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Freudenthal suspension theorem

Contents

Statement

Lemma

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. Kochmann 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

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 1.

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

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

Proposition

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.

Properties

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.

References

Due to

  • Hans Freudenthal, Über die Klassen der Sphärenabbildungen, Compositio Math., 5:299-314, 1937.

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

Revised on May 10, 2016 10:26:44 by Urs Schreiber (131.220.184.222)