Talk:Homotopy groups of spheres/Definitions

message: Topology is a way of studying geometrical objects. In contrast with other fields such as differential geometry topology deals with a relatively coarse aspect of the structure of the objects in question, namely those features which are stable under continuous operations, which means that cutting objects in two parts and will not be allowed. Otherwise all operations are allowed. For example, topology regards circles of different radii to be structurally equivalent, because magnifying the smaller circle is a continuous operation. Similarly, a circle is equivalent to a square, as one can continuously round off the corners of the square to get back the circle.
An important class of examples of geometrical objects are what modern topology calls n-spheres Sn, where n is a natural number. The spheres are of fundamental interest both in theory and practice. They are highly symmetrical. The first case is the circle or 1-sphere, given by the equation
:x12 + x22=1.
The 2-sphere can be thought of to be the surface of the earth (without the interior). It is given by the equation
:x12 + x22 + x32=1.
For any n, the n-sphere is defined to be the set of points (x1, x2, ..., xn+1) in (n+1)-dimensional space satisfying the equation
:x12 + x22+ ... + xn+12=1.
The dimension of the n-sphere is the dimension of the ambiant space (n+1), reduced by 1, as the sphere is defined by one equation. Hence its dimension is n, as it should be. For example the circle, i.e. the 1-sphere, is 1-dimensional, because it is a curve.
In modern mathematics it is common to study a category by associating to every object of this category a simpler object which still retains a sufficient amount of information about the object in question. Homotopy groups are a certain way of associating groups to topological spaces. A group is a set which allows adding of elements in a suitable sense. For examples, the integers Z form a group. Another example are the finite groups Z/sZ. These are the only types of groups which appear in this article.
The provided link between topology, and in particular spheres, to groups allows the application of group theory to get insights in topology. For example, if two topological objects have different homotopy groups, they can't have the same topological structure, a fact which may be difficult to prove without resorting to non-topological means. This article illustrates this pattern: the nature of spheres becomes clear(er) in the light of their homotopy groups. 
The construction of the homotopy groups is as follows. To fix ideas, we will construct
:π1(S2),
the first homotopy group of the 2-sphere. Consider functions from the 1-sphere, the circle, to the 2-sphere, the surface of a ball:
:f: S1 → S2
(For technical reasons we require them to map a fixed point on the circle, for example (1, 0), to a fixed point on the 2-sphere, for example (1, 0, 0). The concrete choice of these points is not important). The maps may map distinct points to the same points. The only requirement is that they have to be continuous: it is not allowed to snap the circle into two parts and map them individually. A slightly more technical way of saying this: points on the circle which are close need to be mapped to close points on the 2-sphere. After all, we want to get information about the topological structure, which considers cutting and snapping to be irregular.
Any such map can be represented by a loop starting and ending in the point (1, 0, 0) on the 2-sphere, possibly intersecting itself. In accordance to the dogma that objects are only important up to continuous deformations, we call two loops f and g homotopic if they can be transformed into each other. 
-- make an image where most of the loop is equal, just a small part is transformed -- 
Given two loops f and g: S1 → S2, we compose them as follows: start at the initially fixed point (1, 0), follow the loop given by f. By the above choice, it ends in the base point (1, 0, 0), which is also where the loop g starts. Therefore concatenating f and g is a new continuous loop, we write it as f+g. Moreover, we can "invert" a loop f by running it the other way. The inverted loop is written -f. Thirdly we have a zero-"loop" 0, which is just staying in the fixed point (1, 0, 0). To get a group, we need to verify the following identity:
:f+(-f)=0,
which is the abstraction of the same identity when adding integers, for example.
Following our definition, we concatenate the loop f and  the same loop with the opposite orientation -f. Unfortunately it is as such not the zero-loop, because it does not stay constantly at the same point. However, and this is the central point of introducing the homotopy notion, the concatenation is homotopic to, i.e. can be continuously deformed to, the zero-loop:
--make an animated gif as follows: draw one 1-sphere at the left and 3 2-spheres at the right. the 1st 2-sphere shows the concatenated loops, the middle one shows the t=1/2-stage of the homotopy, the right one shows the constant loop. the point x \in S^1 and f_t(x) \in S^2 and the trace of f_t(x) up to x should be highlighted. --
Therefore we obtain a group, it is called the first homotopy group of the 2-sphere, π1(S2). However, it is not very interesting: any loop can be continously shrinked to the zero-loop. One writes π1(S2)=0, meaning that there is essentially only the zero-loop, because every other  loop can be continuously deformed into the zero-loop.
There would be no homotopy theory, if things were always that simple. To get more meaningful groups, we can replace the source S1 and/or the target S2 by higher-dimensional spheres Sk, and Sn, respectively. The groups thus obtained are called k-th homotopy group of the n-sphere, denoted
:πk(Sn).
Jakob.scholbach 19:11, 28 October 2007 (UTC)
thread_title:  yet another version of an "Introduction" section