Hilton's theorem

On the loop space of a wedge of spheres

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

Ω ( Σ X ∨ Σ Y ) ≃ Ω Σ X × Ω Σ Y × Ω Σ ( ⋁ i , j ≥ 1 X ∧ i ∧ Y ∧ j ) . {\displaystyle \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 ∨ S 2 {\displaystyle S^{2}\vee S^{2}} . To put this space in the language of the above formula, we are interested in

Ω ( S 2 ∨ S 2 ) ≃ Ω ( Σ S 1 ∨ Σ S 1 ) {\displaystyle \Omega (S^{2}\vee S^{2})\simeq \Omega (\Sigma S^{1}\vee \Sigma S^{1})} .

One application of the above formula states

Ω ( S 2 ∨ S 2 ) ≃ Ω S 2 × Ω S 2 × Ω Σ ( ⋁ i , j ≥ 1 S i + j ) {\displaystyle \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: Ω S 2 , Ω S 2 , Ω S 3 , Ω S 4 , Ω S 4 {\displaystyle \Omega S^{2},\Omega S^{2},\Omega S^{3},\Omega S^{4},\Omega S^{4}} , giving the result

π 4 ( S 2 ∨ S 2 ) ≃ ⊕ 2 π 4 S 2 ⊕ π 4 S 3 ⊕ ⊕ 2 π 4 S 4 ≃ ⊕ 3 Z 2 ⊕ Z 2 {\displaystyle \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

  • 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


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