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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:
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776:{\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}}
206:{\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).}
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491:{\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)}
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i.e. the direct-sum of a free abelian group of rank two with the abelian 2-torsion group with 8 elements.
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of a wedge of spaces can be written as an infinite product of loop spaces of suspensions of
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348:{\displaystyle \Omega (S^{2}\vee S^{2})\simeq \Omega (\Sigma S^{1}\vee \Sigma S^{1})}
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One version of the Hilton-Milnor theorem states that there is a homotopy-equivalence
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589:{\displaystyle \Omega S^{2},\Omega S^{2},\Omega S^{3},\Omega S^{4},\Omega S^{4}}
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Here the capital sigma indicates the suspension of a pointed space.
802:(1955), "On the homotopy groups of the union of spheres",
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256:{\displaystyle S^{2}\vee S^{2}}
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47:of loop spaces of spheres.
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39:of a wedge of spheres is
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41:homotopy-equivalent
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63:smash products
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910:expanding it
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27:, proved by
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949:Categories
793:References
59:suspension
37:loop space
826:0024-6107
759:⊕
738:⊕
734:≃
715:π
705:⊕
701:⊕
682:π
678:⊕
659:π
649:⊕
645:≃
629:∨
607:π
574:Ω
558:Ω
542:Ω
526:Ω
510:Ω
460:≥
447:⋁
438:Σ
435:Ω
432:×
419:Ω
416:×
403:Ω
400:≃
384:∨
368:Ω
330:Σ
327:∨
314:Σ
308:Ω
305:≃
289:∨
273:Ω
241:∨
188:∧
180:∧
172:∧
159:≥
146:⋁
137:Σ
134:Ω
131:×
125:Σ
122:Ω
119:×
113:Σ
110:Ω
107:≃
98:Σ
95:∨
89:Σ
83:Ω
879:0445484
847:(ed.),
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219:Example
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900:This
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865:ISBN
822:ISSN
55:1972
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