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Group and Lie algebra filtrations and homotopy groups of spheres
Journal of Topology ( IF 0.8 ) Pub Date : 2023-06-01 , DOI: 10.1112/topo.12301
Laurent Bartholdi 1 , Roman Mikhailov 2
Affiliation  

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 $s,d$ the torsion of the homotopy group π s ( S d ) $\pi _s(S^d)$ 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 $p$ , there is some p $p$ -torsion in π 2 p ( S 2 ) $\pi _{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group π 4 ( S 2 ) = Z / 2 Z $\pi _4(S^2)=\mathbb {Z}/2\mathbb {Z}$ . We finally obtain analogous results in the context of Lie rings: for every prime p $p$ there exists a Lie ring with p $p$ -torsion in some dimension quotient.

中文翻译:

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 , d $s,d$ 同伦群的挠率 π ( 小号 d ) $\pi _s(S^d)$ 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 $p$ , 有一些 p $p$ -扭转 π 2个 p ( 小号 2个 ) $\pi _{2p}(S^2)$ 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 π 4个 ( 小号 2个 ) = Z / 2个 Z $\pi _4(S^2)=\mathbb {Z}/2\mathbb {Z}$ . 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 $p$ 存在一个李环 p $p$ - 某些维度商的扭转。
更新日期:2023-06-06
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