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Freudenthal suspension theorem

Contents

Statement

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.

This follows 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.

This follows from the fact that:

Proposition

If XX is an n-connected CW-complex, then the unit

XΩΣX X \longrightarrow \Omega \Sigma X

of the (suspension\dashvloop space)-adjunction is an isomorphism on π 2n\pi_{\leq 2n}.

This follows from applying the Serre long exact sequence (which itself follows from the Serre spectral sequence) to the path space fibration (e.g. Kochmann 96, prop. 3.2.2).

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 February 17, 2016 16:25:33 by Urs Schreiber (195.37.209.180)