Group and Lie algebra filtrations and homotopy groups of spheres. (English) Zbl 1531.16019
The paper under review deals with group and Lie algebra filtrations and homotopy groups of spheres, as especially the authors establish a bridge between homotopy groups of spheres and commutator calculus in groups, and moreover they solve in this manner the “dimension problem” by providing a converse to the well-known Sjogren’s theorem: every abelian group of bounded exponent can be embedded in the dimension quotient of a group. This is proven by some special embeddings and by using a classical result of J. Wu [Math. Proc. Camb. Philos. Soc. 130, No. 3, 489–513 (2001; Zbl 0986.55013)]. In particular, this invalidates some long-standing results in the literature by utilizing a paramount theorem due to
J.-P. Serre [C. R. Acad. Sci., Paris 231, 1408–1410 (1950; Zbl 0039.39702)]. Moreover, the authors explain in this manner Rips’s famous counterexample to the dimension conjecture in terms of the homotopy group \(\mathbb{Z}/2\mathbb{Z}\). They, finally, obtain analogous results in the context of Lie rings, namely that for every prime \(p\), there exists a Lie ring with \(p\)-torsion in some dimension quotient.
The paper is really quite well written and motivated, and it is one very comprehensive study of the presented subject that will be useful for many specialists in this area.
The paper is really quite well written and motivated, and it is one very comprehensive study of the presented subject that will be useful for many specialists in this area.
Reviewer: Peter Danchev (Sofia)
MSC:
| 16S34 | Group rings |
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 20F14 | Derived series, central series, and generalizations for groups |
| 55Q40 | Homotopy groups of spheres |
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