Homotopy groups of spheres

From Wikipedia, the free encyclopedia

The homotopy groups of spheres describe the different ways spheres of various dimensions can be wrapped around each other. They are studied as part of algebraic topology, a branch of mathematics. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere or hypersphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar circle (1-sphere) and sphere (2-sphere).

The number n is the intrinsic dimension of the sphere as an independent topological space; these are hollow spheres, not balls. Thus, for example, the 2-sphere resembles a balloon, not a baseball. As such, it is a 2-dimensional manifold, although the construction embeds it in 3-dimensional space. More generally, the n-sphere is an n-dimensional object.

The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century[1] as a tool for such classification, in effect using the set of mappings from an n-sphere in to a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.

[edit] Definition

The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark intermediate stages in one possible homotopy.

Mappings are continuous functions from one topological space into another; every mapping is a rule which assigns each point in the first space to some point in the second. Two mappings are considered equivalent if one can be continuously deformed into the other, a process called homotopy.

A very simple example is a one-point map, which assigns all the points in the first space to the same point in the second space. Any two one-point maps to a path connected space are equivalent (move the target point continuously along a path joining the two map's points). All mappings equivalent to one-point maps through a homotopy are called trivial.

Mappings of a i-sphere (i here is another integer; we'll need n in a moment) into any path connected topological space, X, can be combined by splicing them together, roughly speaking, and the equivalence classes of these mappings ("homotopy classes") form a group called the i^th homotopy group of X, which is written π_i (X). The equivalence class of the trival maps is the identity element of the group and is called 0. Homotopy groups of spheres studies the cases where X is an n-sphere, and the groups are written π_i (Sⁿ).

[edit] Cases that are easy to visualize

The few cases that can be constructed in ordinary 3-dimensional space allow some sense of the subject to be gained. These visualizations are not, of course, mathematical proofs.

[edit] π₁(S¹) = Z

Elements of π₁(S¹)

The simplest case considers the ways a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a rubber band around one's finger. One can wrap it once, twice, three times and so on. One can do the wrapping in either of two directions and if one wraps a certain number of times in one direction and then wraps the same number of time in the other direction, then wraps in opposite directions will cancel out. The group that is formed is therefore the group of integers, known as the infinite cyclic group, and denoted Z.

[edit] π₂(S²) = Z

The case of 2-spheres can be visualized as wrapping a plastic bag around an ordinary toy ball and then sealing it. The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball. (The bag is allowed to pass through itself). The twist can be in one of two directions and opposite twists can cancel out. Again the group of integers, Z, is formed. These two results generalize: for all n > 0, π_n (Sⁿ) = Z.

[edit] π₁(S²) = 0

A homotopy from a circle around a sphere down to a single point

Any continuous mapping from a circle to an ordinary sphere can be continuously deformed down to a point, i.e. the trivial mapping. This can be visualized as a rubber-band wrapped around a frictionless ball - it can always be slid off. Further, this result generalises for higher dimensions. All mappings from a hypersphere into a sphere of higher dimension are similarly trivial: if i < n, then π_i (Sⁿ) = 0 (i.e., the trivial group).

[edit] π₂(S¹) = 0

All the interesting cases of homotopy groups of spheres involve mappings from an n-sphere dimensional onto spheres of lower dimensions. Unfortunately, the one such case we might easily visualize is uninteresting, as there are no non-trivial mappings from the ordinary sphere to the circle. Hence, π₂(S¹) = 0.

[edit] Higher dimensions

The rest of this article provides a summary of the homotopy groups for spheres for higher dimensions. Computing the homotopy group can be complicated, and this article is restricted to summarizing many of the homotopy groups of spheres that have been computed to date. The results are surprisingly complex, with no pattern discerned to date.

From a geometric point of view homotopy groups are invariants of the n-sphere under any homeomorphism. From the algebraic aspect, there is ample evidence that they involve substantial complexity of structure, and intense study from around 1950 has not completely elucidated that.

[edit] π₃(S²)

When i is less than n, the homotopy group is the trivial group: π_i (Sⁿ) = 0, and when i = n, it is always the infinite cyclic group, π_n (Sⁿ) = Z. It is the case i > n that is of real importance, and historically it came as a great surprise that the corresponding homotopy groups were nontrivial. This is in contrast to the behavior for corresponding results in homology theory, where H_i (Sⁿ) = 0 when i > n. The first nontrivial example concerned mappings from the three-dimensional sphere to the ordinary 2-sphere, and was discovered by Heinz Hopf in 1931. The existence of the Hopf fibration implies that π₃ (S²) = Z.

[edit] The Hurewicz theorem

The Hurewicz theorem links homotopy groups with homology groups, which are generally easier to calculate. In particular it shows that for a simply-connected space X, the first nonzero homotopy group π_n(X) is isomorphic to the first nonzero homology group H_n(X) and π_n(Sⁿ)=Z.

[edit] Stable and unstable groups

As the homotopy groups of spheres turn out to be very difficult to compute, algebraic topologists searched for ways to simplify the problem. A key insight was that the suspension theorem of Hans Freudenthal implies that the groups π_n+k (Sⁿ) depend only on k for n ≥ k + 2. These groups are called the stable homotopy groups of spheres, and are denoted π_k^S. They are finite and abelian for k ≥ 1. They have been computed in numerous cases, but the general pattern is still elusive.

For n < k + 2, the groups are called the unstable homotopy groups of spheres. There are ad-hoc methods of calculating these for the cases of n small. A systematic tool in this context is the J-homomorphism. These groups are abelian, and are all finite except for those of the form π_4r−1 (S^2r) (for integer values of r). In this case, the group is the product of the infinite cyclic group with a finite abelian group.

[edit] History

In the late 19th century Camille Jordan introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory.^[1] A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced.^[2]

Higher homotopy groups were first defined by Eduard Čech ^[3] in 1932 (though his first paper on them was rejected by J. Alexander, who apparently mistakenly thought that they were the same as homology groups). Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper and also for the Hurewicz theorem which can be used to calculate some of the groups.^[4]

An important methods for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was Hans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results. In 1953 G. W. Whitehead showed that there is a metastable range for the homotopy groups of spheres. Jean-Pierre Serre used spectral sequences to show that most groups are finite, the exceptions being π_n(Sⁿ) and π_4n−1(S²ⁿ). Others who worked in this area included José Adem, J.P. May, H. Toda and Lev Semenovich Pontryagin. As of 2002, calculations of stable homotopy groups π_i+n(Sⁿ) are known for i up to 60.^[5]

[edit] Table of homotopy groups

Tables of homotopy groups of spheres are most conveniently organized by only showing π_n+k (Sⁿ) for n > 1 and k > 0. For other cases, the homotopy group π_n+k (Sⁿ), with n > 0, is as follows:

• For k < 0, the group is the trivial group (generally written 0)
• For k = 0, the group is the infinite cyclic group (generally written Z), a consequence of the Hurewicz theorem.
• For k > 0 and n = 1, the group is trivial.

The following table shows many of the groups π_n+k (Sⁿ) that have been computed to date, with the following conventions:

• Where the entry is an integer, m, the homotopy group is the cyclic group of that order (generally written Z_m).
• Where the entry is "Z", the homotopy group is the infinite cyclic group, (Z).
• Where entry is a sum, the homotopy group is the direct sum (equivalently, cartesian product) of the cyclic groups of those orders. Powers indicate repeated summation.

Example: π₁₉ (S¹⁰) = π₁₀₊₉ (S¹⁰) = Z + 2³, the table entry at k = 9, n = 10. The expression Z + 2³ denotes the group Z × Z₂ × Z₂ × Z₂.

Note that when a and b have no common factor, Z_a x Z_b is isomorphic to Z_ab. Using this, entries in the above have been written using fewer coefficients than in Toda's table.

[edit] See also

• 3-sphere#Topological properties for a table of the homotopy groups π_k (S³)
• Winding number. An element in π₁(S¹) can be thought of as the winding number of a loop around the origin.

[edit] References

1. ^ O'Connor, John J., and Edmund F. Robertson. "Camille Jordan". MacTutor History of Mathematics archive.
2. ^ O'Connor, John J., and Edmund F. Robertson. "A history of Topology". MacTutor History of Mathematics archive.
3. ^ E. Čech, "Höherdimensionale Homotopiegruppen" , Verh. Intern. Mathematikerkongress Zürich, 1932 , O. Füssli (1932) pp. 203
4. ^ May, J.P. Stable Algebraic Topology, 1945–1966
5. ^ Hatcher, p384

• Allen Hatcher: Stable homotopy groups of spheres page 339. (2002, Cambridge University Press). ISBN 0-521-79540-0.
• Table of homotopy groups of spheres. From Hirosi Toda: Composition Methods in Homotopy Groups of Spheres (Princeton University. Press, 1962). ISBN 0-691-09586-8.
• Douglas C. Ravenel: Complex cobordism and stable homotopy groups of spheres (2nd edition), Appendix 3. (From online edition). (Ams Chelsea Pub., 2003) ISBN 0-8218-2967-X

[edit] Further reading

• relation with string theory