To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, π n ( X ) {\displaystyle \pi _{n}(X)} , of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but topological spaces that are not homeomorphic can have the same homotopy groups.
The notion of homotopy of paths was introduced by Camille Jordan.[1]
That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they can't have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.
As for the example: the first homotopy group of the torus T is
because the universal cover of the torus is the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} , mapping to the torus T ≅ R 2 / Z 2 {\displaystyle T\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2}} . Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere S 2 {\displaystyle S^{2}} satisfies:
π 1 ( S 2 ) = 0 , {\displaystyle \pi _{1}(S^{2})=0,}
f : S n → X {\displaystyle f:S^{n}\to X}
that map the base point a to the base point b. In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, we can define πn(X) to be the group of homotopy classes of maps g : [ 0 , 1 ] n → X {\displaystyle g\colon [0,1]^{n}\to X} from the n-cube to X that take the boundary of the n-cube to b.
For n ≥ 1 {\displaystyle n\geq 1} , the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product f ∗ g {\displaystyle f\ast g} of two loops f , g : [ 0 , 1 ] → X {\displaystyle f,g:[0,1]\to X} is defined by setting
For the corresponding definition in terms of spheres, define the sum f + g {\displaystyle f+g} of maps f , g : S n → X {\displaystyle f,g\colon S^{n}\to X} to be Ψ {\displaystyle \Psi } composed with h, where Ψ {\displaystyle \Psi } is the map from S n {\displaystyle S^{n}} to the wedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to X that is defined to be f on the first sphere and g on the second.
If n ≥ 2 {\displaystyle n\geq 2} , then π n {\displaystyle \pi _{n}} is abelian.[2] Further, similar to the fundamental group, for a path connected space any two basepoint choices gives rise to isomorphic π n {\displaystyle \pi _{n}} .[3]
It is tempting to try to simplify the definition of homotopy groups by omitting the base points, but this does not usually work for spaces that are not simply connected, even for path connected spaces. The set of homotopy classes of maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of the fundamental group on the homotopy group, and in general has no natural group structure.
Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups
Here the maps involving π0 are not group homomorphisms because the π0 are not groups, but they are exact in the sense that the image equals the kernel.
Example: the Hopf fibration. Let B equal S2 and E equal S3. Let p be the Hopf fibration, which has fiber S1. From the long exact sequence
⋯ → π n ( S 1 ) → π n ( S 3 ) → π n ( S 2 ) → π n − 1 ( S 1 ) → ⋯ {\displaystyle \cdots \to \pi _{n}(S^{1})\to \pi _{n}(S^{3})\to \pi _{n}(S^{2})\to \pi _{n-1}(S^{1})\to \cdots }
and the fact that πn(S1) = 0 for n ≥ 2, we find that πn(S3) = πn(S2) for n ≥ 3. In particular, π 3 ( S 2 ) = π 3 ( S 3 ) = Z . {\displaystyle \pi _{3}(S^{2})=\pi _{3}(S^{3})=\mathbb {Z} .}
In the case of a cover space, when the fiber is discrete, we have that πn(E) is isomorphic to πn(B) for n > 1, that πn(E) embeds injectively into πn(B) for all positive n, and that the subgroup of π1(B) that corresponds to the embedding of π1(E) has cosets in bijection with the elements of the fiber.
Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert–van Kampen theorem for the fundamental group and the Excision theorem for singular homology and cohomology, there is no simple known way to calculate the homotopy groups of a space by breaking it up into smaller spaces. However, methods developed in the 1980s involving a van Kampen type theorem for higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. See for a sample result the 2010 paper by Ellis and Mikhailov.[4]
For some spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called aspherical spaces. However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of S2 one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.
There is also a useful generalization of homotopy groups, π n ( X ) {\displaystyle \pi _{n}(X)} , called relative homotopy groups π n ( X , A ) {\displaystyle \pi _{n}(X,A)} for a pair ( X , A ) {\displaystyle (X,A)} , where A is a subspace of X.
The construction is motivated by the observation that for an inclusion i : ( A , x 0 ) ↪ ( X , x 0 ) {\displaystyle i\colon (A,x_{0})\hookrightarrow (X,x_{0})} , there is an induced map on each homotopy group i ∗ : π n ( A ) → π n ( X ) {\displaystyle i_{*}\colon \pi _{n}(A)\to \pi _{n}(X)} which is not in general an injection. Indeed, elements of the kernel are known by considering a representative f : I n → X {\displaystyle f\colon I^{n}\to X} and taking a based homotopy F : I n × I → X {\displaystyle F\colon I^{n}\times I\to X} to the constant map x 0 {\displaystyle x_{0}} , or in other words H I n × 1 = f {\displaystyle H_{I^{n}\times 1}=f} , while the restriction to any other boundary component of I n + 1 {\displaystyle I^{n+1}} is trivial. Hence, we have the following construction:
The elements of such a group are homotopy classes of based maps D n → X {\displaystyle D^{n}\to X} which carry the boundary S n − 1 {\displaystyle S^{n-1}} into A. Two maps f, g are called homotopic relative to A if they are homotopic by a basepoint-preserving homotopy F : Dn × [0,1] → X such that, for each p in Sn−1 and t in [0,1], the element F(p,t) is in A. Note that ordinary homotopy groups are recovered for the special case in which A = x 0 {\displaystyle A=x_{0}} is the base point.
These groups are abelian for n ≥ 3 but for n = 2 form the top group of a crossed module with bottom group π1(A).
For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other. See Eckmann–Hilton argument.
see Allen Hatcher#Books section 4.1.
Ellis, Graham J.; Mikhailov, Roman (2010). "A colimit of classifying spaces". Advances in Mathematics. 223 (6): 2097–2113. arXiv:0804.3581. doi:10.1016/j.aim.2009.11.003. MR 2601009.
Ronald Brown, Philip J. Higgins, Rafael Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages, European Math. Society, Zürich, 2011. doi:10.4171/083 MR2841564
Hazewinkel, Michiel, ed. (2001) [1994], "Homotopy group", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen, 104 (1): 637–665, doi:10.1007/BF01457962 .
Kamps, Klaus H.; Porter, Timothy (1997). Abstract homotopy and simple homotopy theory. River Edge, NJ: World Scientific Publishing. doi:10.1142/9789812831989. ISBN 981-02-1602-5. MR 1464944.