Editing Torus (section)

== Topology ==
{{No footnotes|section|date=November 2015}}
[[Topology|Topologically]], a torus is a [[closed surface]] defined as the [[product topology|product]] of two [[circle]]s: {{math|''S''<sup>1</sup> × ''S''<sup>1</sup>}}. This can be viewed as lying in [[complex coordinate space|{{math|'''C'''<sup>2</sup>}}]] and is a subset of the [[3-sphere]] {{math|''S''<sup>3</sup>}} of radius {{math|√2}}. This topological torus is also often called the [[Clifford torus]].<ref>{{Cite journal |last=De Graef |first=Marc |date=March 7, 2024 |title=Applications of the Clifford torus to material textures |url=https://journals.iucr.org/j/issues/2024/03/00/iu5046/iu5046.pdf |journal=Journal of Applied Crystallography |volume=57 |issue=3 |pages=638–648|doi=10.1107/S160057672400219X |pmid=38846769 |pmc=11151663 |bibcode=2024JApCr..57..638D }}</ref> In fact, {{math|''S''<sup>3</sup>}} is [[foliation|filled out]] by a family of nested tori in this manner (with two degenerate circles), a fact that is important in the study of {{math|''S''<sup>3</sup>}} as a [[fiber bundle]] over {{math|''S''<sup>2</sup>}} (the [[Hopf bundle]]).

The surface described above, given the [[relative topology]] from [[real coordinate space|{{math|'''R'''<sup>3</sup>}}]], is [[homeomorphic]] to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by [[stereographic projection|stereographically projecting]] the topological torus into  {{math|'''R'''<sup>3</sup>}} from the north pole of {{math|''S''<sup>3</sup>}}.

The torus can also be described as a [[quotient space (topology)|quotient]] of the [[Cartesian plane]] under the identifications
: <math>(x,y) \sim (x+1,y) \sim (x,y+1), \,</math>
or, equivalently, as the quotient of the [[unit square]] by pasting the opposite edges together, described as a [[fundamental polygon]] {{math|''ABA''<sup>−1</sup>''B''<sup>−1</sup>}}.

[[File:Inside-out torus (animated, small).gif|thumb|Turning a punctured torus inside-out]]
The [[fundamental group]] of the torus is just the [[direct product of groups|direct product]] of the fundamental group of the circle with itself:
: <math>\pi_1(T^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathrm{Z} \times \mathrm{Z}.</math><ref>Padgett, Adele (2014). "Fundamental groups: motivation, computation methods, and applications", REA Program, Uchicago. https://math.uchicago.edu/~may/REU2014/REUPapers/Padgett.pdf</ref>

Intuitively speaking, this means that a [[loop (topology)|closed path]] that circles the torus's "hole" (say, a circle that traces out a particular latitude) and then circles the torus's "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

The fundamental group can also be derived from taking the torus as the quotient <math>T^2\cong \mathbb{R}^2/\mathbb{Z}^2</math> (see below), so that <math>\mathbb{R}^2</math> may be taken as its [[universal cover]], with [[deck transformation]] group <math>\mathbb{Z}^2=\pi_1(T^2)</math>.  

Its higher [[Homotopy group|homotopy groups]] are all trivial, since a universal cover projection <math>p:\widetilde{X}\rightarrow X</math> always induces isomorphisms between the groups <math>\pi_n(\widetilde{X})</math> and <math>\pi_n(X)</math> for <math>n>1</math>, and <math>\mathbb{R}^2</math> is [[Contractible space|contractible]].

The torus has [[homology groups]]

<math>H_n(T^2)=\begin{cases}\mathbb{Z},& n=0,2\\

\mathbb{Z}\oplus \mathbb{Z},& n=1\\

0&\text{else.}\end{cases}</math>

Thus, the first homology group of the torus is [[isomorphism|isomorphic]] to its fundamental group-- which in particular can be deduced from [[Hurewicz theorem]] since <math>\pi_1(T^2)</math> is [[abelian group|abelian]].

The cohomology groups with integer coefficients are isomorphic to the homology ones-- which can be seen either by direct computation, [[Universal coefficient theorem|the universal coefficient theorem]] or even [[Poincaré duality]].

If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of a rectangle together, choosing the other two sides instead will cause the same reversal of orientation.