Fundamental group | Wikipedia audio article
Channel: wikipedia tts
Date: 10/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fundamental_group


00:01:36 1 Intuition
00:02:20 2 Definition
00:07:03 3 Examples
00:07:12 3.1 Trivial fundamental group
00:08:00 3.2 Infinite cyclic fundamental group
00:09:30 3.3 Free groups of higher rank
00:10:23 3.4 Knot theory
00:11:00 3.5 Lie groups
00:20:12 4 Functoriality
00:22:57 5 Fibrations
00:26:06 6 Relationship to first homology group
00:27:04 7 Universal covering space
00:28:00 7.1 Examples
00:28:51 7.1.1 Circle
00:28:59 7.1.2 Torus
00:29:12 7.1.3 Real projective spaces
00:29:28 7.1.4 Lie groups
00:29:45 8 Edge-path group of a simplicial complex
00:30:14 9 Realizability
00:30:34 10 Related concepts
00:30:53 10.1 Fundamental groupoid
00:31:01 11 See also
00:32:07 12 Notes
00:33:40 13 References
00:37:20 14 External links
00:38:23 Related concepts
00:39:18 Fundamental groupoid
00:45:25 See also



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SUMMARY
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In the mathematical  field of algebraic topology, the fundamental group is a mathematical group associated to any given pointed topological space that provides a way to determine when two paths, starting and ending at a fixed base point, can be continuously deformed into each other. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a topological invariant: homeomorphic topological spaces have the same fundamental group.
Fundamental groups can be studied using the theory of covering spaces, since a fundamental group coincides with the group of deck transformations of the associated universal covering space. The abelianization of the fundamental group can be identified with the first homology group of the space. When the topological space is homeomorphic to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations.
Henri Poincaré defined the fundamental group in 1895 in his paper "Analysis situs". The concept emerged in the theory of Riemann surfaces, in the work of Bernhard Riemann, Poincaré, and Felix Klein. It describes the monodromy properties of complex-valued functions, as well as providing a complete topological classification of closed surfaces.