nLab homotopy groups of spheres in HoTT



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




This entry is about homotopy groups of spheres formalized in homotopy type theory (HoTT).

The homotopy groups of spheres are a fundamental concept in algebraic topology and homotopy theory. They are the homotopy classes of maps between n n -spheres, as nn varies. These may equivalently be understood as collection of different ways to attach spheres to each other. For instance, the homotopy type of a CW complex is completely determined by the homotopy types of the attaching maps.

The following is a quick reference for the state of the art on formalizing and computing homotopy groups of spheres in homotopy type theory.


n/k 0 1 2 3 4
0 π0(S0) π0(S1) π0(S2) π0(S3) π0(S4)
1 π1(S0) π1(S1) π1(S2) π1(S3) π1(S4)
2 π2(S0) π2(S1) π2(S2) π2(S3) π2(S4)
3 π3(S0) π3(S1) π3(S2) π3(S3) π3(S4)
4 π4(S0) π4(S1) π4(S2) π4(S3) π4(S4)

Formalized proofs

The following is a list of results for which at least one proof has been formalized in HoTT.

Calculuation of π 4(S 3)\pi_4(S^3)

  • Brunerie 2016 has proved that there exists an nn such that π 4(S 3)\pi_4(S^3) is n\mathbb{Z}_n. Given a computational interpretation, we could run this proof and check that nn is 2. Added June 2016: Brunerie now has a proof that n=2n=2, using cohomology calculations and a Gysin sequence argument.

See also at first stable homotopy group of spheres.

Calculuation of π 3(S 2)\pi_3(S^2)

  • Peter LeFanu Lumsdaine has constructed the Hopf fibration as a dependent type. Lots of people around know the construction, but I don’t know anywhere it’s written up. Here’s some Agda code with it in it.

  • Brunerie 2016, proof that the total space of the Hopf fibration is S 3S^3, together with π n(S n)\pi_n(S^n), imply this by a long-exact-sequence argument.

  • This was formalized in Lean in 2016.

π n(S 2)=π n(S 3)\pi_n(S^2)=\pi_n(S^3) for n3n\geq 3

  • This follows from the Hopf fibration and long exact sequence of homotopy groups.
  • It was formalized in Lean in 2016.

See also at second stable homotopy group of spheres.

Freudenthal Suspension Theorem

Implies π k(S n)=π k+1(S n+1)\pi_k(S^n) = \pi_{k+1}(S^{n+1}) whenever k2n2k \le 2n - 2

Calculuation of π n(S n)\pi_n(S^n)

See also at Hopf degree theorem.

Calculuation of π k(S n)\pi_k(S^n) for k<nk \lt n

Calculuation of π 2(S 2)\pi_2(S^2)

Calculuation of π 1(S 1)\pi_1(S^1)

See also at circle type for more.


category: homotopy theory

Last revised on June 15, 2022 at 16:54:07. See the history of this page for a list of all contributions to it.