nLab Freudenthal suspension theorem



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





Let XX be an n-connected topological space. Then the adjunction unit of the (suspension \dashv loop space)-adjunction

XΩΣX X \longrightarrow \Omega \Sigma X

is (2n+1)-connected.

(e.g. Kochman 96, prop. 3.2.2)

The Freudenthal suspension theorem (Freudenthal 37) is the following theorem about homotopy groups of n-spheres:


The suspension homomorphism on homotopy groups of spheres

π n+k(S n)π n+k+1(S n+1) \pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})

is an isomorphism for n>k+1n\gt k+1.

More generally, for XX an n-connected CW-complex, then the suspension homomorphism on homotopy groups

π k(X)π k+1(ΣX) \pi_k(X) \longrightarrow \pi_{k+1}(\Sigma X)

is an isomorphism for k2nk \leq 2n.

The suspension isomorphism is equivalently given by the map in lemma .

e.g. (Switzer 75, 6.26)

An alternative proof proceeds from the Blakers-Massey theorem (e.g. Kochman 96, p. 70).

The following more general statement is also often referred to as the Freudenthal suspension theorem:


For XX an n-connected CW-complex and YY a CW-complex of dimension 2n\leq 2n, then the maps of homotopy classes of continuous functions

[Y,X][ΣY,ΣX][Σ 2Y,Σ 2X] [Y,X]\stackrel{}{\longrightarrow} [\Sigma Y, \Sigma X] \stackrel{}{\longrightarrow}[\Sigma^2 Y, \Sigma^2 X]

are isomorphisms. In particular [Y,X][Y,X] canonically has the structure of an abelian group.

(e.g. Kochman 96, corollary 3.2.3)


As motivation for stable homotopy theory

The Freudenthal suspension theorem motivated introducing the stable homotopy groups of spheres π k(S):=π n+k(S n)\pi_k(S):=\pi_{n+k}(S^n), more generally the stable homotopy groups π k S(Y)=π n+k(Σ nY)\pi_k^S(Y) = \pi_{n+k}(\Sigma^n Y), both independent of nn where n>k+1n\gt k+1, and still more generally the Spanier-Whitehead category, then the stable homotopy category and eventually the stable (infinity,1)-category of spectra.


Due to

Textbook accounts include

A nice expanded version of the latter is in

  • Tengren Zhang, Freudenthal suspension theorem (pdf)

A formalization in homotopy type theory in Agda is in

Discussion in equivariant homotopy theory includes

Last revised on February 26, 2021 at 16:59:18. See the history of this page for a list of all contributions to it.