Hilton's theorem

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

One version of the Hilton-Milnor theorem states that there is a homotopy-equivalence $\mathrm{\Omega }(\mathrm{\Sigma }X\vee \mathrm{\Sigma }Y)\simeq \mathrm{\Omega }\mathrm{\Sigma }X\times \mathrm{\Omega }\mathrm{\Sigma }Y\times \mathrm{\Omega }\mathrm{\Sigma }(\underset{i,j\ge 1}{\bigvee }{X}^{\wedge i}\wedge Y^{\wedge j}).$ Here the capital sigma indicates the suspension of a pointed space.

Example

Consider computing the fourth homotopy group of $S^2\vee S^2$. To put this space in the language of the above formula, we are interested in $\mathrm{\Omega }(S^2\vee S^2)\simeq \mathrm{\Omega }(\mathrm{\Sigma }{S}^1\vee \mathrm{\Sigma }{S}^1)$. One application of the above formula states $\mathrm{\Omega }(S^2\vee S^2)\simeq \mathrm{\Omega }{S}^2\times \mathrm{\Omega }{S}^2\times \mathrm{\Omega }\mathrm{\Sigma }\left(\underset{i,j\ge 1}{\bigvee }{S}^{i+j}\right)$. 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: $\mathrm{\Omega }{S}^2,\mathrm{\Omega }{S}^2,\mathrm{\Omega }{S}^3,\mathrm{\Omega }{S}^4,\mathrm{\Omega }{S}^4$, giving the result ${\pi }_4(S^2\vee S^2)\simeq {\oplus }_2{\pi }_4{S}^2\oplus {\pi }_4{S}^3\oplus {\oplus }_2{\pi }_4{S}^4\simeq {\oplus }_3{\mathbb{\mathbb{Z}}}_2\oplus {\mathbb{\mathbb{Z}}}^2$, i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.

References

• 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