Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Group and Lie algebra filtrations and homotopy groups of spheres   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, since 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.

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The categories of this paper are: adap-org
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph