Editing Fundamental group (section)

==Definition==

[[File:Double_torus_illustration.png|thumb]]
Throughout this article, ''X'' is a topological space. A typical example is a surface such as the one depicted at the right. Moreover, <math>x_0</math> is a point in ''X'' called the ''base-point''. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on ''X'' can be deformed into each other. The precise definition depends on the notion of the homotopy of loops, which is explained first.

===Homotopy of loops===

Given a topological space ''X'', a ''[[Loop (topology)|loop]] based at <math>x_0</math>'' is defined to be a [[continuous function (topology)|continuous function]] (also known as a continuous map)
:<math>\gamma \colon [0, 1] \to X</math>
such that the starting point <math>\gamma(0)</math> and the end point <math>\gamma(1)</math> are both equal to <math>x_0</math>.

[[File:Homotopy_of_pointed_circle_maps.png|Homotopy of loops|thumb]]
A ''[[homotopy]]'' is a continuous interpolation between two loops. More precisely, a homotopy between two loops <math>\gamma, \gamma' \colon [0, 1] \to X</math> (based at the same point <math>x_0</math>) is a continuous map
:<math>h \colon [0, 1] \times [0, 1] \to X,</math>
such that
* <math>h(0, t) = x_0</math> for all <math>t \in [0, 1],</math> that is, the starting point of the homotopy is <math>x_0</math> for all ''t'' (which is often thought of as a time parameter). 
* <math>h(1, t) = x_0</math> for all <math>t \in [0, 1],</math> that is, similarly the end point stays at <math>x_0</math> for all ''t''.
* <math>h(r, 0) = \gamma(r),\, h(r, 1) = \gamma'(r)</math> for all <math>r \in [0, 1]</math>.

If such a homotopy ''h'' exists, <math>\gamma</math> and <math>\gamma'</math> are said to be ''homotopic''. The relation "<math>\gamma</math> is homotopic to <math>\gamma'</math>" is an [[equivalence relation]] so that the set of equivalence classes can be considered:
:<math>\pi_1(X, x_0) := \{ \text{all loops }\gamma \text{ based at }x_0 \} / \text{homotopy}</math>.
This set (with the group structure described below) is called the ''fundamental group'' of the topological space ''X'' at the base point <math>x_0</math>. The purpose of considering the equivalence classes of loops [[up to]] homotopy, as opposed to the set of all loops (the so-called [[loop space]] of ''X'') is that the latter, while being useful for various purposes, is a rather big and unwieldy object. By contrast the above [[quotient set|quotient]] is, in many cases, more  manageable and computable.

===Group structure===
[[File:Homotopy_group_addition.svg|Addition of loops|thumb]]
By the above definition, <math>\pi_1(X, x_0)</math> is just a set. It becomes a [[group (mathematics)|group]] (and therefore deserves the name fundamental ''group'') using the concatenation of loops. More precisely, given two loops <math>\gamma_0, \gamma_1</math>, their product is defined as the loop
:<math>\gamma_0 \cdot \gamma_1 \colon [0, 1] \to X</math>
:<math>(\gamma_0 \cdot \gamma_1)(t) = \begin{cases}
        \gamma_0(2t) & 0 \leq t \leq \tfrac{1}{2} \\
    \gamma_1(2t - 1) & \tfrac{1}{2} \leq t \leq 1.
  \end{cases}</math>
Thus the loop <math>\gamma_0 \cdot \gamma_1</math> first follows the loop <math>\gamma_0</math> with "twice the speed" and then follows <math>\gamma_1</math> with "twice the speed".

The product of two homotopy classes of loops <math>[\gamma_0]</math> and <math>[\gamma_1]</math> is then defined as <math>[\gamma_0 \cdot \gamma_1]</math>. It can be shown that this product does not depend on the choice of representatives and therefore gives a [[Equivalence relation#Well-definedness under an equivalence relation|well-defined]] operation on the set <math>\pi_1(X, x_0)</math>. This operation turns <math>\pi_1(X, x_0)</math> into a group. Its [[neutral element]] is the constant loop, which stays at <math>x_0</math> for all times ''t''. The [[inverse element|inverse]] of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction. More formally,
:<math>\gamma^{-1}(t) := \gamma(1-t)</math>.
Given three based loops <math>\gamma_0, \gamma_1, \gamma_2,</math> the product
:<math>(\gamma_0 \cdot \gamma_1) \cdot \gamma_2</math> 
is the concatenation of these loops, traversing <math>\gamma_0</math> and then <math>\gamma_1</math> with quadruple speed, and then <math>\gamma_2</math> with double speed. By comparison,
:<math>\gamma_0 \cdot (\gamma_1 \cdot \gamma_2)</math> 
traverses the same paths (in the same order), but <math>\gamma_0</math> with double speed, and <math>\gamma_1, \gamma_2</math> with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The [[associativity]] axiom
:<math>[\gamma_0] \cdot \left([\gamma_1] \cdot [\gamma_2]\right) = \left([\gamma_0] \cdot [\gamma_1]\right) \cdot [\gamma_2]</math>
therefore crucially depends on the fact that paths are considered up to homotopy. Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops <math>\gamma_0, \gamma_1, \gamma_2</math> with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn <math>\pi_1(X, x_0)</math> into a group.

===Dependence of the base point===
Although the fundamental group in general depends on the choice of base point, it turns out that, up to [[isomorphism]] (actually, even up to ''inner'' isomorphism), this choice makes no difference as long as the space ''X'' is [[Connected space#Path connectedness|path-connected]]. For path-connected spaces, therefore, many authors write <math>\pi_1(X)</math> instead of <math>\pi_1(X, x_0).</math>