Hilton's theorem

In algebraic topology, Hilton's theorem, proved by {{harvs|txt|first=Peter|last=Hilton|authorlink=Peter Hilton|year=1955}}, states that the loop space of a wedge of spheres is homotopy-equivalent to a product of loop spaces of spheres.

{{harvs|txt|first=John|last=Milnor|authorlink=John Milnor|year=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

$\Omega(\Sigma X \vee \Sigma Y) \simeq \Omega \Sigma X \times \Omega \Sigma Y \times \Omega \Sigma \left( \bigvee_{i,j \geq 1} X^{\wedge i} \wedge Y^{\wedge j} \right).$

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

$\Omega (S^2 \vee S^2) \simeq \Omega ( \Sigma S^1 \vee \Sigma S^1)$ .

One application of the above formula states

$\Omega (S^2 \vee S^2) \simeq \Omega S^2 \times \Omega S^2 \times \Omega \Sigma \left( \bigvee_{i,j \geq 1} S^{i+j} \right)$ .

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: $\Omega S^2, \Omega S^2, \Omega S^3, \Omega S^4, \Omega S^4$ , giving the result

$\pi_4(S^2 \vee S^2) \simeq \oplus_2 \pi_4 S^2 \oplus \pi_4 S^3 \oplus \oplus_2 \pi_4 S^4 \simeq \oplus_3 \mathbb Z_2 \oplus \mathbb Z^2$ ,

i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.

References

{{Citation | last1=Hilton | first1=Peter J. | authorlink=Peter Hilton | title=On the homotopy groups of the union of spheres | doi=10.1112/jlms/s1-30.2.154 | mr=0068218 | year=1955 | journal=Journal of the London Mathematical Society |series=Second Series | issn=0024-6107 | volume=30 | issue=2 | pages=154–172}}
{{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | editor-last=Adams | editor-first=John Frank |editor-link=Frank Adams| title=Algebraic topology—a student's guide | orig-date=1956 | publisher=Cambridge University Press | isbn=978-0-521-08076-7 | doi=10.1017/CBO9780511662584.011 | mr=0445484 | year=1972 | chapter=On the construction FK | pages=118–136}}

Category:1955 in science

Category:Theorems in algebraic topology

Category:20th century in mathematics