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,…,x_n+1) such that x₁²+⋯+x_n+1² = 1, then its equator is the (n−1)-sphere obtained by setting x_n+1 to zero, so that x₁²+⋯+x_n² = 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.

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.

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 π_i(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 π_i(Sⁿ) with no mention of base point.

[edit] yet another version of an "Introduction" section

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, ..., x_n+1) in (n+1)-dimensional space satisfying the equation

x₁² + x₂²+ ... + x_n+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 S^k, and Sⁿ, respectively. The groups thus obtained are called k-th homotopy group of the n-sphere, denoted

π_k(Sⁿ).

Jakob.scholbach 19:11, 28 October 2007 (UTC)