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The higher homotopy groups pi_n(X, x_0) for n >= 2 generalize the fundamental group by replacing loops (maps from S1) with maps from higher-dimensional spheres Sn. The n-th homotopy group consists of homotopy classes of based maps Sn -> X, with group operation given by "stacking" spheres. Unlike the fundamental group, all higher homotopy groups are abelian. They detect n-dimensional "holes" in a space and are the most fundamental topological invariants — but they are notoriously difficult to compute, even for simple spaces like spheres.
The n-th homotopy group pi_n(X, x_0) of a pointed space (X, x_0) is defined as the set of homotopy classes of based continuous maps (Sn, s_0) -> (X, x_0), where all homotopies keep the basepoint fixed. For n = 1, this recovers the fundamental group. For n >= 2, the group operation is defined by "stacking": given [f] and [g] in pi_n(X), form the map f * g : Sn -> X that sends the upper hemisphere to f (rescaled) and the lower hemisphere to g (rescaled), with the equatorial Sn-1 mapping to the basepoint x_0. This operation is well-defined on homotopy classes and satisfies the group axioms.
The most important structural difference from the fundamental group is that pi_n is abelian for n >= 2. The proof uses the Eckmann-Hilton argument: the group operation in pi_n can be performed by stacking along any of the n coordinate directions of the unit cube (since Sn = In / boundary(In)), and these different stacking operations satisfy the interchange law (stacking f*g vertically and h*k horizontally gives the same result as stacking f*h vertically and g*k horizontally). The interchange law forces any two group operations sharing an identity to be equal and abelian. For n = 1, there is only one direction to stack (horizontal concatenation of loops), so no interchange is possible and the group can be non-abelian.
The homotopy groups of spheres illustrate both the power and the difficulty of higher homotopy groups. For n < k, pi_n(Sk) = 0 (this follows from cellular approximation: any map Sn -> Sk with n < k can be homotoped to miss the top cell, hence is null-homotopic). For n = k, pi_n(Sn) = Z, generated by the identity map (this is the content of degree theory). But for n > k, the situation is extraordinarily complex. The most famous example is pi_3(S2) = Z, generated by the Hopf fibration eta : S3 -> S2, which maps each point of S3 to a point on S2 such that preimages are linked circles. This cannot be detected by homology (H_3(S2) = 0) and reveals the fundamentally different character of homotopy groups.
The Hurewicz homomorphism h : pi_n(X) -> H_n(X) connects homotopy groups to homology. It sends the homotopy class of a map f : Sn -> X to the homology class f_*([Sn]) in H_n(X) — the image of the fundamental class of Sn under the induced map on homology. The Hurewicz theorem (studied in detail in the next topic) states that in the "first nontrivial dimension," this homomorphism is an isomorphism. This provides the main bridge between the computable world of homology and the powerful but hard-to-compute world of homotopy groups.
Despite their theoretical importance, higher homotopy groups are notoriously hard to compute. They do not satisfy excision (the Freudenthal suspension theorem gives only a limited version), they have no Mayer-Vietoris sequence, and the homotopy groups of even simple spaces like S2 contain intricate patterns that are only partially understood after decades of research. The tools for computing them — spectral sequences, fibrations, obstruction theory — are among the most sophisticated in all of mathematics. This computational difficulty, combined with their theoretical centrality (they determine the homotopy type of CW complexes), makes higher homotopy groups one of the great frontiers of algebraic topology.