In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the nth 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 nth 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 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.
Introduction In modern mathematics it is common to study a category by associating to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating groups to topological spaces.
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 cannot 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 {\displaystyle T} is
π 1 ( T ) = Z 2 , {\displaystyle \pi _{1}(T)=\mathbb {Z} ^{2},}
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}\left(S^{2}\right)=0,}
because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not homeomorphic to the sphere.
Definition In the n-sphere S n {\displaystyle S^{n}} we choose a base point a. For a space X with base point b, we define π n ( X ) {\displaystyle \pi _{n}(X)} to be the set of homotopy classes of maps
f : S n → X ∣ f ( a ) = b {\displaystyle f:S^{n}\to X\mid f(a)=b}
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, define π n ( X ) {\displaystyle \pi _{n}(X)} to be the group of homotopy classes of maps g : [ 0 , 1 ] n → X {\displaystyle g:[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
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