nLab Freudenthal suspension theorem

Statement

Lemma

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

$$X \longrightarrow \Omega \Sigma X$$

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:

Proposition

The suspension homomorphism on homotopy groups of spheres

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

is

1. an isomorphism for $n \gt k+1$,

2. an epimorphism for $n = k + 1$.

Equivalently,

$$\pi_{i}(S^n) \longrightarrow \pi_{i+1}(S^{n+1})$$

is

1. an isomorphism for $i \lt 2n - 1$,

2. an epimorphism for $i = 2n -1$.

(cf. Toda 1962 (3.2) p. 32)

More generally:

Proposition

For $X$ an n-connected CW-complex, then the suspension homomorphism on homotopy groups

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

is an isomorphism for $k \leq 2n$.

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

(cf. Switzer 1975, 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:

Proposition

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

$$[Y,X]\stackrel{}{\longrightarrow} [\Sigma Y, \Sigma X] \stackrel{}{\longrightarrow}[\Sigma^2 Y, \Sigma^2 X]$$

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

(e.g. Kochman 96, corollary 3.2.3)

Properties

As motivation for stable homotopy theory

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

Related concepts

• suspension isomorphism

• stem of homotopy groups of spheres

References

Due to:

• Hans Freudenthal: Über die Klassen der Sphärenabbildungen, Compositio Math., 5 (1937) [numdam:CM_1938__5__299_0]

Early further discussion:

• Hirosi Toda; p. 32 of: Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies 49, Princeton University Press (1962) [doi:10.1515/9781400882625, jstor:j.ctt1bgzb5t]

Textbook accounts include

• Robert Switzer, theorem 6.26 in Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

• Stanley Kochman, section 3.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Allen Hatcher, Algebraic Topology, chapter 4 (pdf)

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

• Peter Lumsdaine, Dan Licata, Freudenthal.agda

Discussion in equivariant homotopy theory includes

• John Greenlees, Peter May, p. 7,8 of Equivariant stable homotopy theory (pdf)