Homology of the pronilpotent completion and cotorsion groups

For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

1. Introduction 1.1. Main result. Given a group G, denote by {γ i (G)} i≥1 its lower central series is called the pronilpotent completion of G. In this paper we study free pronilpotent completionsF , that is the pronilpotent completions of free groups F . Our main result is the following: Theorem A. For a non-cyclic free group F , the second integral homology H 2 (F ) is not a cotorsion group.
[For integral homology we omit the coefficients, i.e. H * (−) = H * (−, Z).] Recall that an abelian group A is called cotorsion, if Ext(Q, A) = 0. The class of cotorsion groups coincides with the class of values of lim ← − − 1 -functor for inverse sequences of abelian groups [23]. As a corollary, we see that H 2 (F ) can not be presented as lim ← − − 1 of an inverse sequence of abelian groups A 1 ← A 2 ← . . . (1.1) In particular, for the case of pronilpotent completion there is no chance to get a Milnor-type exact sequence We also prove the following statement which shows that the property to be a cotorsion group appears naturally in the context of homology of the pronilpotent completions.
Theorem B. Let G be a finitely generated group and F ↠ G be a presentation, where F is a free group of finite rank. Then the cokernel of the map is cotorsion.  Next we give some motivation for the studying the free nilpotent completion and in particular its homological properties.

Group theory.
In 1960-s G. Baumslag initiated a study of parafree groups. A group is called parafree if it is residually nilpotent and has the same lower central quotients as a free group [6], [3], [4], [5]. What are the properties parafree groups inherit from free groups? More general, what group-theoretical properties can one extract from the structure of the lower central quotients? These problems were studied by G. Baumslag during decades. For a free group F , all parafree groups with the same lower central quotients as F , are contained in the free pronilpotent completionF . If F is finitely generated,F is itself parafree.
The free pronilpotent completionF is an extremely complicated group. A.K. Bousfield proved in [8] that its integral second homology H 2 (F ) is uncountable. What is the cohomological dimension ofF ? Is it true that cd(F ) = 2? Is it true that H 3 (F ) = 0? These are open problems. These problems can be viewed as a part of Baumslag's program of the study of properties of parafree groups. The problem about the structure of H 2 (F ) also is intriguing. As a part of the proof of uncountablility of H 2 (F ) in [8], it is shown that there exists an epimorphism from H 2 (F ) to the exterior square of 2-adic integers. The exterior square of 2-adic integers is an uncountable divisible torsion-free group. In [19] it is shown that H 2 (F ) is not divisible. It seems that nothing more is published about the structure of H 2 (F ).
The homology of a group are related to its lower central quotients by the classical Stallings theorem [22]: a homomorphism of two groups which induces an isomorphism of H 1 and an epimorphism of H 2 induces isomorphism of the lower central quotients. In particular, for a group with a free abelian H 1 and zero H 2 , the lower central quotients are free nilpotent. Here is a simple remark which gives a generalization of the above statement. Remark 1.1. Let G be a group with a free abelian H 1 (G) and Hom(H 2 (G), Z) = 0. Then there exists a free group F and a homomorphism F → G which induces the isomorphisms of the lower central quotients.
[We give the proofs of the remarks from this section at the end of the paper.] In particular, a residually nilpotent group G with a free abelian H 1 (G) and Hom(H 2 (G), Z) = 0 is parafree. Is there a chance to convert this statement? In particular, can one show that Hom(H 2 (F ), Z) = 0? We leave these questions in the form of conjectures. Conjecture 1. For a non-cyclic free group F the following holds Conjecture 2. Let G be a residually nilpotent group. Then G is parafree if and only if H 1 (G) is free abelian and Hom(H 2 (G), Z) = 0.
Conjecture 1 would imply that H 2 (F ) is not free abelian, and hence, the cohomological dimension ofF would be greater than two. The class of cotorsion groups is a subclass of groups with the property Hom(−, Z) = 0. Our Theorem A shows that H 2 (F ) is not from this subclass.
1.3. Low-dimensional topology. In order to find transifinteμ-invariants of links, K. Orr introduced the space K ∞ [21]. The space K ∞ is the mapping cone of the natural map K(F, 1) → K(F , 1). The homotopy groups π i (K ∞ ), i ≥ 3, are repositories of potential invariants of links (see [10]). In the case of classical links, the invariants lie in π 3 (K ∞ ). The group π 3 (K ∞ ) is infinite [11], however its structure is far from being clear. In particular, the Hurewicz homomorphism π 3 (K ∞ ) → H 3 (F ) is an epimorphism. That is, the existence of non-zero elements in the (higher) homology of free pronilpotent completions may have application in low-dimensional topology.
1.4. Bousfield-Kan theory. The free pronilpotent completionF appears naturally in the theory of Bousfield-Kan [9], constructions of localizations and completions for spaces.
Recently Barnea and Shelah proved that, for any sequence of epimorphisms G i+1 → G i , i ≥ 1, the kernel and cokernel of the natural map are cotorsion groups [1]. The next remark is a consequence of this statement.
Remark 1.2. For a connected space X, the cokernel of the natural map is a cotorsion group. Here Z ∞ X is the integral Bousfield-Kan completion of X.
For a free group F , Z ∞ K(F, 1) = K(F , 1) (see [9], Section IV). Theorem A implies that, the cokernel of the natural map is not a cotorsion group. That is, the above remark can not be extended to the second homology.
1.5. Lie algebras. For a Lie algebra g over Z, consider its lower central series {γ i (g)} i≥1 and its pronilpotent completionĝ ∶= lim ← − − g/γ i (g). Most of the discussed above problems for groups can be acked for Lie algebras as well. In particular, for the moment we do not understand the structure of homology of a free pronilpotent completion in the case of Lie algebras. We are able to prove the following: Lie analog of Theorem A. For a non-cyclic free Lie algebra f over integers, H 2 (f) is not a torsion group.
In [17] it is shown that H 2 (f) is uncountable. The method used in [17] is similar to one from [15], the authors present explicit cycles in the Chevalley-Eilenberg complexf ∧f ∧f →f ∧f →f and show that these cycles are not boundaries. In particular, divisible elements in H 2 (f) are constructed in [17] in this way. Let f be a free Lie algebra on generators a 1 , . . . , a n . All the elements in H 2 (f) constructed in [17] are of the form (1.4) α 1 ∧ a 1 + ⋅ ⋅ ⋅ + α n ∧ a n , for some (infinite) α i ∈f. Remark 1.3. In the above notation, the subgroup in H 2 (f) generated by elements of the form (1.4) is cotorsion.
That is, the group H 2 (f) contains a huge uncountable cotorsion subgroup.
1.6. Structure of the paper. The paper is organized as follows. In Section 2 we recall the properties of cotorsion groups and give a proof of Theorem B. It turns out that Theorem B follows from the generalized Hopf formula and the result of Barnea  ]. We will also menstion a result of Warfield and Huber [23].
An abelian group C is called cotorsion, if Ext(Q, C) = 0. The following properties are equivalent.
Examples of cotorsion groups: divisible groups; finite groups; bounded groups; moreover, a reduced torsion group is cotorsion iff it is bounded; for any abelian groups A, B the group Ext(A, B) is cotorsion; a quotient of a cotorsion group is cotorsion; product of a family of groups is cotorsion iff each of them is cotorsion; inverse limit of reduced cotorsion groups is a reduced cotorsion group; in particular, the group of p-adic integers Z p = lim ← − − Z/p i is a reduced cotorsion group; for any sequence of abelian groups Non-examples of cotorsion groups: Z is not cotorsion; moreover, any group A with a non-trivial homomorphism A → Z is not cotorsion; the infinite direct sum indexed by all primes ⨁ p Z/p is not cotorsion.
The notion of cotorsion group is closely related to the notion of algebraically compact group. A group C is called algebraically compact, if Pext(A, C) = 0 for any A. Since for a torsion free abelian group A we have Ext(A, C) = Pext(A, C), we see that any algebraically compact group is cotorsion. On the other hand any cotorsion group is a quotient of an algebraically compact group. Moreover, a torsion free abelian group is cotorsion if and only if it is algebraically compact.

Cotorsion quotients of inverse limits.
A theorem of Hulanicki [14], [12,Cor. 42.2] says that for a sequence of abelian group A 1 , A 2 , . . . the quotient of the infinite product by the infinite direct sum (∏ A i )/(⨁ A i ) is algebraically compact, and hence, it is cotorsion. Barnea and Shellah in [1, Th 2.0.4] proved a version of this result about inverse limits of non-abelian groups. For an inverse sequence of groups and epimorphisms G 1 ↞ G 2 ↞ . . . a subgroup of the inverse limit U ⊆ lim ← − − G i is dense in the inverse limit topology if and only if the maps U → G i are surjective. Then the theorem of Barnea and Shelah can be reformulated as follows.

Generalised
Hopf 's formula. The classical Hopf's formula says that for a group presented as a quotient of a free group G = F /R its second integral homology can be computed as follows The following generalization of this formula is seems to be well known (c.f. [19, Lemma 2.3]) but we could not find a reference.

Proposition 2.2 (Hopf's formula). Let U be a normal subgroup of a group G, then there is an exact sequence
Proof. Consider the short exact sequence 1 → U → G → G/U → 1 and the corresponding five-term exact sequence of the Lyndon-Hochschild-Serre spectral sequence Simple computations show that there are isomorphisms induced by embedding maps. Therefore, the kernel of this map is iso- If we apply the limit to the short exact sequence and use that the functor of limit is exact for towers of epimorphisms, we obtain an isomorphism and the fact that the mapF →Ĝ is an epimorphism. Therefore by the generalised Hopf's formula (Proposition 2.2) there is an exact sequence We also have an isomorphism Therefore it is sufficient to prove that the cokernel of the map Prove it. For a finitely generated free group F we have an isomorphism F ab ≅F ab (see [8,Th. 13.3 (iv)], [2, Th.2.1]). So, if we take the limit of the short exact sequence and use that the functor of limit is left exact, we obtain an exact sequence It follows that Note that the image of R ∩ F 2 is dense in R ∩F 2 with respect to the inverse limit topolgy. Denote this image by I. Then the subgroup The assertion follows.

A technical result about power series
3.1. Formulation. For a commutative ring R we denote by R x be the ring of formal power series in the variable x over R. Its elements will be called just power series. We also denote by R × the group of invertible elements of R. It is well known The ring of polynomials R[x] is a subring of R x . We will also consider the following subring Elements of this ring are called rational power series. For two subrings S, T of a commutative ring we denote by S ⋅ T the subring consisting of elements of the form s 1 t 1 + ⋅ ⋅ ⋅ + s n t n , s i ∈ S, t i ∈ T. We will be interested in the following subrings of the ring of power series on two variables The subring R x ⋅ R y is known as the subring of series of finite rank. It has a subring Sym(R x ⋅ R y ) of power series F (x, y) having a finite rank and satisfying F (x, y) = F (y, x).
The aim of this section is to prove the following technical theorem about integral power series which will play a key role in the proof of the fact that H 2 (F ) is not cotorsion.
is not cotorsion.
3.2. Power series of finite rank. Let A be a commutative ring and R be its subring. Then A has a natural structure of R-algebra. We consider the following subring in the ring of power series in one variable x over A which is the product of the subrings A and R x of A x .
Lemma 3.2. Let R ⊆ A be an extension of commutative rings and F = ∑ f i x i be a power series from A x . Then the following statements are equivalent.
for some a ∈ A, G ∈ R x , then ⟨F ⟩ is a submodule of Ra, and hence, F ∈ A. Therefore, using that A is an additive subgroup, we obtain Then there exists a finite collection a 1 , . . . , a n ∈ A such that f i = ∑ n j=1 r i,j a j for some r i,j ∈ R.
If R = k is a field, then we can define the k-rank of a power series . . is finitely generated. For any n the abelian group generated by f 0 (x), f 1 (x), . . . is isomorphic to the abelian group generated by nf 0 (x), nf 1 (x), . . . . Hence nF ∈ Z x ⋅ Z y implies F ∈ Z x ⋅ Z y . The assertion follows.

Sieves.
In this subsection we assume that k ⊂ K is a proper field extension and A is a K-algebra Definition 3.6. Let n, d be positive integers and F = ∑ f j x j ∈ A x be a formal power series. We say that F has an n, d-sieve if there exists m ≥ 0 such that • for any l = 0, 1, . . . , n and i = 1, . . . , d − 1 we have f m+ld+i = 0 and • the coefficients f m+d , f m+2d , . . . , f m+nd ∈ A are linearly independent over K.
As we will see below, the concept of a sieve provide a measurement that allows to indicate if a formal series belongs to the denominator in Theorem 3.1. More formally, we have the following lemma, which is the central ingredient in the proof of the theorem.
Then the series F defined as Proof. Without loss of generality we can assume that β = 1, so that The element (1 − αx) ∈ K x is invertible with the inverse given by (1 − αx) −1 = ∑ j≥0 α j x j . Multiplying (3.7) by this inverse we obtain Let V ⊂ A denote the K-span of the set g 0 , g 1 , g 2 , ⋅ ⋅ ⋅ ∈ A. Due to the assumption 2 of the lemma we have dim k V ≤ n − 1.
Recall that α ∈ K, hence ∑ s+t=j g s α t ∈ V for any j. Using this observation and (3.9) we find out the following relation for any j ≥ 0: Assume now that F has an n, d-sieve and let m be the corresponding integer from Definition 3.6. Set a = ∑ s+t=m f s α t . The definition of the sieve together with (3.10) for j = m, m + 1, . . . , m + n 2 + n − 1 boil down to the following: Due to the assumption on α, β (recall that we set β = 1) we have λ l ∈ K × for any l. Moreover, relations (3.11) imply (line-by-line) relations (3.12) This immediately implies that f m+d , f m+2d , . . . , f m+nd ∈ V . But on the other hand f m+d , f m+2d , . . . , f m+nd are linearly independent over K due to Definition 3.6 of an n, d-sieve, while dim k V ≤ n−1 by the construction. Thus we obtain a contradiction.
3.4. Existence of a divisible series with an arbitrary sieve. Let p be a prime number and F p be a field of order p. Recall that F p (x) denotes the field of rational functions over F p and F p x denotes the field of Laurent series of one variable x. In this section we will always apply Definition 3.6 of an n, d-sieve for and for a power series in two variables from (3.14) First, we need a tool to verify a linear independence over K.
Lemma 3.8. Fix a prime number p. Let A be an arbitrary set and g α (x) = ∑ i∈Z a α,i x i ∈ F p x be a family of power series enumerated by elements α ∈ A. Assume that the following property holds for this family: for any N > 0 and α ∈ A there exists m = m(α, N ) ∈ Z such that a α,m ≠ 0, a β,m = 0 for any β ≠ α and moreover a β,m+i = 0 for any β (including β = α) and 0 < |i| < N . Then the family {g α } α∈A is lineraly independent over F p (x).
Note that one can assume that all r i 's are polynomials (if not, just multiply the equation above by the common denominator of r i 's). Let d i = deg r i and N > max{d 1 , . . . , d n }. Let m = m(α n , N ) be the index promised by the assumption made in the lemma. Let us also write r n (x) = bx d n + . . . where b ≠ 0 is the leading coefficient of r n . Then it is easy to see that the formal power series r 1 (x)g α 1 (x) + r 2 (x)g α 2 (x) + . . . + r n (x)g α n (x) has its coefficient at x m+d n equal to a α n ,m ⋅ b ≠ 0 which contradicts with the fact that this series vanishes. Now we can present the main construction that will be used as an obstruction for the group from Theorem 3.1 to be a cotorsion. Lemma 3.9. There exists a power series F ∈ Z x, y such that: • the element of Z x,y Z x ⋅Z y defined by F is divisible by any prime p; • for any prime p and any d ≥ p the power series F (x, y)−F (y, x) considered as an element of (F p x ) y has an p, d-sieve.
Proof. We construct the series F (x, y) explicitly. For any integers i, k, let us define The second equality in (3.15) reads as By the construction, for any prime p the series g k (x) mod p vanishes if k ≥ p, which implies by Lemma 3.2 that F (x, y) is equal to a series of a finite rank modulo p, so that F satisfies the first part of the lemma. Moreover, by the second and the third equality from (3.15) one has Fix a prime number p and d ≥ p. Note that we have and moreover t d,k −t d,k+1 = d for any k and t d,0 +d ≤ s d,0 . It follows that the element in (F p x ) y defined by F (x, y) − F (y, x) has an p, d-sieve if the elements in F p x defined by h 0 (x), h 1 (x), . . . , h p−1 (x) are linearly independent over F p (x). But it is straightforward to check that these elements satisfy assumptions of Lemma 3.8, so they are independent by this lemma. Proof. By assumptions of Theorem 3.1 we have 3.5. Proof of Theorem 3.1. In order to prove that a group A is not cotorsion, it is sufficient to construct a non-split short exact sequence 0 → A → B → C → 0 with torsion free C. Set We want to prove that the group (Z x ⋅ Z y )/S is not cotorsion. Consider the following short exact sequence By Lemma 3.5 the group Z x,y Z x ⋅Z y is torsion free. Therefore the theorem follows from Proposition 3.11. Proposition 3.11. Under assumptions of Theorem 3.1 the following epimorphism of abelian groups does not split: Proof. Let F (x, y) ∈ Z x, y be the series defined in Lemma 3.9 and let [F ] ∈ Z x,y Z x ⋅Z y denote the corresponding element. The proposition will be proved if we show that for any lift of [F ] under the morphism from the proposition there exists a prime p that does not divide this lift. Any such lift can be represented by a series of the form F (x, y)−G 1 (x, y), where G 1 (x, y) has a finite rank. Fix such a lift. Note that G 1 (x, y) − G 1 (y, x) has a finite rank too. Write G 1 (x, y) − G 1 (y, x) = ∑ j≥0 g j (x)y j ∈ Z x, y . By Corollary 3.4 the sequence g 0 (x), g 1 (x), . . . spans a finitely generated abelian group of some rank n 0 . Let p 0 be the constant from Lemma 3.10 and p ≥ max{n 0 + 2, p 0 } be a prime number. The fact that F (x, y) − G 1 (x, y) is divisible by p as an element of Z x,y (V (x)−U(x)y)⋅Z x ⋅Z y +Z[x] l ⋅Z[y] l +Sym(Z x ⋅Z y ) means that one can find some series is divisible by p as an element of Z x, y . This can be reformulated as Note that there is a natural embedding F p x, y ↪ (F p x ) y , in particular, we can assume that (3.18) holds in (F p x ) y . Using that S(x, y) = S(y, x) by the definition and the polynomial P (x, y) = (V (x) − U (x)y) is either symmetric or anti-symmetric we find out that • α ∈ K × corresponds to U (x) and β ∈ K × corresponds to V (x) (note that the fact that α, β ≠ 0 is guaranteed by the fact that p ≥ p 0 and Lemma 3.10); • G(y) ∈ A y corresponding toG 1 (x, y) +G 2 (x, y) and H(y) ∈ A y corresponding toH 0 (x, y); • d ≥ p be any integer greater than the F p -rank of H(y) and n 1 to be the K-rank of G(y); note that d < ∞ and n 1 ≤ n 0 + 1 by the construction, since the coefficients ofG 2 (x, y) considered as a series of y, are rational functions of x. Now it is straightforward that H, G satisfy assumptions of Lemma 3.7 with the chosen d and and n = p since p ≥ n 0 + 2 ≥ n 1 + 1. Moreover, α and β satisfy assumptions of Lemma 3.7 due to Lemma 3.10 and the choice of p. Therefore, with all this setup we can apply Lemma 3.7 and conclude from (3.19) that the element in (F p x ) y defined by F (x, y) − F (y, x) cannot have an p, d-sieve. But it has p, d-sieve by the construction (recall that F (x, y) is the series from Lemma 3.9), so we obtain a contradiction.
3.6. Some related results. In this subsection we prove some results related to the topic of this section but which are unnecessary for the main topic of the article (Theorem A and Theorem B).
We set The group P is known as Baer-Specker group. By the Hulanicki theorem the group P /S is cotorsion. Moreover, any at most continuous cotorsion group is a quotient of P /S [13, Th.7].
Proposition 3.12. The groups are not cotorsion.
Proof. The group Λ 2 P Λ 2 S is a quotient of P ⊗P S⊗S . Therefore, it is enough to prove the statement for Λ 2 P Λ 2 S . Note that there are isomorphisms of abelian groups P ≅ Z x and S ≅ Z[x] that respect inclusions. So it is enough to prove that is not cotorsion. Consider the homomorphism The image of θ is Z x ⋅ Z y and the image of the subgroup generated by the elements f ⊗ f lies in Sym(Z x ⋅ Z y ). Therefore we have an epimorphism This epimorphism induces an epimorphism By Theorem 3.1 a quotient of the last group is not cotorsion. The assertion follows.
For a prime number p we set where Z p = lim ← − − Z/p i is the group of p-adic integers. The group P p is cotorsion.
Proposition 3.13. The group P p ⊗ P p is not cotorsion.
Proof. Note that there is an isomorphism P p = Z p x . The group Z p x ⊗ Z p x maps onto the group Z p x ⋅ Z p y ⊆ Z x, y . So it is enough to prove that Z p x ⋅ Z p y is not cotorsion. Consider the short exact sequence So it is sufficient to prove that: (1) Z p x,y Z p x ⋅Z p y is torsion free and (2) the short exact sequence (3.26) does not split.
(1) By Lemma 3.2 a power series f (x, y) = ∑ a i (x)y i ∈ Z p x, y lies in Z p x ⋅ Z p y if and only if the Z p -submodule generated by a 0 (x), a 1 (x), . . . is finitely generated. On the other hand the submodule generated by na 0 (x), na 1 (x), . . . is isomorphic to the submodule generated by a 0 (x), a 1 (x), . . . . Therefore nf ∈ Z p x ⋅ Z p y if and only if f ∈ Z p x ⋅ Z p y .
(2) Consider the power series f (x, Z p x ⋅Z p y is nontrivial (because the Z p -submodule generated by 1, px, p 2 x 2 , . . . is not finitely generated) and it is divisible by any power of p. On the other hand there is no a non-trivial element divisible by all powers of p in Z p x, y . The assertion follows.
Remark 3.14. Proposition 3.13 shows that the tensor product of cotorsion groups is not necessarily cotorsion. Proposition 3.15. There exists a family of continuous cardinality (g α ) α∈2 ℵ 0 of power series g α ∈ Z x such that its image in F p x is linearly independent over F p (x) for any prime p.
Proof. Consider a family (X r ) r∈R of subsets of natural numbers X r ⊆ N indexed by real numbers r ∈ R such that for any s < r we have X s ⊆ X r and X r \ X s is infinite. For example, we can renumber all rational numbers {a 1 , a 2 , . . . } = Q and define X r = {n | a n < r}. Then we define g r = ∑ n∈X r x 2 n and consider the family (g r ) r∈R . Fix some prime p and denote byḡ r the image of g r in F p x and prove that (ḡ r ) r∈R is linearly independent over F p (x).

Integral lamplighter group
4.1. Definition of the lamplighter group. The classical lamplighter group can be defined as the restricted wreath product Z/2 ≀ Z. Here we consider its integral version Denote by C = ⟨t⟩ the infinite cyclic group written multiplicatively generated by an element t. Note that ]. Then is not cotorsion (Theorem 4.4).

The completion.
Denote by Z x the ring of power series and consider the ring homomorphism We will consider the ring Z x as a module over Z[t, t   Proof. It follows from [18,Prop.4.7] It is easy to see that there is an isomorphism The assertion follows. We also denote by Λ 2 M its exterior square over Z considered as a module over C with the diagonal action: t(m 1 ∧ m 2 ) = tm 1 ∧ tm 2 .
Lemma 4.2. For any C-group U and any n there is a short exact sequence Proof Proof. It follows from Lemma 4.2 for n = 2 and the the fact that the second homology of an abelian group is naturally isomorphic to its exterior square.
is isomorphic to the cokernel of (4.9). Consider a ring homomorphism If we define an action of C on Z x, y by multiplication on the polynomial (1 + x)(1 + y), then θ becomes a homomorphism of C-modules. Note that Im(θ) = Z x ⋅ Z y . Moreover, if we denote by D the subgroup of Z x ⊗2 generated by the elements of the form f ⊗ f, where f ∈ Z x , we obtain θ(D) ⊆ Sym(Z x ⋅ Z y ). So θ induces an epimorphism . .
⟶ Z x, y is a subring of Z x, y generated by x, y, (1 + x) −1 , (1 + y) −1 . Then we obtain a well defined epimorphism By Theorem 3.1 the image of this epimorphism is not cotosion. It follows that A is not cotorsion.

Proof of Theorem A
We prove that for a free group F of rank at least 2 the group H 2 (F ) is not cotorsion.
The lamplighter group L is a 2-generated group. Then for any free group F of rank ≥ 2 there is an epimorphism F ↠ L. By Theorem B we obtain that is cotorsion. On the other hand by Theorem 4.4 the group is not cotorsion. Consider the exact sequence We claim that the group Im(ϕ) is not cotorsion. Indeed, B is an extension of a cotorsion group C by Im(ϕ). If Im(ϕ) was cotorsion, B would be also cotorsion. Then Im(ϕ) is not cotorsion, and hence, H 2 (F ) is not cotorsion.

Lie algebras
In this paper by a Lie algebra we always mean a Lie algebra over Z. In this section we discuss versions of theorems A and B for the case of Lie algebras. We are not going to give detailed proofs because they are very similar to the case of groups. We just write down some preliminary results for Lie algebras which form a basis for the similar proofs.
There are several non-equivalent definitions of homology of a Lie algebra over Z (see [20]) but all of them coincide if we are interested only in the second homology [20,Th.8.4]. For example, we can define H n (g) = Tor Ug n (Z, Z), where U g is the universal enveloping algebra. Then for any presentation of a Lie algebra as a quotient of a free Lie algebra g = f/r, we have an analogue of Hopf's isomorphism which is natural in the short exact sequence 0 → r → f → g → 0. Moreover, this can be generalized to the following proposition.
which is natural in the short exact sequence 0 → u → g → g/u → 0.
Proof. Consider a presentation g = f/r. Denote by s the preimage of u in f. By the second and the third isomorphism theorems we have In the proof of Theorem B we used the fact that for a finitely generated free group F there is an isomorphism F ab ≅F ab that was proved in [8,Th. 13.3 (iv)] and [2, Th.2.1]. Here we prove an analogue of this result for Lie algebras. Lemma 6.2 (cf. [7]). Let g be a finitely generated Lie algebra. Then the map η ∶ g →ĝ induces an isomorphism (6.3) g/γ n (g) ≅ĝ/γ n (ĝ).
In particular, g ab ≅ĝ ab .
The proof is by induction on n. For n = 1 the statement is obvious. Prove the step.
Theorem B for Lie algebras. Let g be a finitely generated Lie algebra (over Z) and f ↠ g be a presentation, where f is a free Lie algebra of finite rank. Then the cokernel of the map is cotorsion.
Proof. The proof is similar to the proof of Theorem B for groups (see Section 2.4), so we provide only a short sketch here. Set r to be the kernel of f → g and h n = γ n (h) for any h. Arguing as in the proof of Theorem B for groups (with Proposition 6.1 used instead of the Hopf's formula) one can see that it is enough to prove that the cokernel of the map r ∩ f Due to Lemma 6.2 we have f ab ≅f ab , hence the same arguments as in the proof of Theorem B for groups imply that  is not cotorsion.
Proof. The proof is very similar to the proof of Theorem 4.4. In this proof we have to use Theorem 3.1 with P (x, y) = x + y (while in the case of groups we used P (x, y) = x + y + xy.) Theorem A for Lie algebras. Let f be a free Lie algebra (over Z) with at least two generators. Then H 2 (f) is not cotorsion.
Proof. The proof repeats literally the proof of Theorem A for groups (see Section 5).

Proofs of Remarks
Proof of Remark 1.1 First pick a free group F and a homomorphism F → G which induces an isomorphism H 1 (F ) → H 1 (G). The statement then follows by induction on n. Suppose that the above homomorphism induces an isomorphism F /γ n (F ) → G/γ n (G). There is a natural exact sequence H 2 (G) → H 2 (G/γ n (G)) → γ n (G)/γ n+1 (G) → 1.
Here lim ← − − 1 M n (π 1 (X)) is a certain quotient of lim ← − − 1 M n (π 1 (X)). Any quotient of a lim ← − − 1 of abelian groups is a cotorsion group (see Section 2.1 for the properties of cotorsion groups). That is, the group lim ← − − 1 M n (π 1 (X)) is cotorsion. The right hand side of (7.1) also is cotorsion by the result of Barnea and Shelah [1]. The class of cotorsion groups is closed under extensions and the needed statement follows.
Proof of Remark 1.3. Let f denote the Lie algebra freely generated by a 1 , . . . , a n .
Then H 2 (f) is the homology of the complex (7.2) Λ