Klein bottle as the union of two Mobius bands

In summary, the Mayer-Vietoris sequence states that the homology groups of the Klein bottle can be calculated by decomposing it into two Mobius bands (A and B) which are homotopic equivalent to circles. The union of A and B is also homotopic equivalent to a circle, despite the intersection being the disjoint union of two Mobius bands. This is because the intersection can be thought of as wrapping around twice.
  • #1
math8
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I am trying to calculate the homology groups of the Klein bottle. I want to use the Mayer-Vietoris sequence with the Klein bottle decomposed as the union of two Mobius bands (A and B which are homotopic equivalent to circles), now AUB is the Klein bottle, but I don't understand how according to

http://en.wikipedia.org/wiki/Mayer–Vietoris_sequence#Klein_bottle

AnB is also homotopic equivalent to a circle, I would think that since the intersection is the disjoint union of two Mobius bands, it is h.e. to the disjoin union of 2 circles, hence [tex]H_n(AnB)[/tex] should be [tex]Z \oplus Z[/tex] .

Am I thinking this all wrong?
 
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  • #2
It's been a long time since I looked at this stuff but I think that it has something to do with wrapping around twice.

Mat
 

FAQ: Klein bottle as the union of two Mobius bands

1. What is a Klein bottle?

A Klein bottle is a non-orientable surface that has only one side and can be created by joining two Möbius bands together, essentially making a three-dimensional version of a Möbius strip.

2. How is a Klein bottle different from a regular bottle?

Unlike a regular bottle, a Klein bottle has no distinguishable inside or outside, and it cannot be filled with liquid. It also does not have a top or bottom, and it cannot be turned inside out without breaking it.

3. How are two Möbius bands joined to create a Klein bottle?

To create a Klein bottle, the two Möbius bands are joined together along their edges, with one band twisted 180 degrees before being attached. This creates a continuous surface with no boundary or edges.

4. What are some real-world applications of the Klein bottle?

The Klein bottle is mostly studied and researched for its mathematical properties and implications. However, it has also been used in art and design, such as in creating unique and visually interesting sculptures and objects. It has also been used in topology and geometry to study non-orientable surfaces.

5. Can a Klein bottle be physically created?

While a Klein bottle cannot exist in three-dimensional space as we know it, it can be represented and created in four-dimensional space. Some artists and mathematicians have created physical models of the Klein bottle using materials such as glass, clay, and wire. However, these models still have a boundary or edge, as it is impossible to create a true Klein bottle in our three-dimensional world.

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