The Freudenthal suspension theorem (Freudenthal 37) is the following theorem about homotopy groups of n-spheres:
The suspension homomorphism on homotopy groups of spheres
is an isomorphism for $n\gt k+1$.
More generally, for $X$ an n-connected CW-complex, then the suspension homomorphism on homotopy groups
is an isomorphism for $k \leq 2n$.
This follows from the Blakers-Massey theorem (e.g. Kochmann 96, p. 70).
The following more general statement is also often referred to as the Freudenthal suspension theorem:
For $X$ an n-connected CW-complex and $Y$ a CW-complex of dimension $\leq 2n$, then the maps of homotopy classes of continuous functions
are isomorphisms. In particular $[Y,X]$ canonically has the structure of an abelian group.
This follows from the fact that:
If $X$ is an n-connected CW-complex, then the unit
of the (suspension$\dashv$loop space)-adjunction is an isomorphism on $\pi_{\leq 2n}$.
This follows from applying the Serre long exact sequence (which itself follows from the Serre spectral sequence) to the path space fibration (e.g. Kochmann 96, prop. 3.2.2).
The Freudenthal suspension theorem motivated introducing the stable homotopy groups of spheres $\pi_k(S):=\pi_{n+k}(S^n)$, more generally the stable homotopy groups $\pi_k^S(Y) = \pi_{n+k}(\Sigma^n Y)$, both independent of $n$ where $n\gt k+1$, and still more generally the Spanier-Whitehead category, then the stable homotopy category and eventually the stable (infinity,1)-category of spectra.
Due to
Textbook accounts include
Robert Switzer, theorem 6.26 in Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Stanley Kochmann, section 3.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Alan Hatcher, Algebraic Topology, chapter 4 (pdf)
A nice expanded version of the latter is in
A formalization in homotopy type theory in Agda is in
Discussion in equivariant homotopy theory includes