invitation à la topologie lacanienne (page 2) :

la lettre de topologie, , quelques figures rudimentaires, et surtout pour ceux qui veulent en savoir plus, des adresses et autres links intéressants... selon le parcours proposé par Lacan, un détour qu'imposent les questions proposées par la clinique.

Espaces index

 
 
 

- attention :  la plupart des schémas ou images ne s'affichent pas actuellement ; en attendant que ce problème technique soit réglé, je laisse néanmoins la trame de ces pages "topologie" qui proposent déjà quelques liens sur d'autres sites de topologie.
 
 
 



 Klein Bottle

The Klein bottle is the non-orientable surface with Euler characteristic equal to 0.

 A Klein bottle can be made from a rectangular piece of the plane by identifying the top and bottom edges using the same orientation, but identifying the left and right edges with opposite orientation (as in the formation of a Möbius band). The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly.

Another way to make the Klein bottle is to take two Möbius bands and join them along their boundaries (each band has a single boundary curve). Finally, the Klein bottle is the connected sum of two real projective planes, since the projective plane minus a disk is just a Möbius band.

 The Klein bottle can not be embedded in three-space, but it can be immersed there.


 

The Klein Bottle in Four-Space

The Klein bottle is a non-orientable surface obtained by identifying the ends of a cylinder with a twist. This representation is constructed from two pieces, one a tube around a figure eight curve and the other a surface of revolution of a piece of that curve. There is a curve of self-intersection in three-dimensional space, but the intersection points have different fourth coordinate so the surface is actually embedded, without self-intersection, in four-space.

To view the surface rotating in three-space, select the picture.



To see how rotation in four-space can produce local self-intersections, select the second picture.



The Klein bottle can be thought of as two Möbius bands identified along their boundaries. To see the Klein bottle separate into these pieces and recombine, select the third picture.
 
 
 
 
 
 

un noeud borroméen (cf. bienvenue en toposie)


 

 

tore
cross-cap

The Möbius Band is the building block for non-orientable surfaces. It is formed by joining the ends of a rectangle with a twist (180 degrees). Any non-orientable surface contains a Möbius Band, and any surface which contains a Möbius Band is necessarily non-orientable. It is not closed--hence it is not a surface, and it contains one boundary curve. The Möbius Band cannot be embedded in the plane, R^2. 

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Description and pictures of the standard image of the Möbius band embedded in R^3.
 

Description and pictures of a Möbius band embedded in R^3 with three twists instead of one. 

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