Hilton's theorem

In algebraic topology, Hilton's theorem, proved by, states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres.
showed more generally that the loop space of the suspension of a wedge of spaces can be written as an infinite product of loop spaces of suspensions of smash products.

Explicit Statements

One version of the Hilton-Milnor theorem states that there is a homotopy-equivalence
Here the capital sigma indicates the suspension of a pointed space.

Example

Consider computing the fourth homotopy group of. To put this space in the language of the above formula, we are interested in
.
One application of the above formula states
From this one can see that inductively we can continue applying this formula to get a product of spaces, of which only finitely many will have a non-trivial third homotopy group. Those factors are:, giving the result
,
i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.