Editing Fundamental group (section)

===The circle===
[[File:Fundamental_group_of_the_circle.svg|Elements of the homotopy group of the circle|thumb]]
The [[circle]] (also known as the 1-sphere)
:<math>S^1 = \left\{(x, y) \in \R^2 \mid x^2 + y^2 = 1\right\}</math>
is not simply connected. Instead, each homotopy class consists of all loops that wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop that winds around ''m'' times and another that winds around ''n'' times is a loop that winds around ''m'' + ''n'' times. Therefore, the fundamental group of the circle is [[group isomorphism|isomorphic]] to <math>(\Z, +),</math> the additive group of [[Integer#Algebraic properties|integers]]. This fact can be used to give proofs of the [[Brouwer fixed point theorem]]<ref>{{harvtxt|May|1999|loc=Ch. 1, §6}}</ref> and the [[Borsuk–Ulam theorem]] in dimension 2.<ref>{{harvtxt|Massey|1991|loc=Ch. V, §9}}</ref>