arXiv:0802.2390v2 [math.GT] 30 Sep 2009 HOMOLOGICAL STABILITY OF SERIES OF GROUPS TIM COCHRAN† AND SHELLY HARVEY† Abstract. We define the stability of a subgroup under a class of maps, and establish the basic properties of this notion. Loosely speaking, we will say that a normal subgroup, or more generally a normal series {An } of a group A, is stable under a class of homomorphisms H if whenever f : A → B lies in H, we have that f (a) ∈ Bn if and only if a ∈ An . This translates to saying that each element of H induces a monomorphism A/An ֒→ B/Bn . This contrasts with the usual theories of localization wherein one is concerned with situations where f induces an isomorphism. In the literature, the most commonly considered class of maps are those that induce isomorphisms on (low-dimensional) group homology. The model theorem in this regard is the 1963 result of J. Stallings that (each term of) the lower central series is preserved under any Z-homological equivalence of groups [9]. Various other theorems of this nature have since appeared, involving different series of groups- variations of the lower central series. W. Dwyer generalized Stallings’ Z results to larger classes of maps [7], work that was completed in the other cases by the authors. More recently the authors proved analogues of the theorems of Stallings and Dwyer for variations of the derived series [4] [6] [5]. The above theorems are all different but clearly have much in common. We interpret all of these results in the framework of stability. 1. Introduction Loosely speaking we will say that a function, Γ, that assigns to each group A a normal subgroup Γ(A) ⊳ A is stable under a class of maps H if whenever f : A → B lies in H, we have that f (a) ∈ Γ(B) if and only if a ∈ Γ(A). Precise definitions are given in Section 2. This translates to saying that each element of H induces a monomorphism A/Γ(A) ֒→ B/Γ(B). This contrasts with the usual theories of localization wherein one is concerned with situations where f induces an isomorphism. The stabilization, ΓS , of Γ under H is the function assigning the “smallest” normal subgroup that contains Γ(A) and has the property that each f ∈ H induces a monomorphism A/ΓS (A) ֒→ B/ΓS (B). We are motivated by the general question: “What subgroups of a group are unchanged, or stable, under homology equivalences”? The model theorem in this regard is the landmark 1963 result of J. Stallings (below) that (each term of) the lower central series of a group A is preserved under any homological equivalence. Recall that An is defined recursively by A1 ≡ A and An+1 = [A, An ]. Theorem 1.1. [9, Theorem 3.4](Stallings’ Integral Theorem) If f : A → B is a group homomorphism that induces an isomorphism on H1 (−; Z) and an epimorphism on H2 (−; Z) then for each n, f induces an isomorphism A/An ∼ = B/Bn . Therefore a ∈ An if and only if f (a) ∈ Bn . Stallings’ Integral Theorem implies that (each term of) the lower central series is stable under the set of Z-homologically 2-connected maps. Stallings had analogous theorems for Q and Zp (our convention is that Zp denotes the integers modulo p) that involved different series- variations of the lower-central series [9, Theorem 7.3]. William Dwyer improved on Stallings’ theorem by weakening the hypothesis on H2 and finding the precise class of maps † Both authors were partially supported by the National Science Foundation. The second author was partially supported by a fellowship from the Sloan Foundation and by an NSF CAREER grant. 1 2 TIM COCHRAN† AND SHELLY HARVEY† f which, for a fixed n, yield isomorphisms modulo the nth term of the lower central series [7, Theorem 1.1]. This was placed in a larger context by A. Bousfield [1]. Dwyer’s work implies that the lower central series is stable under a larger class of maps. However, the lower central series fails to be stable if one considers Q-homologically 2-connected maps, as may be seen by considering the map Z2 → {e}. This situation is remedied by enlarging the lower central series slightly to form the rational lower central series (see Section 2). The rational lower central series is stable under rational homology equivalences as a consequence of Stallings’ Rational Theorem (see Section 3). More recently the authors found analogues of the theorems of Stallings and Dwyer for the derived series [4] [6]. However to do so it was necessary to expand the derived series and use a larger series, the torsion-free-derived series, introduced by the second author [8, Section 2]. All of these theorems and series will be reviewed as necessary in later sections. The derived series fails dramatically to have the stability property under homological equivalences. For example if we consider the abelianization map below f Z[t, t−1 ]/ < t2 − t + 1 > ⋊Z ≡ A −→ B ≡ Z we see that it is a homologically 2-connected map. But any non-zero element of the commutator subgroup of A, (Z[t, t−1 ]/ < t2 − t + 1 >), is not itself in the second derived subgroup of A, yet maps trivially under f and hence lies in the second term of the derived series of Z. Can the n-th term of the derived series be enlarged until it is “stable” under homologically-2-connected maps? In this case expanding to the “rational derived series” fails. Certainly it can be so enlarged since we could enlarge it all the way to the 2n -th term of the lower-central series which is stable under 2-connected maps. What is the minimum it needs to be enlarged to become stable? These questions motivate the definitions to follow. 2. Stability of Series of Groups Let G be the category of groups and let C be a subcategory. Definition 2.1. A subgroup function (short for normal subgroup function) for C is a function Γ : C → C, assigning to each A ∈ C a normal subgroup Γ(A) of A. Definition 2.2. A series (short for normal series) for C, {Γn }, is a collection of subgroup functions Γn : C → C, n ≥ 0, such that {e} ⊂ · · · ⊂ Γn+1 (A) ⊂ Γn (A) · · · ⊂ Γ1 (A) ⊂ Γ0 (A) = A. The important examples to keep in mind are the lower central series {An }; the rational lower central series, {Arn }, defined by Ar1 = A, Arn+1 = {x | xk ∈ [A, Arn ] for some positive integer k}; and, for a fixed prime p, the Zp -lower central series, {Ap,n }, (also called the p-lower central series or the lower central p-series) which is defined by Ap,1 = A, Ap,n+1 = (Ap,n )p [Ap,n , A]. This is the fastest descending central series whose successive quotients are Zp -vector spaces [9]. For economy of words we make the following definition. Definition 2.3. For R = Z, Q or Zp where p is prime, respectively, let the R-lower central series, {AR n }, be the lower central series, rational lower central series, or Zp -lower central series, respectively. HOMOLOGICAL STABILITY OF SERIES OF GROUPS 3 One also has the derived series, {A(n) }, given by A(0) = A, A(n+1) = [A(n) , A(n) ]; (n) the rational derived series, {Ar }, defined by (n) A(0) = A, A(n+1) = {x | xk ∈ [A(n) r r , Ar ] for some positive integer k}), (n) and, the Zp -derived series, {Ap }, (also called the p-derived series or the derived p-series) which is defined by p (n) (n) (n+1) A(0) = (A(n) p ) [Ap , Ap ]. p = A, Ap This is the fastest descending series whose successive quotients are Zp -vector spaces [9]. Definition 2.4. A class of maps, H, for C is a subset of the morphisms of C that contains all isomorphisms, is closed under composition and is closed under nudge-outs, where by the latter we mean that if f , f ′ ∈ H, as in the diagram below, then there exist g and g ′ in H that make the diagram commute. f A −−−−→ B     f ′y yg g′ B ′ −−−−→ C Clearly being closed under push-outs implies being closed under nudge-outs. Definition 2.5. A subgroup function Γ is H-invariant with respect to the class of maps H if whenever f : A → B is an element of H then f (Γ(A)) ⊂ Γ(B) (This is the same as saying that, for each n, Γ is functorial on the category C with morphisms restricted to lie in H.) A series {Γn | n ≥ 0} is H-invariant with respect to a collection of classes of maps H = {Hn | n ≥ 0} ⊆ Hn ⊆ . . . ⊆ H1 ⊆ H0 ⊂ M orph(C) if whenever f : A → B is an element of Hn then f (Γn (A)) ⊂ Γn (B). Remark 2.6. Each of the versions of the lower central series and derived series defined above, consisting (essentially) of verbal subgroups, is H-invariant with respect to any class of maps. Definition 2.7. Suppose Γ is an H-invariant subgroup function. The stabilization of Γ with respect to H, denoted ΓS , is the subgroup ΓS (A) = {a ∈ A | ∃f : A → B, f ∈ H, such that f (a) ∈ Γ(B)}. We say that Γ is stable under H if ΓS (A) = Γ(A) for each A in C. Suppose {Γn } is an H-invariant series. The stabilization of {Γn } with respect to H = {Hn }, denoted {ΓnS }, is the series wherein ΓnS is the stabilization of Γn with respect to Hn . We say that the series {Γn } is stable under H if each term is stable under the corresponding class of maps. e then clearly ΓS,H ⊂ Γ e . Remark 2.8. If H ⊂ H S,H We first verify that ΓS is itself a subgroup function. Proposition 2.9. If Γ is an H-invariant subgroup function then ΓS is an H-invariant subgroup function. If {Γn } is an H-invariant series then {ΓnS } is an H-invariant series. 4 TIM COCHRAN† AND SHELLY HARVEY† Proof of Proposition 2.9. First we show that ΓS (A) is a normal subgroup of A. Clearly the identity e lies in ΓS (A). Suppose a, a′ ∈ ΓS (A), so there exist f , f ′ ∈ H such that f (a) ∈ Γ(B), f ′ (a′ ) ∈ Γ(B ′ ). Since H is closed under nudge-outs, there exists C, as shown below, with g, g ′ ∈ H. f A −−−−→   f ′y g′ B   yg B ′ −−−−→ C Since H is closed under composition, g ◦ f = g ′ ◦ f ′ ∈ H. Since Γ is H-invariant, we also have g ′ (f ′ (a′ )) ⊂ g ′ (Γ(B ′ )) ⊂ Γ(C) g(f (a)) ⊂ g(Γ(B)) ⊂ Γ(C). Thus (g ◦ f )(aa′ ) = (g ◦ f )(a) · (g ◦ f )(a′ ) = g(f (a)) · g ′ (f ′ (a′ )) implying that (g ◦ f )(aa′ ) ∈ Γ(C). Thus aa′ ∈ ΓS (A). Moreover if f (a) ∈ Γ(B) then f (a−1 ) ∈ Γ(B) since Γ(B) is a subgroup, so ΓS is closed under taking inverses. Similarly, if f (a) ∈ Γ(B) then f (xax−1 ) = f (x)f (a)(f (x))−1 lies in Γ(B) since Γ(B) is a normal subgroup. Therefore ΓS (A) is a normal subgroup of A. Next we show that ΓS is H-invariant. Suppose f : A → B, f ∈ H and a ∈ ΓS (A). Then there exists f ′ ∈ H such that f ′ (a) ∈ Γ(B ′ ). Since H is closed under nudge-outs, there exists C, as below with g and g ′ in H and g ◦ f = g ′ ◦ f ′ ∈ H. f A −−−−→   f ′y g′ B   yg B ′ −−−−→ C ′ ′ Since Γ is H-invariant, g (f (a)) ∈ Γ(C). Thus (g ◦ f )(a) ∈ Γ(C). Since g ∈ H, it follows that f (a) ∈ ΓS (B) as desired. Finally suppose that {Γn } is an H = {Hn }-invariant series. First we show that Γn+1 (A) ⊂ ΓnS (A). S n+1 Suppose a ∈ ΓS (A) so there exists f ∈ Hn+1 such that f (a) ∈ Γn+1 (B) then, by our nesting requirement on H, f ∈ Hn and f (a) ∈ Γn (B), so a ∈ ΓnS (A). Finally, if Γ0 (A) = A then, since Γ0 (A) ⊂ Γ0S (A),  A = Γ0 (A) = Γ0S (A). The following establishes the salient properties of the stabilization of a subgroup function. Theorem 2.10. Suppose H is a class of maps and Γ is an H-invariant subgroup. Then 1) for each A ∈ C, Γ(A) is a normal subgroup of ΓS (A) 2) for every f : A → B, f ∈ H, f induces a monomorphism A/ΓS (A) ֒→ B/ΓS (B). e 3) ΓS is the initial H-invariant subgroup function satisfying 1) and 2) above. To be specific, if {Γ} is an H-invariant subgroup function such that: e 3.1) Γ(A) ⊂ Γ(A) ∀A ∈ C, and e e 3.2) every f ∈ H induces a monomorphism A/Γ(A) ֒→ B/Γ(B), e then ΓS (A) ⊂ Γ(A) ∀A ∈ C. Proof of Theorem 2.10. Given x ∈ Γ(A), since the identity map iA is in H, x ∈ ΓS (A). Normality was established in the proof of Proposition 2.9. This establishes 1). Suppose f : A → B, f ∈ H. Since ΓS is H-invariant by Proposition 2.9, f induces a map f : A/ΓS (A) → B/ΓS (B). Now suppose f (a) ∈ ΓS (B). Then, by definition, there exists g ∈ H, g : B → C HOMOLOGICAL STABILITY OF SERIES OF GROUPS 5 such that g ◦f (a) ∈ Γ(C). Since H is closed under composition, g ◦f ∈ H. Hence a ∈ ΓS (A), establishing 2). e is another H-invariant series satisfying 3.1 and 3.2. Suppose a ∈ ΓS (A), so there Now suppose Γ e Γ(B) ⊂ Γ(B), e e exists f ∈ H, f : A → B, such that f (a) ∈ Γ(B). By property 3.1 for Γ, so f (a) ∈ Γ(B). e f induces a monomorphism By property 3.2 for Γ e e f : A/Γ(A) ֒→ B/Γ(B). e e Hence a ∈ Γ(A). Thus ΓS (A) ⊂ Γ(A) establishing 3).  Now we arrive at our major tool for determining the stabilization. Corollary 2.11. If Γ is an H-invariant subgroup function for which every f ∈ H induces a monomorphism A/Γ(A) −→ B/Γ(B), then, ∀A ∈ C, ΓS (A) = Γ(A) , that is, Γ is stable under H. If {Γn } is an H-invariant series for which every f ∈ Hn induces a monomorphism A/Γn (A) −→ B/Γn (B), then {Γn } is stable under H. Proof of Corollary 2.11. Since Γ satisfies 3.1 and 3.2 above, ΓS (A) ⊂ Γ(A). By property 1) of Theorem 2.10, Γ(A) ⊂ ΓS (A).  Corollary 2.12. (Idempotency) The stabilization of an H-invariant subgroup function (or series), with respect to H, is itself stable with respect to H. Proof of Corollary 2.12. By Proposition 2.9 and part 2) of Theorem 2.10, ΓS satisfies the hypothesis of Corollary 2.11. Thus ΓS is itself stable under H.  3. Stabilizations of the R-Lower Central Series Let R = Z, Q or Zp where p is prime. Suppose C is the category of all groups and consider H = HR the set of all homologically 2-connected maps with R-coefficients; that is, homomorphisms that induce isomorphisms on H1 (−; R) and epimorphisms on H2 (−; R). It is an easy exercise to show that these n classes are closed under push-outs. Let Γn A = AR n+1 , i.e. Γ is the (n + 1)-st term of the R-lower central R series. Results of Stallings establish that maps in H induce monomorphisms modulo any term of the R-lower central series (for R = Z see Theorem 1.1, for R = Q see [9, Theorem 7.3], for R = Zp see [9, Theorem 3.4]). Therefore, using Corollary 2.11, these can be reinterpreted as: Proposition 3.1. The R-lower central series is stable with respect to all 2-connected maps with Rcoefficients, that is, ΓnS (A) = Γn (A) under HR . We will show that the R-lower central series is stable under a much larger class of maps than HR . The work of Dwyer suggested the following filtration of H2 (A; R). Definition 3.2. For any group A and positive integer n, and R = Z, Q or Zp , let ΦR n (A) denote the kernel of the natural map H2 (A; R) → H2 (A/An ; R). R (A) ⊂ Φ Note that, if m ≥ n then ΦR n (A). m This in turn suggests the following class of maps. R R Definition 3.3. For R = Z, Q or Zp , let HDwyer ≡ {(HDwyer )n } be the class of maps, called the Dwyer R )n if it induces an isomorphism on H1 (−; R) and an epimorphism R-class, wherein a map is in (HDwyer R R H2 (A; R)/Φn (A) → H2 (B; R)/Φn (B). TIM COCHRAN† AND SHELLY HARVEY† 6 One easily checks that these classes are closed under push-outs. R Theorem 3.4. The R-lower central series is stable with respect to Dwyer’s R-class of maps, HDwyer . R Proof. By Remark 2.6, the R-lower central series Γn (A) = AR n+1 is HDwyer -invariant. By Corollary 2.11, R n it suffices to show that any map f : A → B such that f ∈ (HDwyer ) induces a monomorphism R A/AR n+1 −→ B/An+1 . In the case R = Z this was established by Dwyer [7, Theorem 1.1]. More recently the authors showed this in the cases R = Q and R = Zp [6, Theorem 3.1][5, Theorem 3.1].  Despite these positive results there are simple questions that are unanswered: Question 3.5. Is the stabilization of the lower central series under HR (for R = Zp or Q) equal to the R-lower central series? We do not know the answer to the above question but we can show: R Proposition 3.6. The stabilization of the lower central series under HDwyer is the R-lower central R series. More generally, the stabilization, under HDwyer , of any series that is contained in the R-lower central series, is the R-lower central series. This is the first case we have discussed where the stabilization is strictly larger than the original series, and we can calculate the stabilization precisely. e n (A) = AR Proof. Suppose Γn (A) ⊂ Γ n+1 . Thus property 3.1) of Theorem 2.10 holds. By the above mentioned theorems ([7, Theorem 1.1][6, Theorem 3.1][5, Theorem 3.1]) property 3.2) also holds. Hence, e n (A). by Theorem 2.10, ΓnS (A) ⊂ Γ n e We must show that Γ (A) ⊂ ΓnS (A). This is trivially true for n = 0 so assume that n > 0. Suppose e n (A) = AR a∈Γ n+1 . Consider the projection map: f : A → A/hai ≡ B. n R Since f (a) = 1, f (a) ∈ Γ (B). Hence, if we can establish that f ∈ (HDwyer )n then, by definition, n R a ∈ ΓS (A). Since n + 1 ≥ 2, a ∈ A2 . Note that H1 (A; Zp ) = A/Ap,2 = A/AR 2 when R = Zp , and H1 (A; Q) = A/Ar2 ⊗ Q = A/AR 2 ⊗ Q when R = Q. Thus in all cases it follows that f induces an isomorphism on H1 (−; R). It is fairly easy to show that, since the kernel of f is contained in AR n+1 , f induces an isomorphism R ∼ A/AR n+1 = B/An+1 . R To show that f ∈ (HDwyer )n it now suffices to show that f induces an epimorphism: H2 (A; R) → H2 (B; R)/ΦR n (B). In the cases R = Z, Q, Zp this follows from [7, Theorem 1.1],[6, Theorem 3.1] and [5, Theorem 3.1] respectively, which are converses to the part of those theorems that were utilized above. In the case of [5, Theorem 3.1] this is not explicitly stated, but the reader can see that it follows directly from the last commutative diagram in the proof of [5, Theorem 3.1].  HOMOLOGICAL STABILITY OF SERIES OF GROUPS 7 4. The Stabilization of the p-Derived Series The derived series in not stable under homological equivalences. However, we find that the p-derived series behaves more like the lower central series. The underlying reason for this is that, if A is finitely (n) (n) generated, then A/Ap is a finite p-group and hence nilpotent. In this section let Γn (A) = Ap Proposition 4.1. The Zp -lower central series is stable with respect to all 2-connected maps with Zp coefficients between finitely generated groups, that is, ΓnS (A) = Γn (A) under HZp (but restricting to finitely generated groups). Proof of Proposition 4.1. This is a direct consequence of Corollary 2.11 and the recent result of the authors [5, Corollary 4.3].  In fact, this recent work of the authors allows us to also show that the Zp -lower central series is stable with respect to a larger class of maps that induce only monomorphisms on H1 (−; Zp ). This leaves the realm of the usual homological localization theory, which is concerned with homological equivalences (see Section 6). Proposition 4.2. The Zp -lower central series is stable with respect to the class of all maps between finitely generated groups that induce a monomorphism on H1 (−; Zp ) and an epimorphism on H2 (−; Zp ). We will now show that the Zp -lower central series is stable under a much larger class of maps. First we define a filtration of H2 (A; Zp ) analogous to Dwyer’s but appropriate for the derived series. (n) Definition 4.3. For a group A and a non-negative integer n, let Φp (A) denote the image of the inclusion-induced map H2 (A(n) p ; Zp ) −→ H2 (A; Zp ). This in turn suggests the following class of maps. Definition 4.4. Let HpCH ≡ {(HpCH )n } be the class of maps whose elements are maps f : A → B where A is finitely generated, B is finitely presented and where f induces an isomorphism on H1 (−; Zp ) and (n−1) (n−1) an epimorphism H2 (A; Zp )/Φp (A) → H2 (B; Zp )/Φp (B). One easily checks that this class is closed under push-outs. Proposition 4.5. The Zp -lower central series is stable with respect the HpCH , and in fact is stable with respect to the larger class of maps which induce a monomorphism on H1 (−; Zp ) rather than an isomorphism. Proof of Proposition 4.5. This is a direct consequence of Corollary 2.11 and [5, Theorem 4.2].  5. The Stabilization of the Derived Series The derived series and the rational derived series, unlike the lower central series and the p-derived series, are highly unstable under homology equivalences. Indeed, until quite recently, nothing much was known or even suspected about their properties under homological equivalences. In 2003, the second (n) author introduced a superseries of the derived series, the torsion-free derived series, GH [8, Section 2], for which the authors were able to prove analogues of the theorems of Stallings and Dwyer [CH1] [CH2]. Here we examine the relationships between the torsion-free derived series and the stabilizations of the ordinary derived series with respect to several natural classes of maps. 8 TIM COCHRAN† AND SHELLY HARVEY† (0) (n) We first recall the recursive definition of the torsion-free derived series. Let GH ≡ G. Suppose GH (n) (n) (n) (n) has been defined as a normal subgroup of G. Then GH /[GH , GH ] is a right Z[G/GH ]-module (G (n) acts by conjugation). Since Z[G/GH ] is an Ore domain [8, Prop. 2.1], this module has a well-defined (n+1) is defined to be the inverse image of T under the natural map torsion submodule T . Then GH (n) (n) (n) (n) π : GH −→ GH /[GH , GH ]. (n) (n) An easy induction shows G(n) ⊂ Gr ⊂ GH . We define several natural classes of maps and then investigate the stabilization of the derived series with respect to these classes. Recall that the classes HZ and HQ have been previously defined as those maps that are homologically 2-connected with Z or Q coefficients respectively. But in this section we will use these symbols to designate these same classes but restricted to those 2-connected maps f : A → B wherein A is finitely generated and B is finitely presented. (n) Definition 5.1. For a group A and a non-negative integer n, let Φ(n) (A) (respectively ΦQ (A)) denote the image of H2 (A(n) ; Z) → H2 (A; Z) (respectively H2 (A(n) ; Q) → H2 (A; Q)). The analogy between this filtration and Dwyer’s filtration involving the lower central series will not be apparent. See [6, Section 1] for a discussion. This suggests the following classes of maps. CH CH n Definition 5.2. Let HCH ≡ {(HCH )n } (respectively HQ ≡ {(HQ ) }) be the class whose elements are maps f : A → B where A is finitely generated, B is finitely presented and where f induces an isomorphism on H1 (−; Z) and an epimorphism H2 (A; Q)/Φ(n−1) (A) → H2 (B; Z)/Φ(n−1) (B) (respectively, (n−1) (n−1) an isomorphism on H1 (−; Q) and an epimorphism H2 (A; Q)/ΦQ (A) → H2 (B; Q)/ΦQ (B)). One easily checks that these classes are closed under composition and push-outs. It is also clear that CH CH HZ ⊂ HCH , HQ ⊂ HQ and HCH ⊂ HQ . The following gives an “upper bound” on the stabilization of the derived series. Theorem 5.3. Let Γn (A) = A(n) , the derived series. Let ΓnS be the stabilization of the derived series (n) CH . Then A(n) ⊂ ΓnS (A) ⊂ AH ∩ Ar2n . The same holds for stabilization with respect with respect to HQ CH to any class of maps contained in HQ , such as HZ , HQ and HCH . (Indeed the same holds if enlarge CH the class HQ by relaxing the isomorphism condition on H1 to a monomorphism condition). CH Proof of Theorem 5.3. It follows from [6, Theorem 2.1] that the torsion-free derived series is HQ (n) (n) CH n ) induces a monomorphism A/AH ֒→ B/BH . Since the torsion-free invariant, and that any f ∈ (HQ derived series contains the derived series, we conclude that the torsion-free derived series satisfies 3.1 (n) and 3.2 of Theorem 2.10 (letting Γ̃n (A) = AH ) and hence the stabilization of the derived series is no (n) larger than the torsion-free derived series, i.e. ΓnS (A) ⊂ AH . On the other hand, if we consider the series given by Γ̃n (A) = Ar2n , the 2n -th term of the rational lower central series, then we claim that this satisfies 3.1 and 3.2 of Theorem 2.10. Once having shown this, ΓnS (A) ⊂ Γ̃n (A) = Ar2n . n Q CH n )2 −1 . ) . We shall show that this implies that f ∈ (HDwyer Now we prove the claim. Suppose f ∈ (HQ Then the claim follows directly from [6, Theorem 3.1]. CH n Since f ∈ (HQ ) , by Definitions 5.2 and 5.1, it induces an epimorphism (n−1) f∗ : H2 (A; Q) → H2 (B; Q)/ΦQ (B). HOMOLOGICAL STABILITY OF SERIES OF GROUPS 9 and we need to show (by Definitions 3.3 and 3.2) that it induces an epimorphism f∗ : H2 (A; Q) → H2 (B; Q)/ΦQ 2n −1 (B). (n−1) Therefore it suffices to show that ΦQ (B) ⊂ ΦQ 2n −1 (B). For this it suffices to show that the composition π i ∗ ∗ H2 (B/B2n −1 ; Z) H2 (B; Z) −→ H2 (B (n−1) ; Z) −→ is the zero map. Since B (n−1) ⊂ B2n−1 , this follows from Lemma 5.4 below (setting k = 2n−1 , then 2k − 1 = 2n − 1). Lemma 5.4. For any group B and integer k, the map π i ∗ ∗ H2 (B/B2k−1 ; Z) H2 (B; Z) −→ H2 (Bk ; Z) −→ is the zero map. Proof of Lemma 5.4. Suppose Bk is presented by < F ′ | R′ >. Of course i(F ′ ) ⊂ Bk , but we can choose a generating set F for B so large that i(F ′ ) ⊂ Fk . Suppose B is presented by < F | R >. It follows that B/B2k−1 is presented by < F | R > where R = < R, F2k−1 >. Then consider the following commutative diagram where the vertical maps are isomorphisms by Hopf’s theorem [2, Theorem 5.3 p.42]. H2 (Bk ) i∗ ✲ H2 (B) ∼ ∼ = = ❄ ❄ R′ ∩ [F ′ , F ′ ] ✲ i R ∩ [F, F ] ′ ′ [R , F ] [R, F ] π✲ ∗ H2 (B/B2k−1 ) ∼ = ❄ π✲ R ∩ [F, F ] [R, F ] Now note that i∗ ([F ′ , F ′ ]) ⊂ [Fk , Fk ] ⊂ F2k . But F2k = [F2k−1 , F ] ⊂ [R, F ]. Hence π ◦ i = 0 and the result follows.  CH Finally, for any class H contained in HQ , ΓnS,H ⊂ ΓnS,HCH by Remark 2.8.  Q We seek to characterize the stabilization of the derived series more precisely. Although the class of CH maps HQ may appear be the most natural extension to the derived series of Dwyer’s class of maps, the following slightly larger class may be even more natural. Indeed for this class we are able to better characterize the stabilization of the derived series by providing both a “lower bound” and “an upper bound.” Both of these bounds are related to torsion elements of the module A(n) /A(n+1) . (n) Definition 5.5. For any group B and non-negative integer, let ΦH (B) ⊆ H2 (B; Q) be the image of (n) (n) H2 (BH ; Q) → H2 (B; Q), where BH is the torsion-free derived series. (n) (n) Definition 5.6. Let HH = {HH } where HH is the set of homomorphisms f : A → B where A is finitely generated, B finitely-related where f induces an isomorphism on H1 (−; Q) and induce an (n−1) (n−1) epimorphism H2 (A; Q)/ΦH (A) → H2 (B; Q)/ΦH (B). It is easy to see that HH is closed under composition and, using [4, Proposition 2.3], closed under push-outs. (0) (n+1) (n) Definition 5.7. Let AC ≡ A and let AC be the subgroup generated by the set of elements x ∈ AC (n) (n) (n) (n) that represent torsion elements of the Z[A/AC ]-module AC /[AC , AC ] that are annihilated by some (n) (n) γ ∈ Z[A/AC ] whose image under the augmentation Z[A/AC ] → Z is non-zero. TIM COCHRAN† AND SHELLY HARVEY† 10 (n) Proposition 5.8. {AC }n≥0 is a normal series for A whose successive quotients are Z-torsion free and, for each n, (n) (n) A(n) ⊂ AC ⊂ AH ∩ Ar2n . Proof of Proposition 5.8. The proof is by induction on n. Suppose the Proposition holds for all values (n) less than or equal to n. An element x ∈ AC represents a torsion element as above precisely when there exists γ = Σki gi ∈ ZA such that Σki 6= 0 and [x] ∗ [γ] = 0, which translates to the condition: Y (5.1) (n) (n) gi−1 xki gi ∈ [AC , AC ]. i (n+1) is a product of such x. To Note that if satisfies x 5.1 then so does x−1 . A general element of AC (n+1) −1 is normal in A it suffices to show that, for any g ∈ A, g xg also satisfies the condition show that AC above. But this is easily seen by setting hi = g −1 gi and observing that Y Y −1 gi−1 xki gi . h−1 xg)ki hi = i (g i i Clearly A(n+1) ⊂ the diagram below: (n+1) AC by definition. To show (n) AC   iy (n) AH (n+1) AC (n+1) ⊂ AH π (n) (n) (n) π (n) (n) (n) (n) (n) assuming that AC ⊂ AH , consider −−−C−→ AC /[AC , AC ]  i y∗ −−−H−→ AH /[AH , AH ]. (n) (n+1) It suffices to show that, for any x ∈ AC with property 5.1, i(x) ∈ AH . But the image, γ̄, of γ (n) in Z[A/AH ] clearly annihilates πH (i(x)) and so it is only necessary to remark that γ̄ 6= 0 since it has (n+1) by definition. non-zero augmentation. Thus i(x) ∈ AH (n+1) (n) r To show that AC ⊂ A2n+1 , assuming AC ⊂ Ar2n , it again suffices to consider a single x satisfying 5.1. We will prove x ∈ Ar2n+1 by another induction. Suppose, by induction, that x ∈ Arm+2n for some (n+1) (n) 0 ≤ m ≤ 2n − 1. Certainly this is true for m = 0 since x ∈ AC ⊂ AC ⊂ Ar2n by our other inductive hypothesis. Consider our hypothesis (n) (n) Σgi−1 xki gi ∈ [AC , AC ] ⊂ [Ar2n , Ar2n ] ⊂ Ar2n+1 ⊂ Arm+1+2n as a statement in the torsion-free abelian group Arm+2n /Arm+1+2n . Since x ∈ Arm+2n , gi−1 xki gi ≡ xki i modulo Arm+1+2n . Thus our hypothesis simplifies to (Σki ) · [x] = 0 implying that [x] = 0 in this quotient and hence that x ∈ Arm+1+2n . Iterating this process yields that x ∈ Ar2n+1 .  (n) Theorem 5.9. On the class of finitely-presented groups, the stabilization, AS , of the derived series with respect to the Harvey class of maps HH satisfies (n) (n) (n) AC ⊂ AS ⊂ AH (and each of these series is HH -invariant). HOMOLOGICAL STABILITY OF SERIES OF GROUPS 11 Proof. Assume that A is finitely-presented. The proof is by induction on n. The case n = 0 is clear since each of the series above is defined to be A itself in that case. Supposing that the theorem holds for all integers ≤ n, we establish it for n + 1. We again invoke the main theorem of [6] (in a stronger from than that used previously). (n) Theorem 5.10. (Cochran-Harvey [6, Theorem 2.1]) If f : A → B and f ∈ HH then f induces a monomorphism (n+1) (n+1) . ֒→ B/BH A/AH (n+1) Corollary 5.11. AH (n−1) is HH (n) (n+1) -invariant (and hence HH -invariant and HH -invariant). (n−1) HH , Proof of Corollary 5.11. If f ∈ f : A → B, then by Theorem 5.10, f induces a monomorphism (n+1) (n+1) (n) (n) . Thus the (n+1)-st ) ⊂ BH A/AH → B/BH . It then follows from [4, Proposition 2.3] that f (AH (n−1) (n+1) (n) (n−1) term of the torsion-free derived series is HH -invariant. Since HH ⊂ HH ⊂ HH , the other statements follow immediately.  (n) It follows that the series Γ̃n (A) = AH satisfies properties 3.1 and 3.2 of Theorem 2.10 for Γn (A) = (n) (n) A(n) with respect to HH and so, by Theorem 2.10, AS ⊆ AH . (n+1) (n+1) (n+1) (n+1) . It suffices to consider ⊂ A/AS assuming that A/AC ⊂ AS Now we prove that AC (n+1) x ∈ AC that satisfies (5.1) since a general element is a product of such x. Consider the projection map: f : A → A/ < x >≡ B. Note that B is also finitely presented. Since f (x) = 1, f (x) ∈ B (n+1) . Hence, if we can establish that (n+1) (n+1) f ∈ HH then, by definition, x ∈ AS with respect to HH . Since n ≥ 1, x ∈ Ar2 by Proposition 5.8, so a multiple of x lies in the commutator subgroup of A. Hence f induces an isomorphism on H1 (−; Q). Thus it suffices to show that f induces an epimorphism (n) H2 (A; Q) → H2 (B; Q)/ΦH (B). (n) (n) Choose a surjection φ : F̄ → AH where F̄ is free (a generating set for AH ) and extend this to a surjection φ : F → A where F ⊂ F and F is free (a generating set for A). Then say A is presented by < F | R > and B is presented by < F | R, x >. The cokernel of H2 (A; Q) → H2 (B; Q) under Hopf’s identifications < R, x > ∩[F, F ] R ∩ [F, F ] ⊗Q∼ ⊗Q = H2 (A; Q) → H2 (B; Q) ∼ = [R, F ] [< R, x >, F ] is generated by the class of x ∈ F . Since gi−1 xki gi ≡ xki modulo [< R, x >, F ], Y gi−1 xki gi ≡ xΣki . y= i Thus y also generates the cokernel of H2 (A; Q) → H2 (B; Q) (under the indentifications above). It suffices (n+1) (n) , by [4, Proposition 2.5], the map f now to show that y is in the image of H2 (BH ). Since x ∈ AH above induces an isomorphism (n) (n) A/AH ∼ = B/BH . TIM COCHRAN† AND SHELLY HARVEY† 12 (n) (n) (n) (n) In particular this implies that AH → BH is surjective. Hence F̄ → AH → BH is surjective and we (n) may suppose that BH ∼ =< F̄ | R̄ >. Hence < R̄ > ∩[F̄ , F̄ ] (n) H2 (BH ; Q) ∼ ⊗ Q. = [< R̄ >, F̄ ] Since by hypothesis (n) (n) (n) (n) y ∈ [AC , AC ] ⊂ [AH , AH ], y is represented by an element of [F̄ , F̄ ]. Moreover this element is surely in R̄ since y represents the (n)  trivial element in B. Thus the class represented by y is in the image of H2 (BH ). 6. Relationship with homological localization of groups In this section we show that if one considers the class of homologically 2-connected maps then the stabilization is related to certain homological localizations that were previously in the literature. However, this fact does not seem to assist in calculating the stabilization. Suppose R = Q or Z and let HR be the set of homomorphisms f : A → B where A is finitely generated, B is finitely presented, and f induces a 2-connected map on R-homology. For any finitely generated group A there exists a group  and a functorial assignment θ : A → Â, called the R-closure of A [3, Theorem 6.1], with the following properties: 1. For any f : A → B in HR f induces an isomorphism fˆ :  ∼ = B̂, and 2. For any finitely presented group A, there is a sequence of finitely presented groups Ai and maps hi ∈ HR h h h A →0 A1 →1 . . . Ai →i . . .  (6.1) such that lim Ai ∼ = Â. −→ Proposition 6.1. Suppose {Γn } is an HR -invariant series. Suppose also that {Γn } commutes with direct limits of maps in HR . Then, for any finitely presented group A, the stabilization, ΓnS (A) with respect to HR is the kernel of θ π A  → Â/Γn (Â). A→ e n (A) denote this kernel. We must first verify that Γ e n (A) is an HR -invariant series. For Proof. Let Γ e n (Â)) ⊂ Γ e n (B̂). Suppose a ∈ Γ e n (Â). Then this it suffices to show that, for any f : A → B in HR , f (Γ n n θA (a) ∈ Γ (Â). By 6.1 and since {Γ } commutes with direct limits, (6.2) n Γn (Â) = − lim → Γ (Ai ) so there is some i such that hi (a) ∈ Γn (Ai ). Now consider the composition fˆ Ai →  → B̂. Since B̂ = limBj and Ai is finitely generated, there is some j such that this composition factors through −→ Bj as below hi A ✲ Ai ✲  f f fˆ ❄ ❄Hj ❄i B ✲ Bj ✲ B̂ HOMOLOGICAL STABILITY OF SERIES OF GROUPS 13 Since hi , f and Hj are in HR , fi ∈ HR . Since Γn is HR -invariant, fi (hi (a)) ∈ Γn (Bj ). Thus Hj (f (a)) ∈ Γn (Bj ). Since {Γn } commutes with direct limits of maps in HR , Γn (B̂) = lim Γn (Bj ). −→ n n e e n (A) is an HR -invariant series. Hence θB (f (a)) ∈ Γ (B̂) so f (a) ∈ Γ (B̂) as desired. Thus Γ n n By 6.2, θA (Γ (A)) ⊂ Γ (Â) so e n (A), Γn (A) ⊂ Γ e n satisfies 3.1 of Theorem 2.10. We next show that Γ e n satisfies 3.2 of that theorem. So and thus Γ R ˆ suppose f : A → B lies in H . By property 1 of the R−closure, f is an isomorphism. We claim that fˆ(Γn (Â)) ⊂ Γn (B̂). For, if α ∈ Γn (Â) then, as above α is represented by some αi ∈ Γn (Ai ). The argument above then shows that fˆ(α) ∈ Γn (B̂). Hence fˆ induces a map fˆn : Â/Γn (Â) −→ B̂/Γn (B̂). It will follow that this map is an isomorphism, if we can verify that (fˆ)−1 induces a map on these same quotients (going the other way). This is accomplished by establishing that (fˆ)−1 is necessarily induced by a family of maps Bj → Aij , which are in HR , and then proceeding as above. Therefore fˆn is an isomorphism. Finally consider [a] in the kernel of fn e n (A) → e n (B). A/Γ B/Γ e n (B) implies that πB (θB (f (a))) = 0. From the diagram below and the fact that By definition, f (a) ∈ Γ ˆ e n (A). Thus fn above is a monomorphism and so fn is an isomorphism, it follows that a ∈ Γ θA✲ πA n A  ✲ Â/Γ (Â) f fˆ fˆn ❄ ❄ ❄ θB B ✲ B̂ π✲ B B̂/Γn (B̂) e n satisfies 3.2 of Theorem 2.10. By that theorem then Γ e n (A). ΓnS (A) ⊂ Γ Finally, we show that e n (A) ⊂ Γn (A), Γ S e n = Γn . Suppose a ∈ Γ e n (A) so θA (a) ∈ Γn (Â). We saw earlier in the which complete our proof that Γ S proof that this implies that there is some i such that hi (a) ∈ Γn (Ai ). Since hi ∈ HR , by definition, a ∈ ΓnS (A).  References [1] A. K. Bousfield. Homological localization towers for groups and Π-modules. Mem. Amer. Math. Soc., 10(186):vii+68, 1977. [2] Kenneth S. Brown. Cohomology of groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1982. [3] Jae Choon Cha. Injectivity theorems and algebraic closures of groups with coefficients. Proc. Lond. Math. Soc. (3), 96(1):227–250, 2008. 14 TIM COCHRAN† AND SHELLY HARVEY† [4] Tim Cochran and Shelly Harvey. Homology and derived series of groups. Geom. Topol., 9:2159–2191, 2005. [5] Tim D. Cochran and Shelly Harvey. Homology and derived p-series of groups. J. London Math. Soc., 78(3):677–692, 2008. [6] Tim D. Cochran and Shelly Harvey. Homology and derived series of groups. II. Dwyer’s theorem. Geom. Topol., 12(1):199–232, 2008. [7] William G. Dwyer. Homology, Massey products and maps between groups. J. Pure Appl. Algebra, 6(2):177–190, 1975. [8] Shelly L. Harvey. Homology cobordism invariants and the Cochran-Orr-Teichner filtration of the link concordance group. Geom. Topol., 12(1):387–430, 2008. [9] John Stallings. Homology and central series of groups. J. Algebra, 2:170–181, 1965. Rice University, Houston, Texas, 77005-1892 E-mail address: cochran@rice.edu, shelly@rice.edu