What is...a covering space?
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- čas přidán 27. 07. 2024
- Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What is...a covering space? Or: A topological Galois correspondence.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
www.dtubbenhauer.com/youtube.html
Website with exercises.
www.dtubbenhauer.com/lecture-a...
Picture on the second slide which I forgot to come back to, stupid me.
A 2-sheet covering of the Klein bottle by a torus.
Material used.
Hatcher, Chapter 1
en.wikipedia.org/wiki/Coverin...
Torus and Klein bottle.
en.wikipedia.org/wiki/Torus
en.wikipedia.org/wiki/Klein_b...
math.stackexchange.com/questi...
Helix.
en.wikipedia.org/wiki/Helix
Finite fundamental groups.
math.stackexchange.com/questi...
Galois correspondence.
en.wikipedia.org/wiki/Fundame...
Pictures used.
en.wikipedia.org/wiki/Torus#/...
en.wikipedia.org/wiki/Klein_b...
en.wikipedia.org/wiki/Klein_b...
en.wikipedia.org/wiki/Coverin...
Awesome fundamental group animations (and other great stuff).
math.unice.fr/~indira/Mygifs....
Hatcher’s book (I sometimes steal some pictures from there).
pi.math.cornell.edu/~hatcher/...
Always useful.
en.wikipedia.org/wiki/Counter...
#algebraictopology
#topology
#mathematics
I don't think I've ever got so many answer from one video. Thank you.
Thank you for the kind feedback, you are very welcome!
great videos,long live visual math!
Thanks! But what about more generally "Long live math" ;-)
From Grothendieck’s Galois theory the Galois coverings of Riemann surfaces (ie Riemann sphere CP^1) correspond to function fields (with transcendental degree 1). The Riemann sphere is a Kähler manifold. Kähler manifolds include submanifolds of complex projective space CP^n which are nonsingular projective varieties because manifolds are nonsingular. Nonsingular projective varieties are motivic. 0-dimensional motives (aka Artin motives) correspond to the Galois theory of number fields. I wonder if the Galois correspondence for number fields is the 0 degree transcendental field extension of a base field F on a 0-dimensional smooth projective variety over F or an equivalent Artin motive over F? I wonder if this generalizes to a Galois correspondence for every n-dimensional motive over a field F corresponding to n degree transcendental field extension of F equivalent to it by a contravariant functor (eg reverses arrows).
Ah, an algebraic geometer ☺
I have no idea, sorry, but I feel the answer must be known since this sounds like a classical question.
@@VisualMath Yes, but there is also some complex geometry by Kodaira-Spencer. For example, Teichmüller spaces of Riemann surfaces have a complex structure.
Maybe it’s known. I forgot to mention the duality of lines (1D) and points (0D) in projective space. Assuming my previous conjecture holds, function fields and number fields being in some sense “duals” of each other would help explain the many analogous objects between them.
In "Complex Geometry - An Introduction” (2004) by Huybrechts at Proposition 2.1.9 a theorem of Siegel defines the algebraic dimension of X is defined to be the transcendental degree of the function field K(X), the field of meromorphic functions on X. It also states that the algebraic dimension of a compact connected manifold is not greater than the complex dimension.
To be more precise this should be between complex pure motives over a field K with their complex dimensions. Note function fields by definition are finitely generated fields with n transcendental degree over a base field K for n >= 1. Pure motives over a field K are (functorial with) complex nonsingular projective varieties when the field K embeds into the complex numbers C. Kähler manifolds are complex nonsingular projective varieties. Moieshzen manifolds are compact complex manifolds with a field of meromorphic functions having their transcendental degree equal to its own complex dimension. Moieshzen manifolds have a Kähler metric and are Kähler manifolds when they are projective spaces. Thus, the field of meromorphic functions C(M) that correspond to the complex dimension of a complex pure motive over a field K as a complex nonsingular projective variety is equal to the transcendental degree of the function field on M. In complex degree 1 this is function fields of Riemann surfaces and in degree 0 this applies to number fields since they are all algebraic extensions.
The Galois correspondence holds in all complex dimensions of this setup because it can be independently derived as a contravariant functor between commutative rings (ie fields) and (ie Galois) groups using a formal group functor.
Excellent, I do not know the text of Huybrechts, but I will have a look.
@@VisualMath Thank you 👍. This is similar to Serre’s GAGA principle and its related theorems used to show that holomorphic functions on a projective varieties are algebraic.
There is a line in Hatcher's, around page 56, 2nd paragraph.
"We allow p^{-1}(U) to be empty, so the covering map p need not be surjective"
I haven't seen a non surjective covering map in his book tho
Yes, that is a good one, thanks!
But in some sense it is even "worse": I can't see a clear advantage of not requiring surjectivity.
See also math.stackexchange.com/questions/1842577
@VisualMath I more see it as, by definition the covering map locally has constant cardinality, thus it is surjective on any connected component.
And if you want a surjective covering space then just union all covering spaces for each connected component.
So it gives you more control, I reckon?
@@juiteko8375 You have a point, and some authors would even require surjectivity. Hatcher goes for the mildly more general definition.
I personally feel its a matter of taste.