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. It was proved in 1937 by Hans Freudenthal.

The theorem is a corollary of the homotopy excision theorem.

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_k(X) → π_k(Ω(X ∧ S¹))

on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product. 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

π_k(Ω(X ∧ S¹)) ≅ π_k+1(X ∧ S¹)

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

π_k(X) → π_k+1(X ∧ S¹),

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

Proof

As mentioned above, the Freudenthal suspension theorem follows quickly from homotopy excision; this proof is in terms of the natural map ${\displaystyle \pi _{k}(X)\to \pi _{k+1}(\Sigma X)}$, where $\Sigma$ is the suspension. If a space $X$ is $n$-connected, then the pair of spaces ${\displaystyle (CX,X)}$ is $(n+1)$-connected, where ${\displaystyle CX}$ is the cone over $X$; this follows from the relative homotopy long exact sequence. We can decompose $\Sigma X$ as two copies of ${\displaystyle CX}$, say ${\displaystyle (CX)_{+},(CX)_{-}}$, whose intersection is $X$. Then, homotopy excision says the inclusion map of ${\displaystyle ((CX)_{+},X)\subset (\Sigma X,(CX)_{-})}$ 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. ${\displaystyle i\leq 2n}$, as claimed above; in the case ${\displaystyle i=2n+1}$, the left and right maps are isomorphisms, regardless of how connected $X$ is, and the middle one is a surjection by excision, so the composition is a surjection as claimed.

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.

References

• Freudenthal, H. (1938), "Über die Klassen der Sphärenabbildungen. I. Große Dimensionen", Compositio Mathematica, 5: 299–314.
• Goerss, P. G.; Jardine, J. F. (1999), Simplicial Homotopy Theory, Progress in Mathematics, 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.