nLab
Freudenthal suspension theorem

Contents

Statement

The Freudenthal suspension theorem (Hans Freudenthal, 1937) is the following theorem about homotopy groups of spheres:

The suspension homomorphism σ:π n+k(S n)π n+k+l(S n+l)\sigma :\pi_{n+k}(S^n)\to \pi_{n+k+l}(S^{n+l}) is an isomorphism for n>k+1n\gt k+1.

In this statement, one can replace S nS^n with any (n1)(n-1)-connected space YY while replacing S n+lS^{n+l} with the corresponding suspension Σ lY\Sigma^{l} Y.

This theorem justifies introducing the stable homotopy groups of spheres π k(S):=π n+k(S n)\pi_k(S):=\pi_{n+k}(S^n), as well as 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.

Relation to the Blakers-Massey theorem

The Frudenthal suspension theorem is a special case of the Blakers-Massey theorem.

References

A formalization in homotopy type theory in Agda is in

Revised on September 29, 2013 16:23:33 by Urs Schreiber (89.204.130.128)