Freudenthal suspension theorem in nLab

Contents

Contents

Statement

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

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

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:

(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 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.

• suspension isomorphism

• stem of homotopy groups of spheres

References

Due to

• Hans Freudenthal, Über die Klassen der Sphärenabbildungen, Compositio Math., 5:299-314, 1937 (numdam:CM_1938__5__299_0)

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)