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Home , Cokernel, Inverse limit

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instance, [W, p. 94, prop. 3.5.8]): Let Ci be a tower of chain complexes (of abelian groups) such that it satisfies degree-wise the Mittag-Leffler condition. Then there is the following short :

0 → lim1H (C ) → H (lim C ) → lim H (C ) → 0 ←− n+1 i n ←− i ←− n i

So, it is natural to ask if the same statement true for homologies of spaces with integer coefficients for a tower of connected pointed spacs Xi, such that maps Xi+1 → Xi are fibrations. (Un)fortunately, the answer is no for two reasons. First, the kernel might be non-zero, while lim1H (X ) = 0. To see it let us consider X = K(F/γ (F )), ←− n+1 i i i where F is 2-generated free group. Using fibrant replacments, we can assume that all maps X → X are fibratons. Then, using Milnor sequences, we see that lim X = i+1 i ←− i K(lim F/γ (F ), 1). This case was studied by Bousfield and Kan deeply because of its ←− n connection with Bousfield’s completion of spaces. It is known that lim1H (X )=0 ←− n+1 i for every n ≥ 0 [BK, p.123]. But it is proven by Bousfied in [Bous] that H2(F/γn(F )) =6 0 (and it implies that wedge of two circles is Z-bad space; actually wedge of two circles is also Q-bad and Z/p-bad space for p > 2, but it was proven much more later by Ivanov and Mkhailov in the papers [IM2], [IM3]). Second, the cokernel might be non- zero. Corresponding example actually was provided by Dwyer in [D] (Example 3.6). See Corollary 4 from my work for more details on this example. Definition. We call an A a cotorsion group if A = lim1B for ←− i some tower of abelian groups Bi. Remark. Usually people define cotorsion groups in a different way but these definitions coincide because of [H] (Theorem 1). More detailed, there is the following equivalent definition of cotorsion groups: Definition. We call an abelian group A a cotorsion group if Ext(C, A)=0 for any torsion- C. There is the following result which was proven by Shelah and Barnea ([BS], Corol- lary 0.0.9):

Theorem. Let Xi be an inverse system of pointed connected spaces, such that all maps Xi+1 → Xi are Serre fibrations, π1(Xi) and π2(Xi) satisfies the Mittag-Leffler condition. Then the natural map H (lim X ) → lim H (X ) is surjective and its kernel 1 ←− i ←− 1 i is a cotorsion group.

This result might give one a hope that, assuming towers π1(Xi), π2(Xi),...,πk(Xi) for large enough (maybe infinite) k satisfy Mittag-Leffler condition, we can ”fix” usual Milnor sequences and provide ”Milnor sequences for Hn”. (Un)fortunately, the answer is no already for n = 2. Moreover, it is false even for homologies of abelian groups and it is proven in this work: Theorem 1. There is an inverse system of abelian groups indexed by N such that all maps A → A are epimorhisms and kernel of the natural map H (lim A ) → i+1 i 2 ←− i lim H (A ) is not a cotorsion group. ←− 2 i This theorem shows that things are very difficult even for abelian groups, so the following result seems quite interesting: HOMOLOGIES OF INVERSE LIMITS OF GROUPS 3

Theorem 2. Let Ai be an inverse system of torsion-free abelian groups indexed by N such that all maps Ai+1 → Ai are epimorhisms. Then for any n ∈ N the natural map H (lim A ) → lim H (A ) is an and its cokernel is a cotorsion group. n ←− i ←− n i Theorem 3. Let Ai be an inverse system of any abelian groups indexed by N such that all maps Ai+1 → Ai are epimorhisms. Then: (1) Cokernel of the natural map H (lim A ) → lim H (A ) is a cotorsion group 2 ←− i ←− 2 i (2) Suppose, additionally, that torsion of Ai is a group of bounded ex- ponent for any i. Then cokernel of the natural map H (lim A ) → lim H (A ) is a 3 ←− i ←− 3 i cotorsion group. In particular, it is true if all Ai are finitely generated. Also in this work there are two results not about homologies but about another and inverse limits of abelian groups:

Statement 2. Let B be any abelian group and Bi be an inverse system of abelian groups with surjective maps fi : Bi+1 → Bi between them. Then kernel of the map Tor(B, lim B ) → lim Tor(B, B ) is trivial. ←− i ←− i Corollary 6. Tor(lim A , lim A ) → lim Tor(A , A ) is embedding for any inverse ←− i ←− i ←− i i system of abelian groups such that all maps Ai+1 → Ai are surjective. The author thanks Roman Mikhailov, Sergei O. Ivanov, Emmanuel Farjoun, Fedor Pavutnitsky and Saharon Shelah for fruitful discussions.

2. Homologies of inverse limits of groups A goal of this chapter is to prove the following results: Theorem 1. There is an inverse system of abelian groups indexed by N such that all maps A → A are epimorhisms and kernel of the natural map H (lim A ) → i+1 i 2 ←− i lim H (A ) is not a cotorsion group. ←− 2 i Theorem 2. Let Ai be an inverse system of torsion-free abelian groups indexed by N such that all maps Ai+1 → Ai are epimorhisms. Then for any n ∈ N the natural map H (lim A ) → lim H (A ) is embedding and its cokernel is a cotorsion group. n ←− i ←− n i Theorem 3. Let Ai be an inverse system of any abelian groups indexed by N such that all maps Ai+1 → Ai are epimorhisms. Then: (1) Cokernel of the natural map H (lim A ) → lim H (A ) is a cotorsion group 2 ←− i ←− 2 i (2) Suppose, additionally, that torsion subgroup of Ai is a group of bounded ex- ponent for any i. Then cokernel of the natural map H (lim A ) → lim H (A ) is a 3 ←− i ←− 3 i cotorsion group. In particular, it is true if all Ai are finitely generated. Proof of Theorem 1. Let us consider ∞ ′ i ′ ′ ′ Ai,p = Z ⊕ ( Z/pZ), Ai := Ai,p, B := Z/pZ, Ai := Ai × B Mi+1 Mp∈P Mp∈P ′ ′ We are going to define maps ψi : Ai → Ai−1 in the most natural way. Let us denote 1 i ′ by ei,p,...,ei,p elements of basis of the free abelian summand of Ai,p and let us denote i+1 i+2 by ei,p , ei,p ,... elements of basis of the Z/pZ-, which is the second direct ′ j j summand of Ai,p. Now let us define ψi(ei,p) := ei−1,p. It is obvious that there is unique 4 DANIL AKHTIAMOV

ψi with such properties. And, finally, let φi : Ai → Ai−1 be the map defined by the ψ 0 matrix i .  0 IdB  ′ Using Kunneth formula and usig that all Ai and B are abelian, we have: H (lim A )= H (lim (A′ × B)) = H ((lim A′ )×B)= H (lim A′ )⊕H (B)⊕((lim A′ )⊗B) 2 ←− i 2 ←− i 2 ←− i 2 ←− i 2 ←− i In the similar way we can get: lim H (A ) = lim H ((A′ × B)) = lim H (A′ ) ⊕ H (B) ⊕ (A′ ⊗ B) = lim H (A′ )⊕H (B)⊕lim (A′ ⊗ B) ←− 2 i ←− 2 i ←− 2 i 2 i ←− 2 i 2 ←− i Thus our map H (lim A ) → lim H (A ) is a map from H (lim A′ ) ⊕ H (B) ⊕ 2 ←− i ←− 2 i 2 ←− i 2 ((lim A′ )⊗B) to lim H (A′ )⊕H (B)⊕lim (A′ ⊗ B). Analyzing maps from the Kunneth ←− i ←− 2 i 2 ←− i formula, we see that this map is given by a diagonal matrix, the corresponding maps H (lim A′ ) → lim H (A′ ) and (lim A′ ) ⊗ B → lim (A′ ⊗ B) coincide with obvious maps 2 ←− i ←− 2 i ←− i ←− i which come from definition of inverse and the corresponding map H2(B) → H2(B) is . Thus kernel of the natural map (lim A′ ) ⊗ B → lim (A′ ⊗ B) is direct ←− i ←− i summand of kernel of the natural map H (lim A ) → lim H (A ). So it is enough to 2 ←− i ←− 2 i prove that kernel of the natural map (lim A′ ) ⊗ B → lim (A′ ⊗ B) is not a cotorsion ←− i ←− i group and now we will do it. Let us note that the map (lim A′ ) ⊗ B → lim (A′ ⊗ B) ←− i ←− i can be decomposed in the following way:

(lim A′ )⊗B = (lim A′ )⊗Z/pZ → lim(A′ ⊗ Z/pZ) → lim( A′ ⊗ Z/pZ) = lim (A′ ⊗ B) ←− i ←− i ←− i ←− i ←− i Mp∈P Mp∈P Mp∈P

It is easy to see that the map lim(A′ ⊗ Z/pZ) → lim( A′ ⊗ Z/pZ) is injective. ←− i ←− i Mp∈P Mp∈P Then kernel of the map (lim A′ )⊗B → lim (A′ ⊗ B) equals Ker[(lim A′ )⊗Z/pZ → ←− i ←− i ←− i Mp∈P ′ ′ lim (Ai ⊗ Z/pZ)]. It is proven at [IM] (Corollary 2.5) that Ker[(lim Ai) ⊗ Z/pZ → ←− ←− ∞ lim (A′ ⊗ Z/pZ)] ∼ lim1(Tor(A′ , Z/pZ)). Let us note that Tor(A′ , Z/pZ)= Z/pZ. ←− i = ←− i i Mi+1 ∞ So lim1(Tor(A′ , Z/pZ)) ∼ lim1( Z/pZ), which is, obviously, p-torsion abelian ←− i = ←− i+1 group and which is nonzero becauseL of [MP] (p. 330, Proposition A.20). Then, fi- nally, we have that kernel of the map (lim A′ ) ⊗ B → lim (A′ ⊗ B) is not a cotorsion ←− i ←− i group because of [B] (Theorem 8.5). Q.E.D. We will need the following lemmas and statements in order to prove Theorem 2 and Theorem 3. All inverse systems supposed to be indexed by N.

Statement 1. Let C be any category with projective limits and let Bi be a tower of objects from C. Let F : C → Ab be a functor such that lim Coker[F (lim B ) → F (B )] = ←− ←− i n n 0. Then cokernel of the natural map F (lim B ) → lim F (B ) is a cotorsion group. ←− i ←− i This statement was proved in the third version of Barnea’s and Shelah’s preprint [BS] in the case when functor F preserves surjection. However, their proof was quite complicated. Our proof of the statement is straightforward. HOMOLOGIES OF INVERSE LIMITS OF GROUPS 5

Proof. Let denote by Φ := Ker(F (lim B ) → F (B )) andby Ψ := Im(F (lim B ) → i ←− i i i ←− i F (Bi)). Then we have the following exact sequences: 0 → Φ → F (lim B ) → Ψ → 0 i ←− i i and 0 → Ψ → F (B ) → Coker[F (lim B ) → F (B )] → 0. i i ←− i i From the second sequence we get by the assumption lim Coker[F (lim B ) → F (B )] = ←− ←− i n n 0 lim Ψ = lim F (B ). From the first we deduce a new exact sequence: ←− i ←− i 0 → lim Φ → F (lim B ) → lim Ψ → lim1Φ → 0 ←− i ←− i ←− i ←− i Thus finally we have the following exact sequence 0 → lim Φ → F (lim B ) → lim F (B ) → lim1Φ → 0 ←− i ←− i ←− i ←− i Now it is enough to note that lim1 of any inverse system of abelian groups is a cotorsion ←− group by [H] (Theorem 1). Q.E.D.

Corollary 1. Let Bi be an inverse system of groups (respectively abelian groups) and any maps between them. Let F : Grp → Ab (respectively F : Ab → Ab) be a functor such that lim Coker[F (lim B ) → F (B )] = 0. Then kernel of the map F (lim B ) → ←− ←− i n ←− i n lim F (B ) equals lim Φ , where Φ = Ker[F (lim B ) → F (B )]. ←− i ←− i n ←− i n Corollary 2. Cokernel of the map Λn(lim B ) → lim Λn(B ) is a cotorsion group ←− i ←− i for any inverse system Bi, such that Bi+1 → Bi are . Proof. Since B → B are epimorphisms, the maps Λn(lim B ) → Λn(B )=0 i+1 i ←− i i are also epimorphisms (it easily follows from constructive description of projective limits in the ). Then Λn(lim B ) → Λn(B ) is , because ←− i i Λn is right-exact. Then Coker[Λn(lim B ) → Λn(B )] = 0 and we are done because of ←− i i Statement 1. Corollary 3. Cokernel of the map H (lim B ) → lim H (B ) is a cotorsion group 2 ←− i ←− 2 i for any inverse system of abelian groups Bi, such that Bi+1 → Bi are epimorphisms. 2 Proof. It is well-known [Breen, section 6] that H2 is naturally isomorphic to Λ in the category of abelian groups, so the corollary follows from Corollary 2.

Definition. Let G be any group. Let γ1(G) := G and γi+1(G) := [G,γi(G)]. Series γ (G) := G are called lower central series of a group G. Let denote G := lim G/γ (G). 1 ←− i Corollary 4. Cokernel of the map H (G) → lim H (G/γ (G)) is a cotorsion 2 ←− 2 i b group for any group G. b Proof. It is clear that the map H2(G) → H2(G/γi(G)) factors through the map H (G) → H (G/γ (G)). Then it is enough to prove that lim Coker[H (G) → H (G/γ (G)] = 2 2 i ←− 2 2 i 0. Let note that maps between groups Coker[H (G) → H (G/γ (G)] are zero. Really, b 2 2 i using that H1(G)= H1(G/γi(G)), we see from the 5-term exact sequence [Brown, p.47, exercise 6a] that these cokernels are equal to γi(G)∩[G,G] = γi(G) . Q.E.D. [γi(G),G] γi+1(G) Following statement is not really necessary for a proof of the Theorems, but it is interesting by itself and makes clearer what is happening. 6 DANIL AKHTIAMOV

Statement 2. Let B be any abelian group and Bi be an inverse system of abelian groups with surjective maps fi : Bi+1 → Bi between them. Then kernel of the map Tor(B, lim B ) → lim Tor(B, B ) is trivial. ←− i ←− i Proof. It is proven in [IM] (Proposition 2.6) that for any free resolution P• of B there are the following exact sequences(take Λ = Z and use a fact which states that Z has a global dimension one):

0 → H (lim P• ⊗ B ) → lim Tor(B, B ) → 0 1 ←− i ←− i So, we understood that H (lim P• ⊗ B ) is isomorphic to lim Tor(B, B ). 1 ←− i ←− i Let take P• be a minimal resolution, i.e. such that Ps = 0 when s > 1. It is possible because global dimension of Z equals 1. Let us note that the maps P ⊗ lim B → lim (P ⊗ B ) are because s ←− i ←− s i of Lemma 1 (s =0, 1). Let consider a short exact sequence of complexes(C• is defined from this sequence):

0 → P• ⊗ (lim B ) → lim (P• ⊗ B ) → C• → 0 ←− i ←− i

Since Ps=0 for s > 1, note that Cs = 0 for s > 1. Thus we have the following exact sequence:

0 → H (P• ⊗ (lim B )) → H (lim (P• ⊗ B )) 1 ←− i 1 ←− i But H (P• ⊗ (lim B )) = Tor(B, lim B ) by definition and we already have got 1 ←− i ←− i that H (lim P• ⊗ B ) = lim Tor(B, B ) so we are done. Q.E.D. 1 ←− i ←− i Corollary 5. Tor(lim A , lim A ) → lim Tor(A , A ) is an embedding for any ←− i ←− i ←− i i inverse system of abelian groups such that all maps Ai+1 → Ai are epimorphic. Proof. Let consider following commutative diagramm with ψ being isomorphism: ϕ Tor(lim A , lim A ) lim Tor(A , A ) ←− i ←− j ←− i i f ψ g lim Tor(A , lim A ) lim lim Tor(A , A ) ←− i ←− j ←− ←− i j But f is because of Statement 3 and g is monomorphism because of Statement 3 and left exactness of lim. Q.E.D. ←− Statement 3. Let B be a cotorsion group and A be another abelian group. Let suppose that there are i : A → B and π : B → A, such that πi = nIdA for n ≥ 1. Then A is a cotorsion group.

Proof. Since Ext(Q, B) = 0, a map πi = nIdA induces zero endomorphism of Ext(Q, A). Let consider the following exact sequences(we denote n-torsion subgroup of A by An):

0 → An → A → nA → 0

0 → nA → A → A/nA → 0 HOMOLOGIES OF INVERSE LIMITS OF GROUPS 7

It is known that any n-torsion group is a cotorsion group (see [B], Theorem 8.5). Thus the maps A → nA and nA → A induce on Ext(Q, −), and then so do nIdA. But it induces zero endomorphism of Ext(Q, A), so we are done. Q.E.D. Proof of Theorem 3. We already proved the first point of this theorem (Corollary 2), so let us prove second point of the theorem. It is known [Breen, section 6] that 2 for any abelian group A there is the following short exact sequence (here L1Λ is first of functor Λ2):

3 2 0 → Λ (A) → H3(A) → L1Λ (A) → 0 Then for A = lim A we have the following sequence: ←− i

0 → Λ3(lim A ) → H (lim A ) → L Λ2(lim A ) → 0 ←− i 3 ←− i 1 ←− i Since Λ3(lim A ) → Λ3(lim A ) is an epimorphism for any i, lim1Λ3(lim A )=0 ←− i+1 ←− i ←− ←− i and we have the following sequence:

0 → lim Λ3(A ) → lim H (A ) → lim L Λ2(A ) → 0 ←− i ←− 3 i ←− 1 i Then Ker[L Λ2(lim A ) → lim L Λ2(A )] ⊆ Ker[Tor(lim A , lim A ) → lim Tor(A , A )] = 1 ←− i ←− 1 i ←− i ←− i ←− i i 0 (here we used Corollary 5). Then, using , we have the following short exact sequence:

0 → Coker[Λ3(lim A → lim Λ3(A )] → Coker[H (lim A ) → lim H (A )] → Coker[L Λ2(lim A ) → lim L Λ2(A )] → 0 ←− i ←− i 3 ←− i ←− 3 i 1 ←− i ←− 1 i It is obvious that any extension of a cotorsion group by a cotorsion group is a cotorsion group. And we already know that Coker[Λ3(lim A → lim Λ3(A )] is a ←− i ←− i cotorsion group. Then it is enough to prove that Coker[L Λ2(lim A ) → lim L Λ2(A )] 1 ←− i ←− 1 i is a cotorsion group. It is known [Breen, sections 4 and 5] that for any abelian group A there are maps 2 2 L1Λ (A) → Tor(A, A) and Tor(A, A) → L1Λ (A), such that their composition equals 2 2 to 2Id 2 . Then they induce natural maps Coker[L Λ (lim A ) → lim L Λ (A )] → L1Λ (A) 1 ←− i ←− 1 i Coker[Tor(lim A , lim A ) → lim Tor(A , A )] and Coker[L Λ2(lim A ) → lim L Λ2(A )] → ←− i ←− i ←− i i 1 ←− i ←− 1 i Coker[Tor(lim A , lim A ) → lim Tor(A , A )], such that their composition is mul- ←− i ←− i ←− i i tiplication by two. Then, using Statement 3, we see that it is enough to prove that Coker[Tor(lim A , lim A ) → lim Tor(A , A )] is a cotorsion group. But if tor- ←− i ←− i ←− i i sion of Ai are groups of bounded exponent, then Tor(Ai, Ai) are torsion groups of bounded exponent. Hence they are cotorsion groups [B, Theorem 8.5], and lim Tor(A , A ) is a cotorsion groups, because it is an of cotorsion groups ←− i i (it is obvious from our second definition of cotorsion groups that an inverse limit of cotorsion groups is itself a cotorsion group). Thus Coker[Tor(lim A , lim A ) → ←− i ←− i lim Tor(A , A )] is a cotorsion group as an of a cotorsion group and we are done. ←− i i Q.E.D. We need the following statement in order to prove Theorem 2: 8 DANIL AKHTIAMOV

Statement 4. Let Ai be an inverse system of torsion-free abelian groups indexed by N such that all maps Ai+1 → Ai are epimorphisms and let B be any torsion-free abelian group. Then the natural map B ⊗ lim A → lim (B ⊗ A ) is an embedding. ←− i ←− i Proof. First let prove the statement for B = Q. Let us consider the following short exact sequence: 0 → Z → Q → Q/Z → 0

After appliying the functor − ⊗ Ai for this sequence we get:

0 → Tor(Q/Z, Ai) → Ai → Q ⊗ Ai → Q/Z ⊗ Ai → 0 Since Tor(Q/Z, A)= t(A) for any abelian group A, we have:

0 → t(Ai) → Ai → Q ⊗ Ai → Q/Z ⊗ Ai → 0 Then, using left-exactness of lim, we have: ←− 0 → lim t(A ) → lim A → lim (Q ⊗ A ) ←− i ←− i ←− i Let us apply the Q ⊗ − for this sequence: 0 → Q ⊗ lim t(A ) → Q ⊗ lim A → Q ⊗ lim (Q ⊗ A ) ←− i ←− i ←− i Let us note that Q ⊗ lim (Q ⊗ A ) ∼ lim (Q ⊗ A ), because lim (Q ⊗ A ) is a Q-vector ←− i = ←− i ←− i space. It means that we got the following: 0 → Q ⊗ lim t(A ) → Q ⊗ lim A → lim (Q ⊗ A ) ←− i ←− i ←− i But in the our case t(Ai)= 0 for any i. So we have: 0 → Q ⊗ lim A → lim (Q ⊗ A ) ←− i ←− i Now let us prove the statement in the case B = Q⊕I , where I is any cardinal. Using the sequence for B = Q we get: ⊕ ⊕ ⊕ 0 → (Q ⊗ lim t(A )) I → (Q ⊗ lim A ) I → (lim (Q ⊗ A )) I ←− i ←− i ←− i Since it is obvious that the map (lim (Q ⊗ A ))⊕I → lim ((Q ⊗ A )⊕I ) is injective, we ←− i ←− i have: ⊕ ⊕ 0 → (Q ⊗ lim t(A )) I → (Q ⊗ lim A ) I → lim ((Q ⊗ A )⊕I) ←− i ←− i ←− i But in the our case t(Ai) = 0 for any i. So we have the exactness of the following sequence: ⊕ 0 → (Q ⊗ lim A ) I → lim ((Q ⊗ A )⊕I ) ←− i ←− i Since the functor −⊗A commutes with direct sums for any A, we proved the statement in the case when B is any Q-vector space.

Now let us prove the statement for any torsion-free B. Let QB be the injective ⊕I hull of B. Since B is torsion-free, QB = Q for some cardinal I. Let us consider the following short exact sequence(B′ is defined from this sequence): ′ 0 → B → QB → B → 0 Since lim A is torsion-free and lim1B ⊗ A = 0 because the maps B ⊗ A → B ⊗ A ←− i ←− i i+1 i are epimorphisms, it gives us two sequences: 0 → B ⊗ lim A → Q ⊗ lim A → B′ ⊗ lim A → 0 ←− i B ←− i ←− i HOMOLOGIES OF INVERSE LIMITS OF GROUPS 9

0 → lim (B ⊗ A ) → lim (Q ⊗ A ) → lim (B′ ⊗ A ) → 0 ←− i ←− B i ←− i Then, using Snake Lemma for this two sequences and natural maps between their elements, we have: 0 → Ker[B ⊗ lim A → lim (B ⊗ A )] → Ker[Q ⊗ lim A → lim (Q ⊗ A )] ←− i ←− i B ←− i ←− B i Since Q is a Q-vector space, we already proved that Ker[Q ⊗lim A → lim (Q ⊗ A )] = B B ←− i ←− B i 0. Then Ker[B ⊗ lim A → lim (B ⊗ A )] = 0 and we are done. Q.E.D. ←− i ←− i Proof of Theorem 2. Let consider following with ψ being isomorphism. ϕ lim A ⊗ lim A lim (A ⊗ A ) ←− i ←− j ←− i i f ψ g lim (A ⊗ lim A ) lim lim (A ⊗ A ) ←− i ←− j ←− ←− i j Then we see that f is embedding because of Statement 4 and g is embeddging because of Statement 4 and left-exactness of lim. This implies that φ is also embedding. ←− 2 Then, since there is a natural embedding Λ (A) → A⊗A and H2 is naturally isomorphic to Λ2 [Breen, section 6] for abelian groups, we proved the theorem for n = 2. n Now let us note that Hn is naturally isomorphic to Λ on the category of torsion- free abelian groups also for any n> 2 [Breen, p. 214, (1.12)]. Let us prove the theorem by induction on n. We can assume that n ≥ 3. Let us note that the map (lim A )⊗n → lim (A )⊗n ←− i ←− i may be decomposed isomorphism in the following way: ⊗ − ⊗ − (lim A )⊗n ∼= lim A ⊗(lim A )⊗(n−1) → lim A ⊗lim (A ) (n 1) → lim lim A ⊗ (A ) (n 1) ∼= ←− i ←− i ←− j ←− i ←− j ←−i ←−j i j ⊗ lim (A ) n. But all maps in the decomposition are monic because of inductional as- ←− i sumption, left-exactness of lim and exactness of − ⊗ B for torsion-free B. Then the ←− map (lim A )⊗n → lim (A )⊗n is also monic, and then so is H (lim A ) → lim H (A ). ←− i ←− i n ←− i ←− n i Q.E.D.

3. Applications to topology

Theorem 4. Let Xi be an inverse system of pointed connected spaces, such that all maps Xi+1 → Xi are Serre fibrations, all π1(Xi) are abelian, all maps π1(Xi+1) → π1(Xi) are epimorphisms and π2(Xi)i∈N satisfy the Mittag-Leffler condition. Then: (1) Cokernel of the natural map H (lim X ) → lim H (X ) is a cotorsion group. 2 ←− i ←− 2 i (2) Suppose, additionally, that all π1(Xi) are torsion-free. Then kernel of the natural map H (lim X ) → lim H (X ) is a cotorsion group. 2 ←− i ←− 2 i (3) Suppose that condition (2) is satisfied and, additionally, π3(Xi)i∈N satisfy the Mittag-Leffler condition. Then the natural map H (lim X ) → lim H (X ) is embedding. 2 ←− i ←− 2 i Proof. Let us denote Π2(X) := Im[π2(X) → H2(X)]. It is obvious that Π2 is a functor. Then we have the following sequence:

0 → Π2(X) → H2(X) → H2(π1(X)) → 0 10 DANIL AKHTIAMOV

Then we get the following sequences, feeding X = lim X and X = X : ←− i i 0 → Π2(Xi) → H2(Xi) → H2(π1(Xi)) → 0 It gives us the following sequence: 0 → lim Π (X ) → lim H (X ) → lim H (π (X )) → lim1Π (X ) → lim1H (X ) → lim1H (π (X )) → 0 ←− 2 i ←− 2 i ←− 2 1 i ←− 2 i ←− 2 i ←− 2 1 i Since π (X ) → Π (X ) is epimorphism, lim1π (X ) → lim1Π (X ) it is well-known 2 i 2 i ←− 2 i ←− 2 i (e.g. see [Brown, p. 42, Theorem 5.2]) that for any space X there is the following exact sequence: π2(X) → H2(X) → H2(π1(X)) → 0 . also epimorphism. Then lim1π (X ) = lim1Π (X ) = 0, because π (X ) satisfy the ←− 2 i ←− 2 i 2 i Mittag-Leffler condition. So we have the following sequence: 0 → lim Π (X ) → lim H (X ) → lim H (π (X )) → 0 ←− 2 i ←− 2 i ←− 2 1 i Also we have the following sequence: 0 → Π (lim X ) → H (lim X ) → H (π (lim X )) → 0 2 ←− i 2 ←− i 2 1 ←− i We have the following sequences, because Xii∈N is a tower of pointed fibrations [GJ, VI Proposition 2.15]: 0 → lim1π (X ) → π (lim X ) → lim π (X ) → 0 ←− 2 i 1 ←− i ←− 1 i Since π2(Xi) satisfy the Mittag-Leffler condition, we have: π (lim X ) ∼ lim π (X ) 1 ←− i = ←− 1 i Finally, we have two sequences: 0 → lim Π (X ) → lim H (X ) → lim H (π (X )) → 0 ←− 2 i ←− 2 i ←− 2 1 i 0 → Π (lim X ) → H (lim X ) → H (lim π (X )) → 0 2 ←− i 2 ←− i 2 ←− 1 i . Let us note that Coker[lim Π (X ) → Π (lim X )] = 0 and Ker[lim Π (X ) → ←− 2 i 2 ←− i ←− 2 i Π (lim X )] ∼ lim1π (X ) . It follows from the following sequence: 2 ←− i = ←− 3 i 0 → lim1π (X ) → π (lim X ) → lim π (X ) → 0 ←− 3 i 2 ←− i ←− 2 i Then, using Snake Lemma for this two sequences and the natural maps between them, we have: 0 → lim1π (X ) → Ker[lim H (X ) → H (lim X )] → Ker[lim H (π (X )) → H (lim π (X ))] → 0 ←− 3 i ←− 2 i 2 ←− i ←− 2 1 i 2 ←− 1 i and Coker[H (lim X ) → lim H (X )] ∼ Coker[H (lim π (X )) → lim H (lim (X )) 2 ←− i ←− 2 i = 2 ←− 1 i ←− 2 ←− i Then point (1) of the Theorem follows from point (1) of Theorem 1. Let us prove points (2) and (3) of the Theorem. It follows from Theorem 2 for n = 2 that Ker[lim H (π (X )) → H (lim π (X ))] = 0. Then we have: ←− 2 1 i 2 ←− 1 i Ker[lim H (X ) → H (lim X )] ∼ lim1π (X ) ←− 2 i 2 ←− i = ←− 3 i So we proved point (3) and point (2). Q.E.D. HOMOLOGIES OF INVERSE LIMITS OF GROUPS 11

∞

Theorem 5. Let Yi be a sequence of spaces. Then cokernel of the map Hk( Yi) → Yi=1 n lim H ( Y ) is a cotorsion group for every k. ←− k i n Yi=1 n ∞

Proof. Since composition of the natural maps Hk( Yi) → Hk( Yi) and Yi=1 Yi=1 ∞ n ∞ n

Hk( Yi) → Hk( Yi) is the identity map, the map Hk( Yi) → Hk( Yi) is sur- Yi=1 Yi=1 Yi=1 Yi=1 jective. So we are done because of Statement 1. Q.E.D. Remark. It is easy to see from this from this proof that the same fact holds for any category which has infinite products instead of category of spaces and for and functor F from this category to Ab instead of Hk. But I formulated it in this way because it seems more natural in this section. We will also prove the following Theorem, which does not follow from our previous results but follows from Shelah’s and Barnea’s. It is connected with Theorem 3, so I found quite natural to formulate and prove it here. This Theorem is a generalization of Shelah’s and Barnea’s [Corollary 0.0.9. BS]. Actually we show that it is not necessary to assume that π2(Xi) satisfies the Mittag-Leffler conditon.

Theorem 6. Let Xi be an inverse system of pointed connected spaces, such that all maps Xi+1 → Xi are Serre fibrations and π1(Xi) satisfies the Mittag-Leffler condition. Then the natural map H (lim X ) → lim H (X ) is surjective and its kernel 1 ←− i ←− 1 i is a cotorsion group. Proof. Let us consider the following exact sequence: 0 → lim1π (X ) → π (lim X ) → lim π (X ) → 0 ←− 1 i 0 ←− i ←− 0 i Then lim X is also connected and we have: ←− i H (lim X ) ∼= π (lim X ) 1 ←− i 1 ←− i ab It follows that: Ker[H (lim X ) → lim H (X )] = Ker[(π (lim X )) → lim (π (X )) ] 1 ←− i ←− 1 i 1 ←− i ab ←− 1 i ab and Coker[H (lim X ) → lim H (X )] = Coker[(π (lim X )) → lim (π (X )) ]. Also 1 ←− i ←− 1 i 1 ←− i ab ←− 1 i ab we have the following sequence: 0 → lim1π (X ) → π (lim X ) → lim π (X ) → 0 ←− 2 i 1 ←− i ←− 1 i Thus we have the following exact sequence from 5-term exact sequence: lim1π (X ) → (π (lim X )) → (lim π (X )) → 0 ←− 2 i 1 ←− i ab ←− 1 i ab Then, since lim1π (X ) is a cotorsion group by the Theorem of Huber(Theorem 1, [H]) ←− 2 i and any of a cotorsion group is itself a cotorsion group, Ker[(π (lim X )) → 1 ←− i ab (lim π (X )) ] is a cotorsion group. Now consider the map (lim π (X )) → lim (π (X )) . ←− 1 i ab ←− 1 i ab ←− 1 i ab It is surjective and its kernel is a cotorsion group by Theorem 0.0.1 of Shelah and Barnea from [BS]. Now note that we can decompose the map (π (lim X )) → lim (π (X )) 1 ←− i ab ←− 1 i ab in the following way: (π (lim X )) → (lim π (X )) → lim (π (X )) 1 ←− i ab ←− 1 i ab ←− 1 i ab 12 DANIL AKHTIAMOV

Then the map is surjective as a composition of two surjective maps and its kernel is an extension of a cotorsion group by a cotorsion group. Thus it is obvious from our second definition of cotorsion groups that the kernel itself is a cotorsion group . Q.E.D.

4. References [B] Reinhold Baer. The subgroups of elements of finite order of an Abelian group, Ann. of Math., 37, (1936), 766-781. [Bous] A. K. Bousfield. Homological localization towers for groups and π-modules, Mem. Amer. Math. Soc, no. 186, 1977. [Brown] K.S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, vol. 87, Springer, Berlin, 1982. [Breen] Lawrence Breen. On the functorial homology of abelian groups, Journal of Pure and Applied Algebra 142 (1999) 199-237 [BS] Ilan Barnea, Saharon Shelah. The abelianization of inverse limits of groups, Israel Journal of Mathematics, August 2018, Volume 227, Issue 1, pp. 455-483. [D] W.G. Dwyer. Homological localization of π-modules, Journal of Pure and Applied Algebra, Volume 10, Issue 2, November 1977, Pages 135-151. [F] L. Fuchs. Infinite Abelian Groups, Academic Press, Jan 1, 1970. [GJ] Goerss P. G., Jardine J. F. Simplicial Homotopy Theory, Progress in Math- ematics, Vol. 174, Birkhauser, Basel, 1999. [H] Warfield, R. B., Jr., Huber, Martin. On the values of the functor lim1, Arch. ←− Math. (Basel) 33 (1979/80), no. 5, 430-436. [Hill] Jonathan Hillman. Algebraic Invariants of Links, Series on Knots and Ev- erything, Vol. 52. [IM] Sergei O. Ivanov, Roman Mikhailov. On a problem of Bousfield for metabelian groups, Advances in Mathematics, Volume 290, 26 February 2016, Pages 552-589. [IM2] Sergei O. Ivanov, Roman Mikhailov. A finite Q-bad space, arXiv:1708.00282, 1 August 2017 [IM3] Sergei O. Ivanov, Roman Mikhailov. On discrete homology of a free pro-p- group, Compositio Mathematica, Volume 154, Issue 10, October 2018 , pp. 2195-2204. [KM] O. Kharlampovich, A. Miasnikov. Implicit function theorem over free groups and genus problem Proceedings of the Birmanfest, AMS/IP Studies in Advanced Math- ematics, v. 24, 2001, 77-83. [MP] Roman Mikhailov, Inderbir Singh Passi. Lower Central and Dimension Series of Groups, Lecture Notes in Mathematics, 2009. [W] Charles A. Weibel. An Introduction to , Cambridge University Press, 1994.

Laboratory of Modern Algebra and Applications, St. Petersburg State University, 14th Line, 29b, Saint Petersburg, 199178, Russia E-mail address: akhtyamoff[email protected]