Let X be an n-connected pointed space (a pointed CW-complex or pointed simplicial set). The map

${\displaystyle X\to \Omega (\Sigma X)}$ 

induces a map

${\displaystyle \pi _{k}(X)\to \pi _{k}(\Omega (\Sigma X))}$ 

on homotopy groups, where Ω denotes the loop functor and Σ denotes the reduced suspension functor. The suspension theorem then states that the induced map on homotopy groups is an isomorphism if k ≤ 2n and an epimorphism if k = 2n + 1.

A basic result on loop spaces gives the relation

${\displaystyle \pi _{k}(\Omega (\Sigma X))\cong \pi _{k+1}(\Sigma X)}$ 

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

${\displaystyle \pi _{k}(X)\to \pi _{k+1}(\Sigma X),}$ 

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

Proof edit

As mentioned above, the Freudenthal suspension theorem follows quickly from homotopy excision; this proof is in terms of the natural map ${\displaystyle \pi _{k}(X)\to \pi _{k+1}(\Sigma X)}$ . If a space ${\displaystyle X}$  is ${\displaystyle n}$ -connected, then the pair of spaces ${\displaystyle (CX,X)}$  is ${\displaystyle (n+1)}$ -connected, where ${\displaystyle CX}$  is the reduced cone over ${\displaystyle X}$ ; this follows from the relative homotopy long exact sequence. We can decompose ${\displaystyle \Sigma X}$  as two copies of ${\displaystyle CX}$ , say ${\displaystyle (CX)_{+},(CX)_{-}}$ , whose intersection is ${\displaystyle X}$ . Then, homotopy excision says the inclusion map:

${\displaystyle ((CX)_{+},X)\subset (\Sigma X,(CX)_{-})}$ 

induces isomorphisms on ${\displaystyle \pi _{i},i<2n+2}$  and a surjection on ${\displaystyle \pi _{2n+2}}$ . From the same relative long exact sequence, ${\displaystyle \pi _{i}(X)=\pi _{i+1}(CX,X),}$  and since in addition cones are contractible,

${\displaystyle \pi _{i}(\Sigma X,(CX)_{-})=\pi _{i}(\Sigma X).}$ 

Putting this all together, we get

${\displaystyle \pi _{i}(X)=\pi _{i+1}((CX)_{+},X)=\pi _{i+1}((\Sigma X,(CX)_{-})=\pi _{i+1}(\Sigma X)}$ 

for ${\displaystyle i+1<2n+2}$ , i.e. ${\displaystyle i\leqslant 2n}$ , as claimed above; for ${\displaystyle i=2n+1}$  the left and right maps are isomorphisms, regardless of how connected ${\displaystyle X}$  is, and the middle one is a surjection by excision, so the composition is a surjection as claimed.

Corollary 1 edit

Let Sⁿ denote the n-sphere and note that it is (n − 1)-connected so that the groups ${\displaystyle \pi _{n+k}(S^{n})}$  stabilize for ${\displaystyle n\geqslant k+2}$  by the Freudenthal theorem. These groups represent the kth stable homotopy group of spheres.

Corollary 2 edit

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.

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• Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, vol. 174, Basel-Boston-Berlin: Birkhäuser.
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.
• Whitehead, G. W. (1953), "On the Freudenthal Theorems", Annals of Mathematics, 57 (2): 209–228, doi:10.2307/1969855, JSTOR 1969855, MR 0055683.