Higher Homotopy Groups

Overview

The higher homotopy groups pi_n(X, x_0) for n >= 2 generalize the fundamental group by considering continuous maps from the n-sphere S^n into X, modulo homotopy. They are abelian for n >= 2 (unlike pi_1). Computing higher homotopy groups is notoriously difficult — even pi_n(S^2) is unknown for large n — but powerful tools like the long exact sequence of a fibration, the Hurewicz theorem, and spectral sequences provide many results.

Intuition

The fundamental group measures 1-dimensional holes by loops. Higher homotopy groups measure higher-dimensional 'voids': pi_2 measures how 2-spheres get trapped, pi_3 measures 3-sphere winding, and so on. The key surprise is that spheres S^n can have nontrivial homotopy groups in dimensions beyond n — for instance pi_3(S^2) = Z, detected by the Hopf fibration.

Formal Definition

Definition

For n >= 1, the n-th homotopy group pi_n(X, x_0) is the set of homotopy classes of maps (S^n, *) -> (X, x_0), where * is the basepoint. For n >= 2, this set has an abelian group structure: two maps f, g: S^n -> X are composed by pinching the equator and applying f on one hemisphere and g on the other. The group operation is commutative for n >= 2 by the Eckmann-Hilton argument.

\pi_n(X, x_0) = [(S^n, *),(X, x_0)]

n-th homotopy group

\pi_n(X) \text{ is abelian for } n \ge 2

Commutativity (Eckmann-Hilton)

\pi_n(X\times Y) \cong \pi_n(X)\oplus\pi_n(Y)

Product formula

\pi_n(S^n) \cong \mathbb{Z}, \quad \pi_k(S^n) = 0 \text{ for } k < n

Homotopy of spheres (low range)

Theorems

Theorem 1: Hurewicz Theorem

If X is (n-1)-connected (meaning \pi_k(X) = 0 for k \le n-1) with n \ge 2, then \tilde H_k(X) = 0 for k < n and the Hurewicz map h: \pi_n(X) \to H_n(X) is an isomorphism.

Theorem 2: Long Exact Sequence of a Fibration

For a fibration F \to E \to B, there is an exact sequence \cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \xrightarrow{\partial} \pi_{n-1}(F) \to \cdots \to \pi_0(E) \to \pi_0(B).

Theorem 3: Freudenthal Suspension Theorem

The suspension map \Sigma: \pi_k(S^n) \to \pi_{k+1}(S^{n+1}) is an isomorphism for k < 2n-1 and a surjection for k = 2n-1. This implies stable homotopy groups stabilize.

Worked Examples

The universal cover of S^1 is R -> S^1 (fiber Z, since deck transformations are translations by integers). This is a fibration with fiber Z (discrete).
Since pi_n(Z) = 0 for n >= 1 (discrete fiber) and pi_n(R) = 0 for n >= 1 (R is contractible), the long exact sequence gives:
$0 = \pi_n(\mathbb{R}) \to \pi_n(S^1) \to \pi_{n-1}(\mathbb{Z}) = 0 \quad\text{for } n\ge 2.$
So pi_n(S^1) = 0 for n >= 2, and pi_1(S^1) = Z by the fundamental group computation.

Answer: pi_1(S^1) = Z, pi_n(S^1) = 0 for n >= 2.

Practice Problems

Difficulty 7/10

Compute pi_2(S^2) using the Hurewicz theorem.

Difficulty 8/10

Use the long exact sequence of the path-loop fibration Omega S^n -> PS^n -> S^n to relate pi_k(S^n) to pi_{k-1}(Omega S^n).

Difficulty 9/10

Prove that pi_n(X × Y) ≅ pi_n(X) ⊕ pi_n(Y) for n >= 1.

Common Mistakes

Common Mistake

pi_n(X) = 0 for n > dim(X).

This is false. For example, pi_3(S^2) = Z even though dim(S^2) = 2. Higher homotopy groups can be nontrivial in all dimensions.

Common Mistake

Higher homotopy groups pi_n are harder to compute than homology because they carry more information.

They are indeed hard to compute, but homology can carry more information in some senses (detects torsion, etc.). The difficulty of computing pi_n comes from lack of excision, not from greater information content alone.

Quiz

Summary

The n-th homotopy group pi_n(X) consists of homotopy classes of maps (S^n, *) -> (X, x_0); it is abelian for n >= 2.
The Hurewicz theorem relates pi_n to H_n for highly connected spaces: pi_n(X) ≅ H_n(X) when X is (n-1)-connected.
The long exact sequence of a fibration is the main computational tool, relating pi_n of total space, base, and fiber.
The Freudenthal suspension theorem shows that homotopy groups of spheres stabilize: pi_{n+k}(S^n) depends only on k for n >> k (stable homotopy groups).
Pi_3(S^2) = Z, detected by the Hopf fibration — a fundamental example of nontrivial higher homotopy.