In algebraic topology, Hilton's theorem, proved by Peter Hilton (1955), states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres.
John Milnor (1972) 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.
One version of the Hilton-Milnor theorem states that there is a homotopy-equivalence
×
∧ i
Here the capital sigma indicates the suspension of a pointed space.
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 (each being a loop space of a sphere), of which only finitely many will have a non-trivial third homotopy group.
Those factors are:
, giving the result
π
π
π
π
, i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.
This topology-related article is a stub.
You can help Wikipedia by expanding it.