Talk:Homotopy groups of spheres/Definitions
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The definition of a homotopy group for spheres depends on topology and algebra.
An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology. Begin with a segment of the real line, which we think of as one-dimensional. Bend it around so that its ends meet, glue the ends together, and smooth over the joint. We have, of course, made a circle, which topology calls a 1-sphere. Place that circle in the xy plane and treat it as the equator of a sphere which we make by sweeping to a point above (creating the northern hemisphere) and to a point below (creating the southern hemisphere. Topology calls this construction a suspension. Thus the ordinary sphere of geometry is a suspension of the 1-sphere, and is called a 2-sphere. Likewise we can take the 2-sphere as the "equator" of a 3-sphere, and so on, creating an endless series of spheres, generically called n-spheres. (If we describe the n-sphere geometrically as points (x1,…,xn+1) such that x12+⋯+xn+12 = 1, then its equator is the (n−1)-sphere obtained by setting xn+1 to zero, so that x12+⋯+xn2 = 1.)
As a topological space, a circle is (by our construction) very much like a line segment, in the small. We speak of a neighborhood of a point on the circle, for example, the points on the circle within a small distance of that point. Since topology ignores the geometry (the curving), it cannot distinguish such a neighborhood of a circle point from a neighborhood of a line point. Likewise, such neighborhoods of a 2-sphere point have the same topology as similar neighborhoods in a plane; the neighborhoods are said to be homeomorphic. Since every point of an n-sphere has the same neighborhood structure as an n-dimensional Euclidean space, we say that the n-sphere is an n-dimensional manifold, or n-manifold for short.
In fact, our "glueing the ends" construction of a circle is instructive; for, another way to make an n-sphere topologically is to take In and glue all its boundary points together. Here I1 is the unit interval I = [0,1], I2 is the (filled in) unit square, I3 is the (solid) unit cube, and so on. The boundary of I1 is its pair of endpoints; of I2, its square perimeter; of I3, its faces; and so on.
Despite the local equivalence, the overall structure of an n-sphere is topologically different from that of an n-dimensional flat space (such as a line or plane). A homotopy group is one way to detect that difference and to quantify it.
The group elements are equivalence classes of maps (continuous functions). If X and Y are two topological spaces, not necessarily distinct, a map ƒ:X→Y from X to Y is a function that "preserves neighborhoods". Specifically, if x is a point in X and y = ƒ(x) is its image in Y, then any neighborhood of y is the image of some neighborhood of x. (The ε-δ definition of continuity used in calculus is an example.) An important special case is the constant map, ƒ(x) = y0 for some fixed point y0.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/6/67/Homotopy_of_pointed_circle_maps.png/640px-Homotopy_of_pointed_circle_maps.png)
Equivalence is defined in terms of homotopy. Whereas a map is a continuous function from one topological space to another, a homotopy is a continuous function from one map to another, parameterized by the unit interval. In formal terms, given two maps ƒ,g:X→Y, a homotopy is a map h:X×I→Y from pairs (x,t) with x∈X and t∈I such that h(x,0) = ƒ(x) and h(x,1) = g(x). As the interval parameter t changes from 0 to 1, ƒ continuously transforms into g. If there is a homotopy from ƒ to g, we say that ƒ and g are homotopic. If ƒ is homotopic to the constant map, we say that ƒ is null homotopic. Note that the relationship of being homotopic satisfies the three requirements for an equivalence relation: it is reflexive, symmetric, and transitive.
Now consider the set of all possible maps from the circle into a space X, and collect equivalent maps into equivalence classes, with each map in a class homotopic to any map in its class. One of the maps must be the constant map, and so if it happens that there is only one homotopy class then all maps S1→X are null homotopic. If we think of the image of the circle as a loop or lasso, this tells us that we can "contract" every loop down to a single point without it getting hung up on any feature of the space X. Examples of this include the line R, the plane R2, the ordinary sphere S2, and in fact any Rn for n ≥ 0, and any Sn for n ≥ 2. But suppose we pluck the origin out of the plane; then the map producing the unit circle is not null homotopic, because any attempt to shrink the loop gets hung up, unable to pass through the missing origin point. We must have more than one homotopy equivalence class. Plucking a point out of a sphere has no such effect, so already this crude tool distinguishes the topology of the sphere from that of the plane.
![Thumb image](http://upload.wikimedia.org/wikipedia/commons/thumb/4/4b/Homotopy_group_addition.svg/320px-Homotopy_group_addition.svg.png)
To further refine the tool of circle map classes, we define an addition operation, as follows. Given two maps ƒ,g:S1→X, we define a map ƒ+g:S1→X, in two steps. First we map the circle to a circle with "pinched equator", a figure-8. Now the top and bottom are both complete circles, and we map the top to X using ƒ and map the bottom to X using g. The composition of two continuous maps is again a continuous map, and so we can now exhibit a map from S1 to X which combines ƒ and g, by way of the pinch. Better still, this addition is well-defined on homotopy classes. On classes addition is associative, has the null homotopic class as identity, and each class has an inverse (a "negative"). Thus homotopy classes with addition satisfy the axioms for a group. In fact, we have just defined the fundamental group of a space X, denoted by π1(X). When there is only one class, the group is the trivial group, denoted by 0. Thus π1(Sn) = 0 for all n ≥ 2. The fundamental group is a homotopy group, and so we have our first examples of homotopy groups of spheres.
To see an example of a non-trivial fundamental group, consider a sphere with opposite points identified. The bottom hemisphere merely duplicates the top, and we may flatten the top hemisphere to a disk whose boundary circle is the equator (still with opposite points identified). For this space we have two classes of circle maps: those homotopic to the constant map, and those homotopic to a diameter. A diameter is a loop because its two endpoints, being opposite points on the rim, are identified; however, it is not possible to pry the loop free from the rim, so such loops are not null homotopic. The sum of two of these loops can escape the rim, so the group has an addition table like that of the integers modulo 2, with 1+1 = 0.
The higher homotopy groups are defined in exactly the same way, with πi(X) based on mappings from Si to X, with classes defined by homotopy equivalence, and with addition through an equator pinch. One subtlety is that for the pinch to work, both maps must carry a distinguished sphere point — say, (1,0,…,0) — to a chosen base point x0 of X. For some spaces different choices of base point can yield different groups; for path-connected spaces like the n-spheres (for n ≥ 1) all base points are equivalent, so we can write πi(Sn) with no mention of base point.