Freudenthal suspension theorem

In mathematics, and specifically in the field of homotopy theory, the Freudenthal Suspension Theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question.

Statement of the theorem

Let X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map X → Ω(X ∧ S¹) induces a map π_i(X) → π_i(Ω(X ∧ S¹)) on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product. Then the induced map on homotopy groups is an isomorphism if i ≤ 2n and an epimorphism if i = 2n + 1.

Note that

π_i(Ω(X ∧ S¹)) ≅ π_i+1(X ∧ S¹)

so that the theorem could otherwise be stated in terms of the map

π_i(X) → π_i+1(X ∧ S¹),

with the small caveat that in this case one must be careful with the indexing.

Corollary 1

Let Sⁿ denote the n-sphere and note that it is (n-1)-connected so that the groups π_n+k(Sⁿ) stabilize for n ≥ k + 2 by the Freudenthal theorem. These groups represent the kth stable homotopy group of spheres.

Corollary 2

More generally, for fixed k ≥ 1, k ≤ 2n for sufficiently large n, so that any n-connected space X will have corresponding stabilized homotopy groups. These groups are actually the homotopy groups of an object corresponding to X in the stable homotopy category.

Reference

P.G. Goerss and J.F. Jardine, "Simplicial Homotopy Theory", Progress in Mathematics Vol. 174, Birkhäuser Basel-Boston-Berlin (1999).