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A two-dimensional representation of the Klein bottle immersed in three-dimensional space. In mathematics, the Klein bottle (/ ˈ k l aɪ n /) is an example of a non-orientable surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle. [ 1 ] It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder D 2 × I {\displaystyle \scriptstyle D^{2}\times I} to the bottom disk by a reflection across a diameter of the disk.
The Klein bottle can be turned into a Klein surface (compact, without boundary); there is a one-parameter family of inequivalent Klein surfaces structures defined on the Klein bottle. Similarly, there is a one-parameter family of inequivalent Klein surface structures (compact, with boundary) defined on the Möbius strip. [2]
In 1961, Professor Cardwell asked Mitsugi Ohno to construct a true glass Klein bottle, a one-sided figure formally described as “an enclosure continuous with its outer surface constructed by twisting a tube through an opening in the side of the tube and joining it to the other end". Klein bottles had previously been made by other glassblowers ...
Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside.
The Klein bottle (fundamental polygon with appropriate edge identifications) decomposed as two Möbius strips A (in blue) and B (in red). A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the Klein bottle X .
What a Klein bottle is to a closed two-dimensional surface, an Alice universe is to a closed three-dimensional volume. The name is a reference to the main character in Lewis Carroll's children's book Through the Looking-Glass.
As a simple example, let be the circle, and be the inversion , then the mapping torus is the Klein bottle. Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied.