introducing the fundamental groupĀ of a topological space S. This is a homeomorphism invariant that is associated to a topological space. Rather than being a number like the Euler characteristic \(\chi (S)\) or a boolean invariant like orientability, the fundamental groupĀ associates a group to S, denoted \(\pi _1(S)\). Furthermore if S is homeomorphic to \(S'\), then the fundamental groups \(\pi _1(S)\) and \(\pi _1(S')\) are isomorphic in the group-theoretic sense. In this chapter, we will build up a set of ideas for defining the fundamental group. For visualization purposes, we will phrase these ideas as if S were a surface; but everything that follows holds mostly unchanged for any topological space.
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Bray, C., Butscher, A., Rubinstein-Salzedo, S. (2021). The Fundamental Group. In: Algebraic Topology. Springer, Cham. https://doi.org/10.1007/978-3-030-70608-1_8
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DOI: https://doi.org/10.1007/978-3-030-70608-1_8
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