homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Let $X$ be an n-connected topological space. Then the adjunction unit of the (suspension $\dashv$ loop space)-adjunction
is (2n+1)-connected.
(e.g. Kochmann 96, prop. 3.2.2)
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 $k \leq 2n$.
The suspension isomorphism is equivalently given by the map in lemma 1.
e.g. (Switzer 75, 6.26)
An alternative proof proceeds 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.
(e.g. Kochmann 96, corollary 3.2.3)
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
Allen 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