Editing Klein bottle (section)

==Properties==
Like the [[Möbius strip]], the Klein bottle is a two-dimensional [[manifold]] which is not [[orientability|orientable]]. Unlike the Möbius strip, it is a ''closed'' manifold, meaning it is a [[compact space|compact]] manifold without boundary. While the Möbius strip can be embedded in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>, the Klein bottle cannot. It can be embedded in '''R'''<sup>4</sup>, however.{{sfn|Alling|Greenleaf|1969}}

Continuing this sequence, for example creating a 3-manifold which cannot be embedded in '''R'''<sup>4</sup> but can be in '''R'''<sup>5</sup>, is possible; in this case, connecting two ends of a [[spherinder]] to each other in the same manner as the two ends of a cylinder for a Klein bottle, creates a figure, referred to as a "spherinder Klein bottle", that cannot fully be embedded in '''R'''<sup>4</sup>.<ref>[[Marc ten Bosch]] - https://marctenbosch.com/news/2021/12/4d-toys-version-1-7-klein-bottles/</ref>

The Klein bottle can be seen as a [[fiber bundle]] over the [[circle]] ''S''<sup>1</sup>, with fibre ''S''<sup>1</sup>, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be ''E'', the total space, while the base space ''B'' is given by the unit interval in ''y'', modulo ''1~0''. The projection π:''E''→''B'' is then given by {{nowrap|π([''x'', ''y'']) {{=}} [''y'']}}.

The Klein bottle can be constructed (in a four dimensional space, because in three dimensional space it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips, as described in the following [[limerick (poetry)|limerick]] by [[Leo Moser]]:<ref name="Darling2004">{{cite book|author=David Darling|title=The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes|url=https://books.google.com/books?id=nnpChqstvg0C&q=get+a+weird+bottle+like+mine&pg=PA176|date=11 August 2004|publisher=John Wiley & Sons|isbn=978-0-471-27047-8|page=176}}</ref>

{{poemquote|text=A mathematician named [[Felix Klein|Klein]]
Thought the Möbius band was divine.
     Said he: "If you glue
     The edges of two,
You'll get a weird bottle like mine."}}

The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle can be given a [[CW complex]] structure with one 0-cell ''P'', two 1-cells ''C''<sub>1</sub>, ''C''<sub>2</sub> and one 2-cell ''D''. Its [[Euler characteristic]] is therefore {{nowrap|1 − 2 + 1 {{=}} 0}}. The boundary homomorphism is given by {{nowrap|&part;''D'' {{=}} 2''C''<sub>1</sub>}} and {{nowrap|&part;''C''<sub>1</sub> {{=}} &part;''C''<sub>2</sub> {{=}} 0}}, yielding the [[cellular homology|homology groups]] of the Klein bottle ''K'' to be {{nowrap|H<sub>0</sub>(''K'', '''Z''') {{=}} '''Z'''}}, {{nowrap|H<sub>1</sub>(''K'', '''Z''') {{=}} '''Z'''×('''Z'''/2'''Z''')}} and {{nowrap|H<sub>''n''</sub>(''K'', '''Z''') {{=}} 0}} for {{nowrap|''n'' > 1}}.

There is a 2-1 [[covering map]] from the [[torus]] to the Klein bottle, because two copies of the [[fundamental region]] of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. The [[universal cover]] of both the torus and the Klein bottle is the plane '''R'''<sup>2</sup>.

The [[fundamental group]] of the Klein bottle can be determined as the [[Deck transformation#Deck transformation group, regular covers|group of deck transformations]] of the universal cover and has the [[presentation of a group|presentation]] {{nowrap|{{angbr|1=''a'', ''b'' {{!}} ''ab'' = ''b''<sup>&minus;1</sup>''a''}}}}. It follows that it is isomorphic to <math>\mathbb{Z} \rtimes \mathbb{Z}</math>, the only nontrivial semidirect product of the additive group of integers <math>\mathbb{Z}</math> with itself.

[[File:Klein bottle colouring.svg|thumb|upright|A 6-colored Klein bottle, the only exception to the Heawood conjecture]]
Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to the [[Heawood conjecture]], a generalization of the [[four color theorem]], which would require seven.

A Klein bottle is homeomorphic to the [[connected sum]] of two [[projective plane]]s.<ref>{{Cite book |last=Shick |first=Paul |title=Topology: Point-Set and Geometric |publisher=Wiley-Interscience |year=2007 |isbn=9780470096055 |pages=191–192}}</ref> It is also homeomorphic to a sphere plus two [[cross-cap]]s.

When embedded in Euclidean space, the Klein bottle is one-sided. However, there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.<ref name=":0">{{Cite book | publisher = CRC Press | isbn = 978-1138061217 | last = Weeks | first = Jeffrey | title = The Shape of Space, 3rd Edn. | year = 2020 | url = https://www.crcpress.com/The-Shape-of-Space/Weeks/p/book/9781138061217 }}</ref>