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.
Explicit Statements[edit]
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[edit]
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
,
i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.
References[edit]
- Hilton, Peter J. (1955), "On the homotopy groups of the union of spheres", Journal of the London Mathematical Society, Second Series, 30 (2): 154–172, doi:10.1112/jlms/s1-30.2.154, ISSN 0024-6107, MR 0068218
- Milnor, John Willard (1972) [1956], "On the construction FK", in Adams, John Frank (ed.), Algebraic topology—a student's guide, Cambridge University Press, pp. 118–136, doi:10.1017/CBO9780511662584.011, ISBN 978-0-521-08076-7, MR 0445484