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$s,d$ the torsion of the homotopy group πs(Sd)$\pi _s(S^d)$ 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 π2p(S2)$\pi _{2p}(S^2)$ by a result of Serre. We explain in this manner Rips's famous counterexample to the dimension conjecture in terms of the homotopy group π4(S2)=Z/2Z$\pi _4(S^2)=\mathbb {Z}/2\mathbb {Z}$ . 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.


INTRODUCTION
The fundamental problem of combinatorial group theory can be phrased as: "Given a group presented as a quotient of a free group, what can be said of quotients of itself?". Of main importance are those quotients produced by universal constructions; prominently the maximal nilpotent In memoriam John R. Stallings, 1935Stallings, -2008 torsion which we expect to be injective under mild finiteness conditions on . Our main result comes from the space = 2 ∨ 2 .
Homotopy groups of spheres are so fundamental objects that they pervade topology, with applications ranging from Brouwer's fixed-point theorem to Rokhlin's theorem on signatures of spin 4-manifolds. Their nontriviality and finiteness (apart from the ( ) and 4 −1 ( 2 )) are among the most profound results of mathematics. Theorem A shows that they are also tightly linked to a question in pure algebra.
Cohen, Wu and their coauthors revealed deep links between combinatorial group theory and homotopy [9-11, 30, 34, 62, 63]. At the heart of our method is a formula by Wu, expressing homotopy groups of spheres as quotients between two subgroups of a finitely generated free group. It is based on simplicial sets and corresponding simplicial groups, see May's fundamental reference [42].
Topological methods are inherent to the modern study of group theory, as witnessed by the monumental treatises by Gromov [21], Bridson and Haefliger [4], and Geoghegan [19]. Stallings [58, p. 117], in a program carried out by Sjogren [57], already recognized the value of homological arguments toward studying dimension quotients.
Nonetheless, Theorem A is the first instance of a classical problem in algebra that is solved using higher algebraic topology, in effect harnessing the powerful instruments of Steenrod algebra and spectral sequences, notably the Adams, Curtis, and May spectral sequences.

History of the dimension problem
The dimension problem has a long history, starting with Magnus's investigation of the lower central series of a free group, and its associated Lie algebra [39]; he showed with Witt that the dimension property holds for free groups, see [40,61], attributing the first proof to Grün [22]. For a small subset of the literature, we refer to [24,25,43,49,59], and for further historical remarks to [52]. Remarkably, incorrect proofs of the dimension problem appeared more than once, by Cohn [12], Losey [37] (Lyndon remarks dryly, in his MathSciNet review, "The main content of this paper is another incomplete proof that the (integral) dimension subgroups of an arbitrary group are the terms of its lower central series"), and even Magnus himself [41, Theorem 5.15(i)]! We note that if one replaces the ring ℤ by a field, then there is an elegant and elementary description of the corresponding dimension subgroup, depending only on the field's characteristic; see [31,32].
The dimension problem is quantitatively studied in terms of the quotients ( )∕ ( ) called dimension quotients. Gupta and Kuzmin proved in [23] that they are all abelian, and Sjogren proved in [57] that they have finite exponent, bounded by a function of only: there exists a minimal ( ) ∈ ℕ (at most ( !) ) such that ( ) ( ) ⊆ ( ) ⩽ ( ) holds for all groups .
This can be improved in the case of metabelian groups: Gupta proved in [26] that ( ) is a power of 2. He then claimed that ( ) is a power of 2 for all groups, a proof is published in [28]; and even that it may be improved to ( ) = 2 for all ⩾ 4, see [27]. However, many parts of his arguments were never fully understood. Now our main result, stated above, shows that the function ( ) is unbounded, and its values cannot even be constrained to a finite collection of primes.

Lie rings
An variant of the dimension problem may be asked for Lie rings; namely, Lie algebras over ℤ. Every Lie ring embeds in its universal enveloping algebra ( ), which also admits an augmen-tation ideal. The dimension subrings are defined analogously by ( ) = ∩ , see [2]. Again ( ) = ( ) when ⩽ 3, and there is a Lie ring with ( )∕ ( ) = ℤ∕2. Sjogren's bound also holds for Lie rings [56], and many details are simpler in the category of Lie rings.
Even though we are not aware of any direct construction of a group from a Lie ring or vice versa that preserves dimension quotients, it often happens that a presentation involving only powers and commutators, which may therefore be interpreted either as group or Lie algebra presentation, yields isomorphic dimension quotients.
It is even possible to write a 3-related Lie algebra, based on [

Homotopy groups of the two-sphere: Main statement and sketch of proof
According to a result of Wu [15,62], we may express the homotopy groups of spheres +1 ( 2 ) as a quotient of two normal subgroups in a free group. More precisely, write = ⟨ 0 , … , | 0 ⋯ = 1⟩ a free group of rank with one redundant generator, and for = 0, … , let denote the normal closure of in . We write iterated commutators as left-normed: [ 1 , 2 ], … ], ], and denote by Σ +1 the symmetric group on {0, … , }. Then We can now state more precisely the main step toward our result: Theorem 1.1′. Given an integer ⩾ 3, there is for all large enough a group and a set-wise map → inducing via (1.2) an injective homomorphism In particular, the exponent of ( )∕ ( ) is divisible by that of +1 ( 2 ).
The spectral sequence * * , converges to +1 ( 2 ), so for = 2 − 1 there is an order-term ∈ 1 +1, that survives as the Serre element in +1 ( 2 ). We are able to write it explicitly in terms of a shuffle product.
Note that only a finite number of roots of group elements are required, though it seems messy to specify exactly which ones. Theorem 1.2. Given an integer ⩾ 3, there are integers , 0 , … , and = 0 + ⋯ + such that, in the Lie ring The must only satisfy some linear inequalities, and every large enough may be obtained.
The constants and are somewhat explicit, based on the exponent of +1 ( 2 ) and the connectivity of certain simplicial groups. We have determined tighter values for = 2 and = 3, see Section 9.
Here is a sketch of the proof in the group case; the Lie algebra case is essentially the same, and slightly simpler. The first claim follows from the fact that is a free product with amalgamation. For this last part, we first invoke Curtis' connectedness theorem [13], from which there is an integer such that ( ) ∩ 0 ∩ ⋯ ∩ ⩽ ∏ [ (0) , … , ( ) ]. It therefore suffices to prove We then use the particular form of the presentation of : consider an element of , identified with its image in , and write it in terms of commutators g = [ 1 , … , ] with each ∈ { 0 , … , }, for some < . This element g seemingly defines an element of ( ), but in it may be rewritten, in the presence of sufficiently high powers of , into a commutator of higher weight by replacing some by the corresponding . If the are chosen such that 0 ⩾ and ∕ −1 ⩾ for all , then in order for g to belong to ( ) either each must have been replaced at least once by a , so the commutator had to belong to some [ (0) , … , ( ) ], or a larger power of is required, and therefore the original term in itself was a power of . In this manner we obtain g ∈ ∏ [ (0) , … , ( ) ] ( ). In effect, we use two filtrations on , by the lower central series and by powers of . Only the substitution ⇝ increases much the degree in the first filtration; but it consumes a high degree in the second. All the other substitutions ⇝ for < involve a trade-off between how much they increase either degree, and each one requires the previous one. Finally, the substitutions ⇝ or ⇝ decrease the first degree too much to be of any use in attaining ( ). We derive in Theorem 8.1 an expression for the -torsion of 2 ( 2 ), first at the level of Lie algebras, namely on the first page of the Curtis spectral sequence, and deduce in Proposition 8.2 some properties of its representation˜as an element of the free group 2 −1 . We make use of an explicit form for = 2 and = 3 to obtain smaller examples, in particular for = 2 we obtain straightforward constructions, for arbitrary ⩾ 4, of Lie algebras in which ∕ contains 2-torsion, and for = 3 we obtain a Lie algebra and a group in which 7 ∕ 7 contains 3-torsion. These examples have also been checked using computer algebra software.

Wedges of spheres
An analogous statement to (1.2) holds for wedges of two-spheres, and even more generally for suspensions of spaces with contractible universal cover. We restrict ourselves here to the space 2 ∨ 2 , which is sufficient to prove Theorem A.
Consider the group = * , and identify the generators of its factors as , for = 0, … , and ∈ {1, 2}; set = ⟨ ,1 , ,2 ⟩ . Then Note that only a finite number of roots of group elements are required, though it seems messy to specify exactly which ones. Note also that the group constructed in Theorem 1.3 is, apart from the adjunction of roots , the free product of two copies of the group constructed in Theorem 1.1.
There is a Lie algebra analogue to Theorem 1.3, which we do not state because it does not seem to have any applications. In particular, we do not know whether there exists a Lie algebra such that one of its dimension quotients contains ℤ∕ 2 ℤ-torsion for some prime . Indeed the torsion in the first page of the spectral sequence converging to * ( 2 ∨ 2 ) has only prime orders. There seems to be a fundamental difference, here, between groups and Lie algebras.
The "bounded exponent" statement is due to Sjogren. In his notation, let denote the least common multiple of {1, … , }, and define integers and recursively by Then for any group we have ( ) ⊆ ( ). By [56], the same result holds in Lie algebras: for any Lie algebra we have ( ) ⊆ ( ). Gupta and Kuzmin prove even that the quotient ( )∕ +1 ( ) is abelian; following [43] the same result is easily seen to hold in Lie algebras. We reproduce the argument, in the Lie algebra case, because of its simplicity: Proof (that ( )∕ +1 ( ) is abelian). Let be nilpotent of class ; we are to show that ( ) is abelian. Let be maximal abelian normal in ; so is an -module via adjunction. For any ∈ , ∈ ( ) we have [ , ] ∈ +1 ( ), so ( ) centralizes . Now as is maximal it is self-centralizing, so ( ) ⩽ and therefore is abelian. □

HOMOTOPY GROUPS OF SPHERES
We describe in this section the group-theoretic and Lie-algebra-theoretic formulations of homotopy groups of spheres. They will be essential in the proofs of Theorems 1.2 and 1.1. We use " " in this section for what is written " " in the rest of the text, to avoid confusion with the degeneracies in simplicial objects.

Groups
Fix an integer ⩾ 1 and let = ⟨ 0 , … , | 0 ⋯ = 1⟩ be a free group of rank . Consider its normal subgroups Note that is the fundamental group of a 2-sphere with + 1 punctures, and contains the conjugacy class of a loop around the th puncture; the operation of filling-in the th puncture induces the map → ∕ on fundamental groups.
Denote by Σ +1 the symmetric group on {0, … , }, and define the symmetric commutator product of the above subgroups by Here and below the iterated commutators are assumed to be left-normalized, namely, 2 ], and so on.
We view the circle 1 as a simplicial set. Milnor's construction produces a group complex, having in degree a free group on the degree-objects of 1 subject to a single relation ( 0 ( * ) = 1) and the same boundaries and degeneracies as 1 . According to a formula due to Jie Wu [15,62], considered in the standard basis of Milnor's [ 1 ]-construction, homotopy groups of the sphere 2 can be presented in the following way: Consider now for = 0, … , the ideals ∶= ( − 1)ℤ[ ] in the free group ring ℤ[ ], and their symmetric product which is also an ideal in ℤ[ ].
Proof. It is shown in [44] that the quotient 0 ∩⋯∩ The lower map is the th Hurewicz homomorphism for the loop space Ω 2 . As all homotopy groups (Ω 2 ) are finite for ⩾ 3, but all homology groups (Ω 2 ) are infinite cyclic ( * (Ω 2 ) is the tensor algebra generated by the homology of 1 in dimension one [3]), we conclude that, for ⩾ 3, the map in the above diagram is zero. □

Lie algebras
One obtains an analogous picture in the case of Lie algebras over ℤ. The homotopy groups of the simplicial Lie algebra are equal to the direct sum of terms in rows of the 1 -term of the Curtis spectral sequence The mod--lower central series spectral sequence is well-studied, see, for example, the foundational paper [7]. The integral case that we consider here has similar properties, see [6,33]. Here we will only need elementary properties of this spectral sequence.
Observe that the 1 -page of the above spectral sequence consists of derived functors in the sense of Dold-Puppe, applied to Lie functors: if L denotes the th Lie functor in the category of abelian groups, then ( Recall the definition of derived functors. Let be an abelian group, and let be an endofunctor on the category of abelian groups. For every , ⩾ 0 the derived functors of in the sense of Dold-Puppe [14] are defined by where * → is a projective resolution of , and is the Dold-Kan transform, inverse to the Moore normalization functor from simplicial abelian groups to chain complexes. We denote by ( , ) the object ( * [ ]) in the homotopy category of simplicial abelian groups determined by ( * [ ]), so that ( , ) = ( ( , )). Consider a free Lie algebra over ℤ with generators 0 , … , and relation 0 + ⋯ + = 0, and the Lie ideals Define their symmetric product by The same arguments as in the group case imply the Lie analog of Wu's formula: In fact, but we shall not need this, each term 1 , may be singled out by filtering via its lower central series: one has Consider the universal enveloping algebra ( ), the corresponding ideals ∶= ( ) in ( ), and their symmetric product: Proof. Similarly to the group case, the natural map → ( ) induces By [53], the 1 , -terms of the lower central series spectral sequence for 2 are finite for all ⩾ 3, while the universal enveloping simplicial algebra ( [ 1 ]) has infinite cyclic homology groups in all dimensions. It follows that the map is 0. □

PROOF OF THEOREM 1.2
We begin with the proof of Theorem 1.2 on Lie algebras, as it is slightly simpler, while conceptually similar, to the corresponding statement for groups. Let an integer ⩾ 3 be fixed throughout this section.

Fourth claim: is injective
It is time to specify more precisely the admissible parameters in the construction of . The parameter is the exponent of ⨁ 1 , , or any multiple thereof. By the Curtis connectivity theorem [13], or more precisely its variant for Lie algebras, there is an integer such that ( ) ∩ 0 ∩ ⋯ ∩ ⩽ [ 0 , … , ] Σ : let us quickly sketch the argument. For any connected free simplicial group , consider the associated Lie algebra We apply this to = [ 1 ] and its Lie algebra We may choose the parameters 0 , … , arbitrarily so long as 0 ⩾ and ⩾ −1 for all = 1, … , . To fix matters, let us choose = +1 .
As is injective, we are to prove −1 ( ( )) ∩ 0 ∩ ⋯ ∩ ⩽ [ 0 , … , ] Σ . We note that ∩ = for all , so 0 ∩ ⋯ ∩ ∩ = ( 0 ∩ ⋯ ∩ ). By the choice of and , it therefore suffices to prove Consider the free graded Lie algebra = ⟨ 0 , … , , 0 , … , ⟩, in which has degree : it admits a natural surjection ∶ ↠ . Given ∈ ⟨ 0 , … , ⟩ ⩽ , consider the collection of expressions in −1 ( ( )) that are obtained by replacing, in an expression of , some terms by the corresponding , adjusting appropriately the power of . We call in x-form, and the corresponding expressions obtained by replacing some by are called in xy-form. We also denote by the natural map ⟨ 0 , … , ⟩ ⊆ ↠ ; we have • = . Let us consider ∈ 0 ∩ ⋯ ∩ , and assume ( ) ∈ ( ). We shall write = 0 + 1 + 2 , with 0 ∈ [ 0 , … , ] Σ and 1 ∈ and 2 ∈ ( ). Now by assumption ( ) may be written as an element̃∈ , all of whose terms have degree at least ; we express this in two steps: first, ( ) gives rise to an x-form ∈ by application of the relation 0 + ⋯ + = 0; and then gives rise to an xy-form̃of by application of the other relations, namely, replacement of by with absorption of in the coefficient. Indeed it follows from the form of as an amalgamated free product that the xy-form̃may be obtained from first by selecting an appropriate x-form using the relation 0 + ⋯ + = 0, and then converting it tõ; thus there is a natural bijection between the summands of and̃.
As is graded and free, we may writẽin a standard basis of free Lie algebras, such as a selection of left-normed commutators. Let us consider in turn all summands of̃, a typical one being of the form̃∶= [ 1 , … , ] with all ∈ { 0 , … , , 0 , … , }; let be the corresponding monomial in .
If ⩾ , we put ( ) in 2 . We may therefore from now on suppose < . On the other hand, say for = 0, … , that of the 's are the generator , and that ∞ of the 's are in { 0 , … , }; then the weight of̃is Combining 0 ⩽ < for all with ⩾ −1 and 0 ⩾ , we see by unicity of the baserepresentation of an integer that either 0 = ⋯ = = 1 or 0 0 + ⋯ + > . In the former case, each of the 0 , … , iñmay be replaced by the corresponding 0 , … , to produce a monomial in with coefficient multiplied by 0 +⋯+ ; and then this summand belongs to ([ 0 , … , ] Σ ). Add ( ) to 0 .
In the latter case, replace again all iñby the corresponding to produce the monomial in with coefficient multiplied by 0 0 +⋯+ . Remembering that its coefficient is divisible by +1 , add ( ) to 1 .
We have in this manner expressed in the required form 0 + 1 + 2 , concluding the proof that is injective.

PROOF OF THEOREM 1.1
The proof of Theorem 1.1 follows closely that of the previous section; the main difference is that the connection between a group and its group ring is not quite at tight as that between an algebra and its universal enveloping algebra. We remedy this issue by adding roots of elements of ⟨ 0 , … , ⟩. Note that only finitely many elements need a root, but we added all for simplicity of the argument. Let an integer ⩾ 3 be fixed throughout this section. We begin by specifying more precisely the admissible parameters in the construction of . The parameter is the exponent of +1 ( 2 ), or any multiple thereof. By [13], there is an integer such that ( ) ∩ 0 ∩ ⋯ ∩ ⩽ [ 0 , … , ] Σ . We then choose 0 , … , as before subject to 0 , ∕ −1 ⩾ , for example, = +1 .

First claim: exists and is injective
The assignment ( 0 , … , ) ↦ ( 0 , … , ) naturally defines a map ∶ → , as 's only relator holds in . Furthermore, is an iterated amalgamated free product, to wit start with ⟨ 0 , … , | 0 ⋯ = 1⟩ and repeatedly amalgamate, for = 0, … , , with ⟨ , (1 ⩽ ⩽ )⟩ along a cyclic subgroup. Then amalgamate, for ∈ ⟨ 0 , … , ⟩, with ⟨ ⟩ along a cyclic subgroup. By the standard normal form theorem for free products with amalgamation (see [ do not quite hold in groups, but in the presence of sufficient roots a close analogue exists. For g in a group we denote by g the normal closure of ⟨g⟩ in , and by [g, ( )ℎ] the iterated commutator [g, ℎ, … , ℎ] with copies of "ℎ":

Fourth claim: is injective
The argument is essentially the same as in the Lie algebra case, so we only indicate the differences. We are to prove We again start with ∈ 0 ∩ ⋯ ∩ and assume ( ) ∈ ( ), and write it as = 0 ⋅ 1 ⋅ 2 with 0 ∈ [ 0 , … , ] Σ and 1 ∈ and 2 ∈ ( ). Again we write ( ) as̃in the free group with generators 0 , … , , 0 , … , , and let be the corresponding x-form of̃. These elements are written as left-normed commutators, more precisely as left-normed commutators of generators if their length is < , and as arbitrary commutators of length ⩾ . As before, those commutators of length ⩾ are gathered in 0 . For each of the remaining ones iñ, again the number of generators in them is denoted by , and the number of other generators is denoted by ∞ ; and (4.1) still holds.
If all ⩾ 1, then we get a term in 0 , while if some = 0 then (4.1) is strict, and we consider the equation coming from the exponents. Now the other generators are either or their roots ; let us write ∞ = ′ ∞ + ′′ ∞ with ′ ∞ the number of generators in { 0 , … , }. Then, in converting a term into its x-form, the exponent gets multiplied by as all are divisible by and ′′ ∞ < . Therefore, we again get a term in 1 .

PROOF OF THEOREM 1.3
We begin by an analogue of Proposition 3.1. Dropping the "overlines" from our notation, consider the group = ⟨ 0,1 , 0,2 , … , ,1 , ,2 | 0,1 ⋯ ,1 = 0,2 ⋯ ,2 = 1 =⟩ and its normal subgroups = ⟨ ,1 , ,2 ⟩ for = 0, … , . Consider also the ideal = ( − 1)ℤ . The lower map is the th Hurewicz homomorphism for the loop space Ω( 2 ∨ 2 ). By [3], the lower right term is the degree-part of the free associative algebra on two generators (the homology of 1 ∨ 1 ). The homotopy group +1 ( 2 ∨ 2 ) is the sum of its torsion and torsion-free part, and the torsion-free part is the degree-part of the free Lie algebra on two generators (also the homology of 1 ∨ 1 ). The lower map, on the torsion-free part, is the natural inclusion of the free Lie algebra into the free associative algebra, so the kernel of the lower map coincides with the torsion subgroup of +1 ( 2 ∨ 2 ). □ Combining Serre's finiteness theorem and Hilton's theorem [29], the torsion of +1 ( 2 ∨ 2 ) is finite, and in particular has bounded exponent . We may also apply Curtis's theorem: there is an integer such that ( ) ∩ 0 ∩ ⋯ ∩ = 1. The proof of Theorem 1.3 then proceeds exactly as that of Theorem 1.1, with Proposition 6.1 used as a replacement of Proposition 3.1.

PROOF OF THEOREM A
Let be an abelian group of bounded exponent. We begin by recalling Prüfer's "first" theorem [50]: every abelian group of bounded exponent is a direct product of cyclic groups. Now clearly so it suffices to prove Theorem A for cyclic . We recall next Hilton's theorem [29]: Therefore, in particular every +1 ( ) is a direct summand of +1 ( 2 ∨ 2 ). We finally recall Gray's theorem [20], proving that the exponent bound of Cohen-Moore-Neisendorfer is optimal: arbitrary cyclic groups appear as subgroups of some +1 ( ).
It follows that for every cyclic group there is an integer such that +1 ( 2 ∨ 2 ) contains a copy of . We then conclude by Theorem 1.3.

THE SERRE ELEMENT IN ( )
Let be a prime. In this subsection, we describe explicitly a copy of ℤ∕ in 2 ( 2 ) due to Serre [55], by computing its (pre)image in the 1 -term of the lower central spectral sequence associated to [ 1 ]. There is a single (ℤ∕ )-term in dimension 2 − 1 of the spectral sequence p-torsion Proof. Consider the free abelian simplicial group (ℤ, 2): it has a single generator in degree 2, and its other generators may be chosen to be all iterated degeneracies of . We will use the dual notation for generators: for > 2 the free abelian group (ℤ, 2) is generated by ordered sequences of two elements  where the left-hand map is given by Applying the functors ↪ −1 ⊗ ↠ to the simplicial abelian group (ℤ, 2 ), and taking the homotopy groups, we get the long exact sequence ) .
It follows from [14, p. 307] that the above sequence has the following form: By [53,Proposition 4.7], the natural epimorphism L ↠ gives a natural isomorphism of derived functors Let us first find a simplicial generator of 2 (ℤ, 2). For this, we observe that the inclusion of the symmetric power into the tensor power ↪ ⊗ induces an isomorphism of derived functors 2 (ℤ, 2) → 2 ⊗ (ℤ, 2).
A simplicial generator of 2 ⊗ (ℤ, 2) can be given by the Eilenberg-Zilber shuffle-product theorem. Using interchangeably the notation ( ) and , this is the element ∑ It follows immediately from the definition of 2 -shuffles that the symmetric group Σ , acting by permutation on blocks {2 , 2 + 1} of size 2, acts on 2 -shuffles. A generator of 2 (ℤ, 2) can be chosen by keeping only a single element per Σ -orbit, and replacing tensor products by symmetric products: The conditions imply (2 − 1) = 2 − 1. For example, for = 3 we get the element Observe that we have with the understanding that ( ) = 0, that we use the same notation ( 1 2 ) for elements of varying degree, and that 0 (0 2 ) = 0 and ( 1 2 ) .

Now the sum in (⋯)
is a symmetric product similar to , but with − 1 instead of factors, so (⋯) is exact. The second terms telescope, so we get (̃) = 0 when < 2 − 2. However,̃is not a cycle in −1 (ℤ, 2) ⊗ (ℤ, 2), because 2 −2 (̃) is not zero: we compute We use the long exact sequence associated with (8.1) to obtain a cycle in (ℤ, 2) 2 −1 . The ascending 2 -shuffles ( (0), … , (2 − 1)) appearing in the sum can in fact be viewed as Up to sign and renumbering, this is exactly our element . □ Note that we considered, in the beginning of this section, a free Lie algebra of rank 2 − 1 with 2 generators 0 , … , 2 −1 subject to the relation ∑ = 0. Any choice of 2 − 1 out of these 2 generators yields a free Lie algebra on 2 − 1 generators, and an expression . The point being made is that every such expression involves one of the generators (here 2 −2 ) twice, and omits another (here 2 −1 ). We summarize the properties of the element that will be useful to us as follows. Proof. The first claim follows from Proposition 3.1, as represents an element of 2 ( 2 ). The second claim holds because this element is nontrivial in 2 ( 2 ). The third claim holds because it has order in 2 ( 2 ). The last claim follows from general facts: L (ℤ, 1) = 0 for odd , and L 2 (ℤ, 1) = L (ℤ, 2). □ The same statement holds for Lie algebras; we omit the proof.

Example 8.5.
Here is a generator of the 3-torsion in 6 ( 2 ). For = 3, we have six [2,2]-shuffles in Theorem 8.1:   Again it is possible (but now with considerably more effort) to lift 3 to a generator of 6 ( 2 ) in terms of free groups. We return to the notation of simplicial free groups: we consider the free group = ⟨ 0 , … , 4 ⟩ and normal subgroups 0 = ⟨ 0 ⟩ , = ⟨ −1 −1 ⟩ for ∈ {1, … , 4} and 5 = ⟨ 4 ⟩ . In other words, we set ∶= 0 ⋯ . Here is a lift of 3 to that defines a simplicial cycle, that is, which lies in the intersection 0 ∩ ⋯ ∩ 5 : it is the product of the following fourteen elements One can directly check that˜3 defines a simplicial cycle and that modulo the seventh term of the lower central series it represents exactly the element 3 . Remark 8.6. We have 6 ( 2 ) = ℤ∕3 × ℤ∕4, and it is also possible to give an explicit generator of the 4-torsion. In the same notation as above, it is Here is a brief explanation of the origin of˜4. The elements of the 1 -page of the spectral sequence can be coded by generators of lambda-algebra. Serre elements, which we study, correspond to the elements 1 . The element˜4 corresponds to 2 1 of the lambda-algebra. The ∞ * ,5 column of 2 has the following nontrivial terms: ∞ 8,5 = ℤ∕2 (generator 2 1 ), ∞ 6,5 = ℤ∕3 (generator 1 for = 3), ∞ 16,5 = ℤ∕2 (generator 3 1 ). The 4-torsion in 6 ( 2 ) is glued from two terms in ∞ : 3 1 and 2 1 . A representative of 3 1 is the bracket (8.5), see, for example, [15]. More generally, each corresponds to an operation on a simplicial group, with 1 corresponding (for = 2) to a simple bracketing ↦ [ 0 , 1 ]. Iterating it three times gives (8.5); see [46] for details. To show that˜4 represents the 4-torsion, we observe first that it is a cycle, namely that it lies in 0 ∩ ⋯ ∩ 5 , and second we show that, modulo 9 2 8 , it represents the element 2

EXAMPLES
The homotopy classes presented above yielded with relatively little computational effort Lie algebras and groups with -torsion in some high-degree dimension quotient. Using more computational resources, we were able to find -torsion in lower degree for = 2 and = 3.
A general simplification (see Propositions 8.2 and 8.3) is that we can start by an element of degree and not 2 , by writing generators in place of [ , ]. Indeed all the computations that express as an symmetrized associative product actually take place in L L 2 (ℤ 2 ) ⊂ L 2 (ℤ 2 ). In fact, this amounts to working in Milnor's simplicial construction [ 2 ], whose geometric realization is Ω 3 , and in its Lie analog [ 2 ]. Observe that, for spheres of dimension > 3, as well as of Moore spaces, there is a description of homotopy groups as centers of explicitly defined finitely generated groups [45]. However, these groups are not as easily defined as in the case of 2 , when we quotient by the symmetric commutator. This is why we concentrated on 2 ∨ 2 in this article.

=
The construction given in the proof of Theorem 1.2 has generators 0 , 1 , 2 and 3 ∶= − 0 − 1 − 2 . The element belongs to 14 ( ) ⧵ 14 ( ). It is possible to be a little bit more economical, by keeping the nilpotency degrees of the more under control: the best we could achieve is = ⟨ 0 , 1 , 2 , 3 , (1) 0 , ( Increasing the degree of the 0 leads, for every ⩾ 4, to a group with 2-torsion in ( )∕ ( ). This is essentially Rips's original example (1.1), except that his example contains more relations that make the group finite.
Proof. The proof is computer-assisted. It suffices to exhibit a quotient of in which the image of does not belong to 7 ( ) but its cube does, and we shall exhibit a finite 3-group as quotient.
To make the computations more manageable, we replace the generators and by generators 0 , … , 3  We compute a basis of left-normed commutators of length at most 6 in that group; notice that may be expressed as [ 3 , 2 , 3 , 1 , 1 , 3 ] 3 5 , and impose extra relations making 6 cyclic and central.
The resulting finite presentation may be fed to the program pq by Eamonn O'Brien [47], to compute the maximal quotient of 3-class 17. This is a group of order 3 3996 , and can (barely) be loaded in the computer algebra system GAP [18] so as to check (for safety) that the relations of hold, and that the element has a nontrivial image in it.
Finally, the order of the group may be reduced by iteratively quotienting by maximal subgroups of the center that do not contain . □ The resulting group, which is the minimal-order 3-group with nontrivial dimension quotient that we could obtain, has order 3 494 .
It may be loaded in any GAP distribution by downloading the ancillary file 3group.gap to the current directory and running Read("3group.gap"); in a GAP session.
We have in this manner verified that does not belong to 10 , but that 2 does belong to 10 . It remains to check, by hand, that belongs to 10 to conclude that indeed the example above has 2-torsion in 10 ∕ 10 .
Note that the same verification could have been made by computing with coefficients ℤ∕2 15 ; but the answer with coefficients ℤ∕2 14 would have been inconclusive.

A C K N O W L E D G M E N T S
The authors are grateful to Jacques Darné for having pointed out a mistake in a previous version of this text, which led to simplify some proofs. The first author is supported by the "@raction"

J O U R N A L I N F O R M AT I O N
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