We give some elementary examples of the torsion-free derived series. Example 2. Thus the torsion-free derived series stabilizes at precisely n. The fundamental group of any 3—manifold that fibers over a circle has this property [12, Proposition 8. By contrast, the derived series of the fundamental group of a knot exterior does not stabilize for any finite n unless the Alexander polynomial is 1 [5, Corollary 4. There are many non- free groups where the torsion-free derived series does not stabilize at a finite ordinal.
Thus, by Theorem 4. It will be important for our main theorem that the reader understand the connection between the torsion-free derived series and group homology.
This is provided by the following basic observations. H Proof Property 1 follows directly from Remark 2. We also give an application to when a set of elements of a group generates a free subgroup similar to that of Stallings. Deeper secondary applications will appear in [11].
Theorem 3. Suppose that A is finitely-generated and B is finitely related. Proof The theorem follows immediately from Theorem 4. Corollary 3. In addition, the ranks of the above modules are new homology cobordism invariants. A common example of this Corollary is the case of the exteriors of concordant links. As above, Harvey uses Corollary 3.
For example, note that the ranks of the above modules are new concordance invariants of links, generalizing the well-known fact that the rank of the Alexander module of a link is an invariant of concordance. The above-mentioned Cheeger—Gromov invariants are used in [11] to show that the concordance group of disk links in any odd dimension has infinite rank even modulo local knotting.
Stallings theorem also gives a beautiful criterion to establish that a set of ele- ments of a group generates a free subgroup. We have our own version, although we do not know an example where it is stronger than Stallings result and there are examples where it is weaker. Proposition 3. Proof Following Stallings, let F be a free group of rank k equipped with the obvious map into G determined by the xi.
Then apply Corollary 4. In the case that B, A admits the structure of a relative 2—complex, the first and third conclusions above remain valid without the finiteness assumptions on A and B. Proof of Theorem 4. It follows from Proposition 2. Hence the diagram below exists and is commutative. Assuming this, we finish the inductive proof. By the assertion, [a] is ZAn —torsion, contradicting the choice of [a].
The injectivity at the first infinite or- dinal follows immediately, finishing the inductive step of the proof of the first part of Theorem 4. The kernel of i is precisely this submodule since tensoring with the quotient field kills precisely the torsion submodule. Therefore it suffices to show that the other two maps in the composition are injective. Lemma 4. Moreover, the KH —rank of the domain of this map equals the KG—rank of the range.
Proof of Lemma 4. Since any KH —module is free [26, Proposition I. This will follow immediately from Proposition 4. The latter is immediate since we have previously observed that KBn is a flat ZBn —module. For the former, note that, since any module over a division ring is free, KBn is a free, and hence flat, KAn module. Moreover KAn is a flat ZAn —module.
This completes the proof of the first claim of Theorem 4. The following proposition is an important result in its own right. The more general result below seems to require a different proof. Proposition 4. Suppose also that A is finitely generated and B is finitely related. Before proving Proposition 4. Thus, if the pair of Eilenberg—Maclane Spaces K B, 1 , K A, 1 has the homotopy type of a relative 2—complex, then we do not need these finiteness assumptions to deduce the first part of the theorem.
The fact that this is a monomorphism follows from the first part of the theorem. By Lemma 4. Hence Proposi- tion 4. Thus the latter has K Bn —rank equal to rA.
Note also that since this argument does not use the argument of the preceding paragraph, it holds without the finiteness assumptions on A and B if B,A is 2—dimensional in the sense described. This concludes the proof of our main theorem, modulo the proof of Proposi- tion 4. Proof of Proposition 4. In [27, page ], Strebel shows that, under the hypotheses of Lemma 4. Consequently, any PTFA group is in this class. Now we continue with the proof of Proposition 4. Moreover the homology of a group G is well known to be the same as that of its associated Eilenberg—Maclane space K G, 1 [13, page ].
Proof of Corollary 4. Then we only need comment on why it is not necessary to require that F be finitely generated. Remark 4. Such groups arise commonly as the fundamental groups of the exteriors in B 4 of a set of ribbon disks for a ribbon link. The same example shows that the part of the conclusion of Theorem 4.
As we explain below, in the context of rational homological localization, this can be viewed as an analogue of the Malcev completion wherein one replaces the lower central series by the torsion-free derived series. We parallel A. At the end of the section we also compare our lo- calization to the universal integral homological localization functor due to P Vogel and J Levine.
This notion differs from ours. It is related to the lower-central series of the commutator subgroup. In particular, it is not invariant under homological equivalence. Theorem 5. However, we have been unable to prove the result, which we expect en is initial in the appropriate sense among functors satisfying is true, that our G the properties of Theorem 5.
Without this fact, the analogy to the Malcev completion is incomplete. The problem seems to be a failure of functoriality since the torsion-free-derived series is not fully invariant. The authors expect this to be repaired by the modifications of Remark 5. Definition 5. In particular, any such group is poly- torsion-free-abelian. Proof that Theorem 5. Property 1 of Theorem 5. Proof of Theorem 5. Since G en is torsion-free-solvable it is poly-torsion-free- e abelian so ZGn is an Ore domain as in Proposition 2.
Then an extension of G en by K is determined as above. In fact we shall see in Lemma 5. Lemma 5. Proof of Lemma 5. By Lemma 5. But this property of a classical quotient field is trivial to check see [26, Proposition 3.
This requires a short induction. Proof Following Stallings, let F be a free group of rank k equipped with the obvious map into G determined by the xi. Then apply Corollary 4. In the case that B, A admits the structure of a relative 2—complex, the first and third conclusions above remain valid without the finiteness assumptions on A and B. Proof of Theorem 4. It follows from Proposition 2. Hence the diagram below exists and is commutative.
Assuming this, we finish the inductive proof. By the assertion, [a] is ZAn —torsion, contradicting the choice of [a]. The injectivity at the first infinite or- dinal follows immediately, finishing the inductive step of the proof of the first part of Theorem 4.
The kernel of i is precisely this submodule since tensoring with the quotient field kills precisely the torsion submodule. Therefore it suffices to show that the other two maps in the composition are injective. Lemma 4. Moreover, the KH —rank of the domain of this map equals the KG—rank of the range. Proof of Lemma 4. Since any KH —module is free [26, Proposition I. This will follow immediately from Proposition 4.
The latter is immediate since we have previously observed that KBn is a flat ZBn —module. For the former, note that, since any module over a division ring is free, KBn is a free, and hence flat, KAn module.
Moreover KAn is a flat ZAn —module. This completes the proof of the first claim of Theorem 4. The following proposition is an important result in its own right. The more general result below seems to require a different proof.
Proposition 4. Suppose also that A is finitely generated and B is finitely related. Before proving Proposition 4. Thus, if the pair of Eilenberg—Maclane Spaces K B, 1 , K A, 1 has the homotopy type of a relative 2—complex, then we do not need these finiteness assumptions to deduce the first part of the theorem. The fact that this is a monomorphism follows from the first part of the theorem. By Lemma 4. Hence Proposi- tion 4.
Thus the latter has K Bn —rank equal to rA. Note also that since this argument does not use the argument of the preceding paragraph, it holds without the finiteness assumptions on A and B if B,A is 2—dimensional in the sense described.
This concludes the proof of our main theorem, modulo the proof of Proposi- tion 4. Proof of Proposition 4. In [27, page ], Strebel shows that, under the hypotheses of Lemma 4. Consequently, any PTFA group is in this class. Now we continue with the proof of Proposition 4. Moreover the homology of a group G is well known to be the same as that of its associated Eilenberg—Maclane space K G, 1 [13, page ]. Proof of Corollary 4. Then we only need comment on why it is not necessary to require that F be finitely generated.
Remark 4. Such groups arise commonly as the fundamental groups of the exteriors in B 4 of a set of ribbon disks for a ribbon link. The same example shows that the part of the conclusion of Theorem 4. As we explain below, in the context of rational homological localization, this can be viewed as an analogue of the Malcev completion wherein one replaces the lower central series by the torsion-free derived series. We parallel A. At the end of the section we also compare our lo- calization to the universal integral homological localization functor due to P Vogel and J Levine.
This notion differs from ours. It is related to the lower-central series of the commutator subgroup. In particular, it is not invariant under homological equivalence. Theorem 5. However, we have been unable to prove the result, which we expect en is initial in the appropriate sense among functors satisfying is true, that our G the properties of Theorem 5. Without this fact, the analogy to the Malcev completion is incomplete. The problem seems to be a failure of functoriality since the torsion-free-derived series is not fully invariant.
The authors expect this to be repaired by the modifications of Remark 5. Definition 5. In particular, any such group is poly- torsion-free-abelian. Proof that Theorem 5. Property 1 of Theorem 5. Proof of Theorem 5. Since G en is torsion-free-solvable it is poly-torsion-free- e abelian so ZGn is an Ore domain as in Proposition 2.
Then an extension of G en by K is determined as above. In fact we shall see in Lemma 5. Lemma 5. Proof of Lemma 5. By Lemma 5. But this property of a classical quotient field is trivial to check see [26, Proposition 3. This requires a short induction. This completes the inductive proof that sm exists and is an isomorphism.
By Proposition 2. But by Lemma 5. Now we verify Property 3. Details are left to the reader. This completes the proof of Theorem 5. Corollary 5. It follows that the torsion-free-solvable completion may be constructed analo- gously to the Malcev completion.
Proof of Corollary 5. Thus, by Theorem 5. Example 5. For we observed in Example 2. Thus by Corollary 5. Note that the algebraic closure of Zm is Zm but the Malcev completion and the torsion-free-solvable completion are Qm. For we saw in Example 2. Thus, by Corollary 5. This is a flaw in the torsion-free derived 1 series that would be corrected by Remark 5.
In particular, as we showed in Theorem 5. He pointed out that, if G is finitely presented, it coincides with a notion previously investigated by P. Vogel see [17].
The Levine—Vogel algebraic closure is a uni- versal integral homological localization in the following sense. The proof follows exactly the proof of Theorem 4. The Lemma would follow directly if B were finitely related. However this hypoth- esis on B is only needed in the inductive step to establish the monomorphism conclusion of Proposition 4.
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