Homotopy GroupsEdit
Homotopy groups are among the central algebraic invariants of topological spaces, distilled from the geometry of maps and deformations. At their heart, they measure how spheres can sit inside a space, up to continuous deformation. By encoding this geometric information into groups, homotopy theory turns shapes and spaces into algebraic data that can be compared, classified, and computed in ways that illuminate both simple and highly intricate spaces.
The basic object is the based n-sphere, denoted n-sphere, and a based space (X, x0). A based map from (S^n, s0) to (X, x0) is a continuous function sending the chosen base point s0 in S^n to x0 in X. Homotopy between such maps is a continuous deformation through based maps. For each n ≥ 1, the set of based homotopy classes of maps, usually written as π_n(X, x0), carries a natural group structure (with the operation described below) that is invariant under based homotopy equivalence. When X is path-connected, these groups can be regarded, up to a choice of base point, as intrinsic invariants of the space.
The first and most familiar homotopy group is the fundamental group, written π_1(X, x0) or simply π_1(X) when the basepoint is understood. It classifies loops at x0 up to based homotopy. Higher homotopy groups π_n(X, x0) generalize this idea by studying maps from the n-sphere into X. A striking feature is that π_n(X, x0) is abelian for n ≥ 2, a consequence of the higher-dimensional ways to concatenate spheres. The groups are functorial: a continuous map f: (X, x0) → (Y, y0) induces a homomorphism π_n(f): π_n(X, x0) → π_n(Y, y0). When basepoints are moved along a path, the groups are naturally isomorphic, so in practice one works with the isomorphism class of the groups for connected spaces.
In addition to their intrinsic definition, homotopy groups connect to several classical constructions in topology. The loop space of X, denoted loop space, encodes based maps from S^1 into X, and there is a fundamental isomorphism πn(X, x0) ≅ π{n-1}(ΩX, basepoint) that reflects a deep loop–space perspective on homotopy theory. The relationship between π_1 and higher π_n is more than a mere curiosity: π_1 acts on each π_n for n ≥ 2, reflecting how loops in X can twist higher-dimensional spheres embedded in X. The study of this interaction leads into more elaborate structures such as Eilenberg–MacLane space and Postnikov tower.
Foundations
Based spaces and based homotopy: The theory builds from spaces with a fixed basepoint and maps preserving that basepoint. The fundamental group π_1(X, x0) reflects the space’s loop structure, while higher π_n(X, x0) capture higher-dimensional deformation patterns of spheres inside X.
Group structure: For n ≥ 2, the operation on π_n(X, x0) comes from gluing hemispheres along a common equator; associativity and invertibility come from suitable deformations. The abelian nature of π_n for n ≥ 2 is a hallmark of the higher-dimensional flexibility of spheres.
Basepoint issues and connectedness: In practice, one often works with path-connected spaces so that all basepoints yield isomorphic groups, modulo canonical isomorphisms provided by paths.
Examples and basic computations: The fundamental group of the circle satisfies π1(S^1) ≅ Z; the higher sphere S^n has π_n(S^n) ≅ Z for n ≥ 1, and π_k(S^n) = 0 for k < n. The first nontrivial higher group of S^n beyond π_n is often nontrivial in surprising ways: for instance, π{n+1}(S^n) is Z for n = 1 but Z/2 for many n ≥ 3, with the Hopf fibration providing a class in π_3(S^2) ≅ Z; these computations illustrate the richness beyond the simplest cases.
Natural frameworks: The theory is usually developed for spaces with reasonable finiteness properties, such as CW complex, where cellular methods and cellular approximations simplify computations. The ideas are deeply categorical as well, sitting inside the framework of homotopy theory and its functors.
Fundamental results and examples
The Hurewicz theorem: For spaces with connectivity conditions, the first nontrivial π_n injects into, and in the simplest form isomorphically maps onto, the corresponding homology group H_n(X). This bridge between homotopy and homology is central to transferring geometric information into computable algebraic invariants. See the Hurewicz theorem for precise statements and generalizations.
The Whitehead theorem: A map f: X → Y that induces isomorphisms on all homotopy groups is a weak homotopy equivalence, meaning it induces isomorphisms on all homotopy groups and, for reasonable spaces, is equivalent up to homotopy. This result underlines how homotopy groups control the essential “shape” of a space.
Suspension and loop spaces: The togetherness of πn and ΩX provides a way to shift dimensions and study the stabilized behavior of homotopy groups. The Freudenthal suspension theorem gives a stability window in which the suspension map π_n(X) → π{n+1}(ΣX) is an isomorphism, illuminating how higher-dimensional homotopy information can become predictable in a range.
Postnikov towers: Any space can be approximated by a sequence of spaces, each determined by a truncated set of homotopy groups and a series of k-invariants that measure the obstruction to extending partial information. This construction allows one to reconstruct spaces from their homotopy groups in a controlled, stepwise way. See Postnikov tower.
Homotopy groups of spheres: The πn(S^m) are among the most studied and difficult invariants. In the stable range, where the dimensions are large relative to one another, patterns emerge that can be manipulated using tools such as spectral sequences and cohomology operations. The classical computations begin with π_n(S^n) ≅ Z and extend into deeper territory such as π{n+1}(S^n) being Z/2 for many n ≥ 3 and more intricate groups for larger shifts.
Computation, methods, and connections
Serre spectral sequence and fibrations: When a space fibers over another with a typical fiber, the Serre spectral sequence relates the homotopy of the total space to the homotopy of the base and fiber. While the spectral sequence is a homology tool, its interplay with the homotopy groups via long exact sequences and universal bundles is a fundamental computational theme.
Adams spectral sequence and stable stems: In the stable range, the deep problem of determining the stable homotopy groups of spheres is approached via the Adams spectral sequence, which translates homotopy-theoretic questions into computations in the cohomology of the Steenrod algebra. This is one of the crown jewels of modern algebraic topology and connects to Eilenberg–MacLane space and cohomology operations.
Cellular and CW methods: For many spaces of interest, especially those built from cells, the computation of π_n(X) reduces to combinatorial data about attaching maps and cellular structure. This makes a bridge to more explicit calculations and to the construction of examples and counterexamples.
Applications to geometry and topology: Homotopy groups inform the study of fiber bundles (through long exact sequences associated to fibrations), the classification of manifolds via obstruction theory, and the study of maps between spaces up to homotopy. The loop space perspective and the corresponding algebraic structures provide a conceptual language for these applications.
History and context
The theory grew from the early work of pioneers who connected geometric intuition to algebraic invariants. The development of the fundamental group as a tool for classifying coverings, along with the subsequent generalization to higher homotopy groups, established a robust framework for comparing spaces beyond mere shape. The synthesis of homotopy theory with homology, the introduction of spectral sequences, and the later elaboration of Postnikov towers and modern stable homotopy theory created a powerful agenda for understanding spaces that arise in geometry, algebraic topology, and mathematical physics. See Whitehead (topologist) and Hatcher (topology) for foundational expositions and historical development.