We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the “dimension problem” by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by embedding for arbitrary $s,d$ the torsion of the homotopy group ${\pi }_s(S^d)$ into a dimension quotient, via a result of Wu. In particular, this invalidates some long-standing results in the literature, as for every prime $p$, there is some $p$-torsion in ${\pi }_{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group ${\pi }_4(S^2)=\mathbb{Z}/2\mathbb{Z}$. We finally obtain analogous results in the context of Lie rings: for every prime $p$ there exists a Lie ring with $p$-torsion in some dimension quotient.

中文翻译：

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群和李代数过滤和球体的同伦群

我们在球面的同伦群和群中的换向子微积分之间架起一座桥梁，并通过提供 Sjogren 定理的逆定理以这种方式解决“维数问题”：每个有界指数的阿贝尔群都可以嵌入到一个群的维商中。这通过嵌入任意$秒,d$同伦群的挠率${\pi }_{秒}({\mathrm{小号}}^d)$通过 Wu 的结果变成维商。特别是，这使文献中一些长期存在的结果无效，对于每个素数$p$， 有一些$p$-扭转${\pi }_{\mathrm{2个}p}({\mathrm{小号}}^{\mathrm{2个}})$Serre 的结果。我们以这种方式用同伦群来解释里普斯著名的维数猜想反例${\pi }_{\mathrm{4个}}({\mathrm{小号}}^{\mathrm{2个}})=\mathbb{Z}/\mathrm{2个}\mathbb{Z}$. 我们最终在李环的背景下获得了类似的结果：对于每个素数$p$存在一个李环$p$- 某些维度商的扭转。