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.

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

induces a map

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

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

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

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

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