page_title: Homotopy groups of spheres
section_title: Homotopy group
hierarchical_section_title: Homotopy groups of spheres / Background / Homotopy group
caption_attribution_description: Homotopy of pointed circle maps.
context_page_description: In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.
  The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sⁿ — generalizes the familiar circle and the ordinary sphere. The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin. The i-th homotopy group πᵢ summarizes the different ways in which the i-dimensional sphere Sⁱ can be mapped continuously into the n-dimensional sphere Sⁿ. This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.
context_section_description: The distinguishing feature of a topological space is its continuity structure, formalized in terms of open sets or neighborhoods. A continuous map is a function between spaces that preserves continuity. A homotopy is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the residue theorem of complex analysis, where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.
  The first homotopy group, or fundamental group, π₁(X) of a (path connected) topological space X thus begins with continuous maps from a pointed circle (S¹,s) to the pointed space (X,x), where maps from one pair to another map s into x. These maps (or equivalently, closed curves) are grouped together into equivalence classes based on homotopy (keeping the "base point" x fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map S¹↦x are called null homotopic. The classes become an abstract algebraic group with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "bouquet of spheres" — two pointed spheres joined at their distinguished point. The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map.
  More generally, the i-th homotopy group, πᵢ(X) begins with the pointed i-sphere (Sⁱ,s), and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for X equal to Sⁿ (for positive n) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some i all maps are null homotopic, then the group πᵢ consists of one element, and is called the trivial group.
  A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism), so that the two spaces have the same topology, then their i-th homotopy groups are isomorphic for all i. However, the real plane has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also make certain computations easier.
caption: Homotopy of two circle maps keeping base point fixed
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