Questions: Higher Homotopy Groups

Question 1

Why are the higher homotopy groups π_n(X, x_0) abelian for n ≥ 2, while π_1 can be non-abelian?

• A. Because S^n is simply connected for n ≥ 2
• B. Because for n ≥ 2, the two maps f, g: S^n → X can be 'slid past each other' using the extra dimensions — there is room to homotope f·g to g·f that does not exist for loops on a circle
• C. Because higher homotopy groups are defined using abelian groups as coefficients
• D. Because the composition in π_n is defined differently from π_1

Answer: (B)

The group operation in π_n 'stacks' two n-spheres (pinches the equator and puts f on one hemisphere, g on the other). For n ≥ 2, the two hemispheres can be continuously slid past each other through the extra dimensions: the equatorial (n-1)-sphere has codimension 1 in S^n, and for n ≥ 2, there is enough room to perform the interchange. This is the Eckmann-Hilton argument: when you have two independent group structures that share an identity and distribute over each other, both must be abelian and equal. For π_1, loops live in dimension 1 and cannot slide past each other without crossing.

Question 2

π_3(S^2) ≅ Z, generated by the Hopf fibration η: S^3 → S^2. This is surprising because S^2 has no 3-dimensional 'hole.'

• T. True
• F. False

Answer: True

This is one of the most remarkable facts in algebraic topology. H_3(S^2) = 0 (no 3-dimensional hole in the homological sense), yet π_3(S^2) ≅ Z — there are infinitely many homotopically distinct ways to map S^3 to S^2. The Hopf fibration η: S^3 → S^2, which maps each point of S^3 to a point of S^2 such that the preimage of each point is a great circle, generates this group. This shows that homotopy groups detect fundamentally different information than homology groups: they capture the complexity of the space of maps, not just the 'holes.'

Question 3

Computing π_n(S^k) for general n and k is one of the deepest open problems in algebraic topology.

• T. True
• F. False

Answer: True

Unlike homology of spheres (completely determined: H_m(S^n) = Z for m = 0, n and 0 otherwise), the homotopy groups of spheres are only partially known. π_n(S^n) = Z for all n (generated by the identity), π_n(S^k) = 0 for n < k (by cellular approximation), but the groups π_n(S^k) for n > k are incredibly complex. They have been computed through a substantial range but show no simple pattern. The computation involves spectral sequences, stable homotopy theory, and the Adams spectral sequence, and remains one of the central research programs in topology.

Question 4

A space X with π_n(X) = 0 for all n ≥ 1 must be contractible (assuming X is a CW complex).

• T. True
• F. False

Answer: True

This is Whitehead's theorem applied to the constant map X → point. If all homotopy groups vanish, the constant map induces isomorphisms on all π_n, and by Whitehead's theorem (for CW complexes), this map is a homotopy equivalence. So X is homotopy equivalent to a point, i.e., contractible. This shows that the homotopy groups collectively form a complete invariant for CW complexes up to homotopy equivalence — a space is completely determined (up to homotopy) by its homotopy groups and their relationships.

Question 5

Explain the relationship between π_n(X) and H_n(X) — in what ways do they agree and disagree?

Model answer: For n = 1: π_1 maps onto H_1 via abelianization (the Hurewicz homomorphism), so H_1 = π_1/[π_1, π_1]. For the first nontrivial dimension: if π_k(X) = 0 for k < n and X is connected, then the Hurewicz theorem says π_n(X) ≅ H_n(X). Beyond that, the two invariants diverge. Homology is always computable and satisfies excision; homotopy groups do not satisfy excision and are very hard to compute. Homology of S^n is concentrated in dimensions 0 and n, but homotopy groups of S^n are highly nontrivial in infinitely many dimensions above n.

The Hurewicz homomorphism h: π_n(X) → H_n(X), sending [f: S^n → X] to f_*[S^n], is always defined. It connects the two invariants but is generally neither injective nor surjective beyond the first nontrivial dimension. The fundamental tension: homotopy groups are 'complete' (they determine the homotopy type) but uncomputable; homology is computable but 'incomplete' (it does not determine the homotopy type).