Covering space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Covering_space


00:01:17 1 Formal definition
00:05:12 1.1 Alternative definitions
00:05:53 2 Examples
00:11:24 3 Properties
00:11:34 3.1 Common local properties
00:12:32 3.2 Homeomorphism of the fibers
00:12:37 3.3 Lifting properties
00:13:31 3.4 Equivalence
00:13:45 3.5 Covering of a manifold
00:13:51 4 Universal covers
00:14:01 5 G-coverings
00:14:25 6 Deck transformation group, regular covers
00:15:09 7 Monodromy action
00:16:22 8 More on the group structure
00:16:56 9 Relations with groupoids
00:17:05 10 Relations with classifying spaces and group cohomology
00:17:50 11 Generalizations
00:17:58 12 Applications
00:18:07 13 See also
00:19:26 14 Notes
00:20:31 15 References
00:20:49 q and f(d)
00:24:27 G-coverings
00:25:58 Deck transformation group, regular covers
00:27:42 zn is a regular cover. The deck transformations are multiplications with n-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group Cn.  Likewise, the map exp : C → C× with exp(z)
00:28:15 Monodromy action
00:28:39 x), and γ : [0, 1] → X is a path with γ(0)
00:31:02 x.
00:31:05 More on the group structure
00:31:16 Let p : C → X be a covering map where both X and C are path-connected. Let x ∈ X be a basepoint of X and let c ∈ C be one of its pre-images in C, that is p(c)
00:36:34 Relations with groupoids
00:37:57 Relations with classifying spaces and group cohomology
00:38:15 0 for all n ≥ 2, then the universal covering space T of X is contractible, as follows from applying the Whitehead theorem to T. In this case X is a classifying space or K(G, 1) for G
00:40:47 Generalizations
00:42:42 Applications
00:44:37 See also



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.879800755986452
Voice name: en-US-Wavenet-D


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space C to a topological space X such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below. In this case, C is called a covering space and X the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.
Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. In Riemannian geometry for example, ramification is a generalization of the notion of covering maps. Covering spaces are also deeply intertwined with the study of homotopy groups and, in particular, the fundamental group. An important application comes from the result that, if X is a "sufficiently good" topological space, there is a bijection between the collection of all isomorphism classes of connected coverings of X and the conjugacy classes of subgroups of the fundamental group of X.