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Algebraic topology Definition

Stable homotopy theory

Builds on SuspensionPointed spaceCW complexHomotopy groupFreudenthal suspension theorem

Definition

Stable homotopy theory is the study of homotopy invariants that are unchanged by suspension, such as the following groups. Let XX and YY be pointed CW complexes, with XX finite, and let Σ\Sigma be the reduced suspension. Let [A,B]∗[A, B]_* be the set of pointed homotopy classes of pointed maps A→BA \to B. The group of stable maps from XX to YY is the colimit {X,Y}=colim⁡k [ΣkX,ΣkY]∗,\set{X, Y} = \operatorname{colim}_{k} \, [\Sigma^k X, \Sigma^k Y]_*, taken along the maps [ΣkX,ΣkY]∗→[Σk+1X,Σk+1Y]∗[\Sigma^k X, \Sigma^k Y]_* \to [\Sigma^{k+1} X, \Sigma^{k+1} Y]_* that send ff to Σf\Sigma f. For n≥0n \ge 0, the nn-th stable homotopy group of YY is πns(Y)=colim⁡k πn+k(ΣkY)={Sn,Y}.\pi_n^s(Y) = \operatorname{colim}_k \, \pi_{n+k}(\Sigma^k Y) = \set{S^n, Y}.

Remarks

For k≥2k \ge 2 the set [ΣkX,ΣkY]∗[\Sigma^k X, \Sigma^k Y]_* is an abelian group, so the colimit is an abelian group. Let YY be (c−1)(c-1)-connected and let XX have dimension dd. The Freudenthal suspension theorem shows that the maps in the colimit are bijective once d+k<2(c+k)−1d + k < 2(c + k) - 1, that is, once k>d−2c+1k > d - 2c + 1. So the colimit is reached at a finite stage.

Example: π0s(S0)=ℤ\pi_0^s(S^0) = \Z, since πk(Sk)=ℤ\pi_k(S^k) = \Z for all k≥1k \ge 1 and suspension preserves degree. Also π1s(S0)=ℤ/2\pi_1^s(S^0) = \Z/2, generated by the stable class of the Hopf map η\eta.

Non-example: the group π3(S2)≅ℤ\pi_3(S^2) \cong \Z, generated by the Hopf map, is not a stable group. Its suspension lands in π4(S3)≅ℤ/2\pi_4(S^3) \cong \Z/2, and the stable value π1s(S0)\pi_1^s(S^0) is ℤ/2\Z/2.

Today stable homotopy theory is usually set in the stable homotopy category of spectra, where {X,Y}\set{X, Y} becomes a group of morphisms between suspension spectra.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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CW complex, concept, no evidence levelCW complexconceptEquivalence relation, concept, no evidence levelEquivalence relationconceptHomeomorphism, concept, no evidence levelHomeomorphismconceptHomotopy group, concept, no evidence levelHomotopy groupconceptPointed space, concept, no evidence levelPointed spaceconceptProduct topology, concept, no evidence levelProduct topologyconceptQuotient topology, concept, no evidence levelQuotient topologyconceptSuspension, concept, no evidence levelSuspensionconceptTopological space, concept, no evidence levelTopological spaceconceptn-connected space, concept, no evidence leveln-connected spaceconceptSuspension, definition, no evidence levelSuspensiondefinitionFreudenthal suspension theorem, statement, no evidence levelFreudenthal suspensio…statementStable homotopy theory, definition, no evidence levelStable homotopy theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. CW complex concept
  2. Equivalence relation concept
  3. Homeomorphism concept
  4. Homotopy group concept
  5. Pointed space concept
  6. Product topology concept
  7. Quotient topology concept
  8. Suspension concept
  9. Topological space concept
  10. n-connected space concept
  11. Suspension definition
  12. Freudenthal suspension theorem statement
  13. Stable homotopy theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stable_homotopy_theory
About this page
Tag5KERJ85BXJ
URIm2://5KERJ85BXJ
Kinddefinition
WritingW0 stub
MSC55P42, 55Q10
DefinesStable homotopy theory
UsesSuspension, Pointed space, CW complex, Homotopy group, Freudenthal suspension theorem
Changed2 November 2026

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