Coating (mathematics)

In Mathematical, and more particularly in Topology, a coating of a topological space X by a topological space C is a continuous application and surjective p : C → X such that any point $x \ in X$ admits a open Voisinage U such as the reciprocal image of U by p is a union disjoined of open of C , each one homeomorphic with U by p .

It is about a particular case of Fibration, with discrete fiber.

Examples

Coating of the circle by a propeller

That is to say

S^1

the circle in the plan

\ mathbb {R} ^2= \ mathbb {C}

. The real line

\ mathbb {R}

is then a coating defined by the application:

$p: \ mathbb {R} \ to S^1, \ quad p (T) = (\ cos (2 \ pi T), \ sin (2 \ pi T))=e^ {2i \ pi T}$ . 2 \ pi wiederholen. -->

Each fiber is infinite countable here ( $p^ {- 1} (p (X))=x+ \ mathbb {Z}$ ).

Construction spreads with the exponential coating of the torus: $\ mathbb {R} ^n \ to \ mathbb {T} ^n= \ mathbb {S} ^1 \ times \ mathbb {S} ^1 \ times \ ldots \ times \ mathbb {S} ^1 \ subset \ mathbb {R} ^ {2n}$

The fiber is countable: ( $p^ {- 1} (0) = \ mathbb {Z} ^n$ ).

the functions power

The

application of the private complex plan of the origin

\ mathbb {C} ^*

p: \ mathbb {C} ^* \ to \ mathbb {C} ^*, \ quad p (Z) =z^n

defines a coating.

Each fiber is here finished and has

n

elements.

the exponential application

The

application of the plan complexes

\ mathbb {C}

p: \ mathbb {C} \ to \ mathbb {C} ^*, \ quad p (Z) =e^z

defines a coating.

Each fiber is infinite countable here (

p^ {- 1} (p (X))=x+2i \ pi \ mathbb {Z}

the band of Möbius

Coating of projective space

The canonical application

\ mathbb {S} ^n \ to \ mathbb {RP} ^n

is a coating of the projective Espace (real), the fiber has two elements.

Constructions of coatings and terminology

Spaces with the top of B

A space with the top of a topological Espace B is a space X provided with an application continues

\ pi: X \ mapsto B

called projection . B is called the bases . For any point

B \ in B

, one calls fiber X with the top of the point B and one notes X (b) under space (closed)

\ pi^ {- 1} (b) \ subset X

. One calls section (continues) X an application continues

\ sigma: B \ to X

such as

\ pi \ circ \ sigma = Id_B

Theorem : Is p a local homeomorphism of which all the fibers are finished of the same cardinal N, then p is a finished coating.

Product fiber, direct Sum, basic Change

August 1st

Discrete groups operating properly and freely

Either G a discrete group operating properly and freely on a space locally compact E, projection

E \ to E/G

defines a fiber G coating.

In particular, if $\ Gamma$ is a discrete Sous-groupe topological group G, projection $G \ to G \ Gamma$ is a fiber coating $\ Gamma$ .

Construction of coatings per sticking together

August 1st

Theory of the coatings

Morphisms and transformations of coatings

A morphism of spaces to the top of B is an application

X \ to X'

which Commute with projections

\ pi

and

\ pi'

Monodromy of the laces and raising of the applications

The fundamental Groupe $\ pi_1 (X, X)$ operates by a action of group on the right on the fiber $\ pi^ {- 1} (X)$ .

Coatings galoisiens and groups of Welshman of a coating

A coating is known as galoisien (or regular or normal ) if it is related by arcs and the group of the automorphisms acts transitively on fiber of a each point.

Universal coatings

A universal coating of a space B is a coating galoisien E such as any coating is isomorphous with a coating associated with E (not necessarily related). I.e. for any coating D of B, there exists a morphism of E on D. Two universal coatings are isomorphous.

Theorem : a coating simply related E is a universal coating.

Theorem : a space (related by arcs) B admits a simply related coating if and only if it is semi-locally simply related.

Classification of the coatings and Welshman theory

August 1st

Applications

Free graphs and groups

Theorem : All Sous-groupe of a free Groupe is a free group.

theorem of Van Kampen

Ramified coatings and surfaces of Riemann

August 1st

Coatings of the topological groups

August 1st

Bibliography (in French)

Dolbeault : Analyze complexes
Jean Dieudonné: Elements of Analysis , volume 3

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