The homotopy groups of spheres: a periodic table of ℤs and ℤ₂s, with cells no human being knows
A table of homotopy groups of spheres reads like a mountain range: zeros below the diagonal, integers along it, and chaos above. Its strangest entry — three-dimensional loops on a two-sphere — is the Hopf fibration.
Toggle the overlays: the silent triangle, the integer diagonal, the Hopf cell, and the stable range. Click a stable cell to see its partner.
The Hopf cell. π₃(S²) = ℤ is the first surprise above the diagonal: the 2-sphere holds infinitely many essential 3-dimensional loops, generated by the Hopf map. See also the Hopf Fibration module for that map in full.
This corner of the table is settled. Far enough off it — high probes into high spheres — the entries are unknown to mathematics: not unpublished, not merely hard, but open. The table has blank cells no human being can fill.
A homotopy group is a probe. The k-th homotopy group of a space collects the maps from the k-sphere into it, up to deformation — and in this setting the definition is short enough to write in one line: take the k-fold loop space and truncate it to a set. The two structural facts about the table both come from suspension. Everything below the diagonal is trivial because suspending raises connectedness, so an n-sphere simply has nothing for a lower-dimensional probe to catch. And the diagonal is always the integers, proved by induction that starts at the circle.
Above the diagonal the values look arbitrary — the integers modulo two, then modulo twelve, then modulo twenty-four — but the Freudenthal suspension theorem imposes real order. Once the probe dimension is small enough relative to the target, suspending stops changing the answer. Each diagonal running down and to the left freezes, so a single computation determines infinitely many entries. The chaotic region is finite; the rest is repetition.
Pour in what is known about the circle and the 3-sphere, and watch exactness squeeze an unknown group into existence.
Ten tanks, one sequence: πₖ(fiber) → πₖ(total) → πₖ(base) → πₖ₋₁(fiber) → … for the Hopf fibration S¹ → S³ → S². Fill the knowns and let exactness do the rest.
Take a map and form its fiber; then take the fiber of that inclusion, and keep going. The sequence does not wander off — it cycles back through loop spaces of everything you started with. Truncating the whole chain to sets converts it into the long exact sequence of homotopy groups, and exactness is a genuinely powerful constraint: knowing the groups on either side of a map often pins down the map, and knowing that both neighbours vanish forces it to be an isomorphism outright.
Applied to the Hopf fibration — circle fibers, total space the 3-sphere, base the 2-sphere — the plumbing does something delightful. The circle’s fundamental group has an integer in it, and the circle has nothing above dimension one. Those zeros clamp the sequence so tightly that the integer has nowhere to go but into the second homotopy group of the 2-sphere, and every group above dimension two on the 3-sphere is forced to match its counterpart on the 2-sphere. Combine that with the integer diagonal and you get the headline: the third homotopy group of the 2-sphere is the integers.
Pick points on the inset 2-sphere to see their fibers. No two circles meet, yet every pair is linked.
Over every point of the 2-sphere hangs a circle — one fiber of the Hopf map. The fibers never touch, yet any two of them are linked like consecutive rings of a chain, and together they fill the entire 3-sphere; a latitude circle of base points sweeps its fibers into a torus, and the nested tori stack up to fill space. See also the Hopf Fibration module for the full story of this map.
What separates this account from the classical one is that nothing is imported. The Hopf fibration is not borrowed from complex geometry; it is built inside the type theory out of the circle’s own multiplication, glued over the two hemispheres of the 2-sphere by exactly the same univalence move that built the helix over the circle. The proofs are constructive and have been checked by machine.
There is a striking footnote. The book computes the fourth homotopy group of the 3-sphere as the integers modulo some number, where the proof itself could in principle be run to output that number. It is, as far as anyone can tell, the first homotopy group ever computed without knowing the answer in advance — a proof that does not merely confirm a fact but produces it.
See also: Hopf Fibration for a full module on the geometry — the Villarceau circles, the nested tori, and the quaternionic picture — and Algebraic Topology for the classical route to the same invariants.
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