Whitehead's Problem is UndecidableAuthor(s): Paul C. EklofReviewed work(s):Source: The American Mathematical Monthly, Vol. 83, No. 10 (Dec., 1976), pp. 775-788Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2318684 .Accessed: 06/03/2012 11:39

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1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 775

2. CBMS Newsletter, 10 (May-June, 1975) 47. 3. , 10 (Oct.-Nov., 1975) 55. 4. Erwin H. Bareiss, The college preparation for a mathematician in industry, this MONTHLY, 79 (1972)

972-984. 5. R. E. Gaskell and M. S. Klamkin, The industrial mathematician views his profession: A report of the

Committee on Corporate Members, this MONTHLY, 81 (1974) 699-716. 6. Charles A. Hall, Industrial mathematics: A course in realism, this MONTHLY, 82 (1975) 651-659. 7. Keith B. Oldham and Jerome Spanier, The Fractional Calculus, Academic Press, New York, 1974.

DEPARTMENT OF MATHEMATICS, CLAREMONT GRADUATE SCHOOL, CLAREMONT, CA 91711.

WHITEHEAD'S PROBLEM IS UNDECIDABLE

PAUL C. EKLOF

1. Introduction. The working mathematician, unless he is studying the foundations of mathema- tics, usually does not find it necessary to make explicit references to axioms of set theory - except perhaps to invoke the Axiom of Choice or the Continuum Hypothesis. As long as his arguments can be carried out within the framework of a commonly accepted system such as Zermelo-Frankel set theory (ZF), his set-theoretic assumptions can remain unexpressed. However, mathematicians have known since the work of Godel that there are mathematical statements that are undecidable (i.e., neither provable nor refutable) on the basis of Zermelo-Frankel set theory. (In fact, any consistent axiomatization of set theory which can be effectively written down will have such undecidable statements: see, for example, Monk [13; Thm 1].) In recent years, a number of concrete examples of undecidable statements have been discovered. Probably the most famous is the Continuum Hypothesis (2No = M1) proved undecidable in ZFC (= ZF + Axiom of Choice) by Godel and Cohen. Other examples belong to topology and analysis. (See Rudin [16] and Shoenfield [18].) In this paper, we are going to discuss an algebraic example: a famous problem which had resisted the best efforts of mathematicians for many years before it was recently proved by Shelah to be undecidable in ZFC. His method of proof is to show that two axioms, each consistent with ZFC, yield contradictory answers to the problem.

In order to state the problem, we need some definitions. Let 1A denote the identity function on A and let Z denote the group of integers. A surjective homomorphism of abelian groups 7: B -> A is said to split if there is a homomorphism p: A -- B such that 7rp = 1A. An abelian group A is called a W-group if it satisfies the property: for all surjective homomorphisms 7 : B -> A, if the kernel of 7 is isomorphic to Z, then r splits. It is not hard to see that a free abelian group is a W-group. (See Corollary 2.4.) Whitehead's Problem asks whether the converse is true. In homological terms, Whitehead's Problem asks whether Ext (A, Z) = 0 implies A is free (see section 3).

The problem also has an equivalent formulation in terms of topological groups: is every compact arcwise-connected abelian group a product of copies of the circle group, R/Z? The equivalence follows from the Pontryagin Duality Theorem.

It was proved by Stein in 1951 that every countable W-group is free. (See Theorem 4.1.) But for uncountable groups only partial results have been obtained in ZFC.

Set-theorists have studied various hypotheses which may consistently be added to ZF. (We shall assume that ZF is consistent.) One of these is Godel's Axiom of Constructibility, denoted V = L. (Godel defined a subclass L of the class V of all sets and proved that the axioms of ZF as well as the Axiom of Choice and the Continuum Hypothesis are true in L. The sets in L are called constructible sets. The Axiom of Constructibility asserts that every set is constructible.) Godel proved the following:

776 P. C. EKLOF [December

1.1 THEOREM [5]. (i) ZF + V = L is consistent; (ii) ZF + V = L implies the Axiom of Choice and the Continuum Hypothesis.

It follows from the theorem that ZFC is consistent (assuming, of course, that ZF is consistent). Another axiom that is important to us is Martin's Axiom (MA), which was introduced originally in

connection with Souslin's Problem (see Shoenfield [18; ?5]). We shall state (a weak form of) MA in section 7. Here we state only the important consistency theorem proved by Solovay and Tennenbaum using the forcing technique introduced by Cohen.

1.2 THEOREM [19]. ZFC + MA + 2o ?> X, is consistent.

Now we can state Shelah's theorem, which shows that at least for cardinality N, Whitehead's Problem is not decidable on the basis of ZFC. (See also Section 8.)

1.3 THEOREM [17]. (i) ZFC + V = L implies that every W-group of cardinality M, is free; (ii) ZFC + MA + 2Mo > Ml implies that there is a W-group of cardinality t, which is not free.

It follows that either the affirmative or the negative answer to Whitehead's Problem is consistent with ZFC. The reader may still ask: which answer is "true"? The question presumes a "platonist" philosophy, i.e., that sets really exist in some sense so that it is meaningful to ask about the "truth" of a given assertion about sets (see Monk [13]). Neither of the hypotheses V = L or MA + 2No > I4 is one which mathematicians would agree is intuitively "true" about sets. Thus Shelah's theorem does not settle the "truth" about Whitehead's Problem. It is conceivable that in the future some new axiom about sets will come to be as accepted as are the axioms of ZFC and that it will yield a definitive solution to Whitehead's Problem (as well as to other undecidable problems, such as the Continuum Hypothesis).

In the remainder of this paper, we shall give the algebraic portion of the proof of the undecidability of Whitehead's Problem. (The set-theoretic pre-requisites, i.e., Theorems 1.1 (i), 1.2, 6.1 and 7.1 will not be proved here.) The two parts of Theorem 1.3 are proved, respectively, in sections 6 and 7.

We begin in the next section with a study of properties of free groups.

2. Free groups. Throughout the rest of the paper "group" will mean abelian group. Functions which are group homomorphisms will be denoted by lower case Greek letters, and functions which are simply set mappings will be denoted by lower case Roman letters.

Recall that a group is free if it has a basis, that is, a linearly independent generating set. One of the most important results about free groups is the following, which is a special case of a theorem about modules over a principal ideal domain. (See, for example, Lang [11; Thm. 4, p. 45].)

2.1 THEOREM. A subgroup of a free group is free.

Another useful result is the so-called Fundamental Theorem of Abelian Groups which in the case of torsion-free groups may be stated as follows. (See, for example, Lang [11; Thm. 7, p. 49].)

2.2 THEOREM. A finitely-generated torsion-free group is free.

We shall prove a characterization of free groups in terms of the notion of splitting which will immediately imply that every free group is a W-group. First, let us call a homomorphism p: A - B a splitting homomorphism for T B -:>A if 7p = 1A. Note that p is necessarily injective since p(a) = 0 implies a = vp(a) = O.

2.3 THEOREM. A group A is free if and only if every homomorphism onto A splits.

Proof. Suppose that A is free and that 7r: B -> A is surjective. If X = {xi: i E I} is a basis of A, choose b, in B for each i in I such that r(bi) = xi. Since X is a basis of A there is one and only one

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 777

homomorphism p: A -> B such that p(xi) = bi for each i E I. Clearly p is a splitting homomorphism for v.

To prove the converse, consider a free group F with basis X = {Xa a E A}. Let 7 : F-> A be the unique homomorphism such that r(xa) = a for all a E A. By hypothesis, there is a splitting homomorphism p: A -> F for v. Since p is injective, A is isomorphic to a subgroup of F; therefore, by Theorem 2.1, A is free.

2.4 COROLLARY. Every free group is a W-group.

2.5 COROLLARY. Suppose B is a subgroup of A such that B and A /B are both free. Then A is free; moreover, any basis of B extends to a basis of A.

Proof. Let r: A --A /B be the canonical surjection (i.e., r(a) = a + B for all a E A). Since A /B is free, the theorem implies that there is a splitting homomorphism p for v. Then A = p(A/B) D B. (Indeed, for any a E A, a = pir(a) + (a - pr(a)), which is the unique representation of a as a sum of elements of p(A/B) and B respectively.) If Y is a basis of A/B, then p(Y) is a basis of p(A/B) since p is injective. Therefore, if X is any basis of B, then p(Y) U X is a basis of A. This proves the corollary.

An important element in the analysis of Whitehead's Problem is the study of ascending chains of groups. Consider an ascending chain of sets

AoCA1C -- CA5C vA<a

indexed by an ordinal a. This chain is called a smooth chain if for every limit ordinal A < a, AA = UV<AAV. The chain is called strictly increasing if for every v < a, A,# A,+,. Finally, the chain is a chain of groups if for every v < a, A, is a group which is a subgroup of A,+,.

2.6 THEOREM. Let A be the union of a smooth chain of groups {A, I v < a} such that Ao is free and for every v <: a, A,+,/A, is free. Then A is free; moreover, for every v < a, A/A, is free.

Proof. Let XO be a basis of Ao. We shall construct by transfinite induction on v < a a smooth chain of sets

XO5X1 C * *C Xv c5 v < a

such that Xv is a basis of A, This will suffice since X = Uv<aXv will be a basis of A and {x + A, I x E X - Xv} will be a basis of A /A,. Suppose we have already exhibited a chain

X C XI .. Xv v r</

with the desired properties, for some At < a. We must produce X,L. If At is a limit ordinal, let X,. = UX<, Xv; then X, is a basis of U <,.A, which equals A,, by the definition of a smooth chain. If At is a successor ordinal, say At = 8 + 1, then A,,1/A, is free by hypothesis, so by Corollary 2.5, X8 extends to a basis X8+? of A,,1. This completes the proof.

3. Properties of W-groups. In this section, and this section only, we shall assume familiarity with the rudiments of homological algebra (e.g., Chapters 1 and 3 of Jans [8]) in order to prove three properties of W-groups. Only the statements of the following three theorems are used in the rest of the paper, so the reader who is willing to accept them on faith - or is able to supply his own proofs -

can skip the remainder of this section.

3.1 THEOREM. A subgroup of a W-group is a W-group.

3.2 THEOREM. Every W-group is torsion-free.

3.3 THEOREM. If Bo is a subgroup of B,, such that B1 is a W-group but Bj/Bo is not a W-group, then there exists a homomorphism q: Bo -- Z which does not extend to a homomorphism: B1 --> Z.

778 P. C. EKLOF [December

In abelian group theory, Ext is usually defined as a group of equivalence classes of short exact sequences (see Fuchs [4; Chapter IX]) but here we shall define it in terms of free resolutions for the benefit of those familiar with an introductory text in homological algebra such as Jans [8] or Northcott [14].

Let us recall some definitions. If A and C are groups, Hom (A, C) is the group of all homomorphisms p :A -- C, where addition is defined by ('Pi + <p2)(a).= <p,(a) + <p2(a). For every group C, a homomorphism o: A,--> A2 induces a homomorphism

cr': Hom (A2, C) ->Hom (A, C)

defined by o-'(<p) = gpo- for >, E Hom (A2, C). Recall that a sequence

-*E Ai+, ---> Ai

is exact if Ker oi = Im 0i?+ for every i. A free resolution of a group A is a short exact sequence

(*) F FoAO>

where Fo (and hence F1) is free. Given a free resolution (*) of A, we define

Ext (A, C) = Hom (F1, C)IIm 8'.

(This is called Ext1 (A, C) in [8], Extl (A, C) in [14].) It may be proved that Ext (A, C) does not depend on the choice of the free resolution (see [8; p. 35]). The following theorem is crucial (see [8; p. 41f] for a proof).

3.4 THEOREM. For any exact sequence

0 --*A 1->A2-*A3 >0

and any group C there is an exact sequence

0-> Hom (A3, C)-4 Hom (A2, C) Hom (A1, C)-> Ext (A3, C)-> Ext (A2, C)-> Ext (Ai, C)->0.

In order to make use of Theorem 3.4, we need to express in terms of Ext the property of being a W-group.

3.5 THEOREM. A group A is a W-group if and only if Ext(A, Z) = 0.

Proof. Suppose A is a W-group and consider a free resolution (*). We must prove that Im 8'=Hom (F1,Z). Given oi: F1->Z, define B =(Z ?D Fo)/I where I={(>pj(y), - 8(y))|y E F1}. Then we have a commutative diagram

O-> F1 Fo A > 0

(3.5.1) (P i {O J1A O >Z '>B r> A >0

where the homomorphisms r, 6 and t are defined, respectively, by: 7((n, x) +I)= c(x); 0(x)= (0, x) + I; t(n) (n, 0) + I. The bottom line of (3.5.1) is exact.

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 779

Since A is a W-group, there is a r: B -> Z such that rt = 1,. If we let 'po = rO, it is easily verified that 8'(qpo) = 'pi. Hence 8' is onto and Ext (A, Z) = 0.

Conversely suppose Ext (A, Z) = 0 and consider a short exact sequence

(3.5.2) O Z->Z BA ->O.

Let Fo be as in (*) and let 0: Fo -> B be a surjective homomorphism such that 70 = E. (Note that 0 exists since Fo is projective; see [8; pp. 7, 8].)

Then there is a homomorphism ,: F1 -> Z such that we have a commutative diagram (3.5.1). By hypothesis there exists >:o: Fo-> B such that (poS = (p,. We claim that 'po(x) = 0 if x E Ker 0. Indeed if x E Ker 0, then s(x) = rO(x) = 0, so x = 8(y) for some y E F1; therefore 'po(x) = 'poa(y) = 'p1(y) = 0, the last since t is injective and upi(y) = 05(y) = 0(x) = 0. Thus (po induces a map r: B -> Z. We claim that rt = 14. Indeed, if n E Z, rn(n) = (po(x) where 0(x) = t(n). Now ?(x) = 7r0(x) = Ir(n) = 0, so by exactness, x = 8(y) for some y E Fl. Hence Spo(x) = 'poS(y) = 'pi(y). Moreover, qpi(y) =08(y) = 0(x) = t(n). Since t is injective, n = Sp,(y) = nt(n). Therefore (3.5.2) splits and A is a W-group. The theorem is proved.

We are now ready to prove the three theorems stated at the beginning of this section.

Proof of 3.1. Suppose A1 is a subgroup of a W-group A2. There is an exact sequence

0 ->Ai- A2-4 A2/A, -O,

where o-, is the inclusion map. By Theorem 3.4 there is an exact sequence

Ext (A2, Z)-> Ext (A,, Z)-> 0.

Since A2 is a W-group, Theorem 3.5 implies Ext (A2, Z) = 0, and therefore Ext (Al, Z) = 0. Again by Theorem 3.5, Al is a W-group.

Proof of 3.2. If A is not torsion-free, then there exists a E A such that (a) is a non-zero finite cyclic group.

By Theorem 3.1 it suffices to prove that Z/nZ is not a W-group for any n > 0. Consider the canonical projection

X : Z ->Z/nZ.

The kernel of ir is isomorphic to Z, but clearly ir does not split (since Z is torsion-free). Hence Z/nZ is not a W-group.

Proof of 3.3. Consider the exact sequence

Bo->B B1-> B/Bo->O,

where o-, is the inclusion map. By Theorem 3.4 there is an exact sequence

Hom (B1, Z)--4 Hom (Bo, Z)-- Ext (B/Bo, Z)-> Ext (B1, Z).

By hypothesis and Theorem 3.5, Ext (B1, Z) = 0 and Ext (B1lBo, Z) # 0. Therefore or is not surjective, which is precisely the conclusion of Theorem 3.3.

4. Countable W-groups. The principal goal of this section is to prove the following result, which is a converse of Corollary 2.4 for countable groups. Shorter proofs than ours may be given, but our proof has the advantage that it is a paradigm for the proof of Theorem 1.3 (i).

780 P. C. EKLOF [December

4.1 THEOREM [20]. Every countable W-group is free.

Before embarking on the proof of 4.1 we shall prove an important characterization of countable free groups. A subgroup B of a torsion-free group A is called a pure subgroup if A /B is torsion-free. (A warning: this definition is not the proper definition of pure if A is not torsion-free; see Kaplansky [10; p. 14].) If B is a subgroup of A the pure closure of B in A is the subgroup B' = {a E A I na E B for some n $ 0}. It is clearly a pure subgroup of A. If A is free and B is finitely-generated, then B' is free by Theorem 2.1. Clearly B' has the same dimension as B so if B is finitely-generated so is B'. We have therefore proved that if A is free then every finitely-generated subgroup of A is contained in a finitely-generated pure subgroup of A. The following result known as Pontryagin's Criterion, says that the converse is true for countable torsion-free groups.

4.2 THEOREM [15]. Let A be a countable torsion-free group such that every finitely-generated subgroup of A is contained in a finitely-generated pure subgroup of A. Then A is free.

Proof. Since A is countable we can enumerate the elements of A in a sequence:

A = {an j n < w}.

We define by induction on n < w a (smooth) chain {B| I n < } of finitely-generated pure subgroups of A. Let Bo = 0. If Bn has been defined, let Bn+1 be a finitely-generated pure subgroup of A containing Bn U {a,}. The union of this chain is clearly A. Consider the quotient Bn+1/Bn: it is torsion-free because Bn is pure in A; it is finitely-generated because Bn+l is finitely-generated. Therefore, by Theorem 2.2, Bn+l/Bn is free. The proof of the theorem is completed by an application of Theorem 2.6.

Theorem 4.2 is not true if we remove the hypothesis of countability. In section 7, we shall exhibit a torsion-free group of cardinality Xi which satisfies a condition even stronger than the hypothesis of Theorem 4.2 and yet is not free.

Let us adopt the convention that whenever C is a set (or group) of the form B x Z, r will denote the projection map onto the first factor, i.e., rr(b, n) = b for all (b, n) E B x Z. In context there will be no ambiguity as to the domain of v. If B is a group we define a (B, Z)-group to be a group C whose underlying set is B x Z such that v-: C-> B is a homomorphism and (0, n) + (0, m) = (0, n + m) for all n, m in Z. The simplest example is B (? Z, i.e., the set B x Z equipped with the group operation. given by (b1, n1) + (b2, n2) = (b1 + b2, ni + n2). Notice that for any (B, Z)-group C, the kernel of v is Z, so if v : C-> B does not split then B is not a W-group.

4.3 LEMMA. Let Bo be a subgroup of B1 such that B1 is a W-group but B1/Bo is not a W-group. Let Co be a (B0, Z)-group and p a splitting homomorphism for v : CO-> Bo. Then there is a (B1, Z)-group C, which is an extension of CO such that p does not extend to a splitting homomorphism for 77: C1-> B1.

Proof. Since 77: Co-> Bo splits, there is an isomorphism r : Bo Z-> Co given by r(b, n)= p(b) + (0, n). Notice that r-'p(b) = (b, 0) for all b E Bo. Hence we may assume that CO = Bo DZ and that p(b) = (b, 0) for all b E Bo. Let Cl = B1 @ Z and let 0: Bo-> Z be the homomorphism given by Theorem 3.3. Define y : CO-> Cl by y(b, n) = (b, n + +(b)). Suppose that there is a splitting homomorphism pl : B1 -> C1 for I : Cl -> B1 such that pl I Bo = y p. Let p = v o pi : B1 -> Z. Then for any b E Bo, p(b) = rri1(b) = rryp(b) = +f(b). Thus (p is an extension of q, which contradicts Theorem 3.3. We conclude that no such 5l exists. If y were inclusion we would be done; since it is not, we must employ a little trick. Define a set map f: C1 --B1 x Z by

(b,n) if b Bo f(b, n)=

(b, n-q,(b)) if bEBo.

Clearly f is a bijection and f o y is the inclusion of BoD? Z into B1 x Z. Let C, be B1 x Z with the

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 781

group structure which makes f a group isomorphism (i.e., for any u, v E B x Z u + v = f(f-1(u) + f-(v)). Then C1 is an extension of CO and by the property of y proved above, there is no splitting homomorphism pi : B1 -> C1 for r7: C1 -> B1 which extends p : Bo-> Co.

Proof of 4.1. By Theorem 3.2, we know that A is torsion-free, so to prove Theorem 4.1 it will suffice to prove that A satisfies the hypothesis of Pontryagin's Criterion. Assume it does not; that is, there is a finitely-generated subgroup Bo of A which is not contained in a finitely-generated pure subgroup of A. Let B be the pure closure of Bo in A. By our assumption B is not finitely-generated. Hence B is the union of a strictly-increasing chain of finitely-generated groups

Bo0B1" *..B .*** n <w.

Note that B/Bo is a torsion group by the definition of the pure closure. We shall construct by induction on n a chain of groups

CO CZCl *CZ..Cnc n n< ,

such that Cn is a torsion-free (Bn, Z)-group. The union C= U n Cn will then be a torsion-free (B, Z)-group. Our goal is to define the Cn's so that r-: C-> B does not split, thereby contradicting Theorem 3.1.

Let S be a finite set of generators for Bo. We claim that if C is torsion-free, any homomorphism p B -> C is completely determined by its values on S. Indeed for any b E B, there is an n X 0 such that nb E Bo. Now p(nb) is clearly determined by the values of p on S and since C is torsion-free, the equation nx = p(nb) has only one solution in C, viz x = p(b).

Let {gn I n < } be a list of all the (set) maps gn S-> S x Z such that ign = is. (There are only a countable number of such maps since S is finite and Z is countable.) Now we proceed to define the Cn 's. Let CO be Bo ?D Z. If Cn has been defined, we consider two cases. In the first case, if gn extends to a splitting homomorphism p for r : Cn -> Bn, let Cn+i be an extension of Cn such that p does not extend to a splitting homomorphism for r: Cn+1 -> Bn+l. (Note that Cn+i exists by Lemma 4.3 since Bn+l/Bn is torsion and therefore not a W-group by Theorem 3.2.) If the first case does not hold, let p be any splitting homomorphism for r: Cn -> Bn and define Cn+i as above. (At least one such p exists by Theorem 2.2 since Bn is finitely generated, torsion-free and therefore free.)

To complete the proof of 4.1 we must show that r7: C-> B does not split. If there is a splitting homomorphism p: B -> C for X then p I S = gn for some n. But then p I Bn is a splitting homomor- phism for r7: Cn -> Bn which is an extension of gn and extends to a splitting homomorphism for

Cn+i- BBn,+ This contradicts the construction of Cn+i and hence the proof is finished. Theorem 4.1 yields some information about arbitrary W-groups. Let us call a group A N,-free if

every subgroup of cardinality less than M, (i.e. countable) is free. Putting together Theorems 3.1 and 4.1 we obtain the following result.

4.4 COROLLARY. Every W-group is Xi-free.

5. Chase's condition. We shall consider a generalization of the hypothesis of Pontryagin's Criterion (Theorem 4.2). First of all, notice that '"H-free" is a natural generalization of "torsion-free" since by Theorem 2.2 "torsion-free" is equivalent to "every finitely-generated subgroup is free." Taking this as our cue, we define a generalization of "pure." If A is Xr-free, call a subgroup B of A an X1-pure subgroup if A/B is Xi-free. We shall refer to the following as Chase's condition.

(5.1) A is an Xi-free group such that every countable subgroup of A is contained in a countable Xr-pure subgroup of A.

(This terminology is not standard. We have chosen it since Chase [1] proved, under the assumption of

782 P. C. EKLOF [December

the Continuum Hypothesis, that every W-group satisfies (5.1).) As we shall see later (Theorem 7.3), Chase's condition does not imply that A is free (unless A is countable). But it does play a central role in our proof of Theorem 1.3. In this section we shall give a necessary and sufficient condition for a group of cardinality Ml satisfying Chase's condition to be free. We begin by expressing the condition in terms of ascending chains. (Here w, is the first uncountable ordinal, i.e. the first ordinal of cardinality xi.)

5.2 LEMMA. If A is a group of cardinality X,, A satisfies Chase's condition (5.1) if and only if A is the union of a smooth chain of countable free groups

AoC .. *C*5A, C * * *, v < ct)

such that AO = 0 and for each ordinal v < w1, A,+1 is Xi-pure in A.

Proof. Suppose A satisfies (5.1). Since A has cardinality Xi we can list all the elements of A in a sequence of length w1:

A = {a, I V < vO.

Define A. by induction on v < wi. Let AO be 0. If A,1 has been defined for all ,u < v consider two cases. First, if v is a limit ordinal, let A. = U,1<VA,!. (A countable union of countable sets is countable.) Second, if v is a succossor ordinal, say v = ,t + 1, let AD be a countable X,-pure subgroup of A which contains A,1 U {a, }. The chain constructed in this fashion clearly satisfies the properties stated in the lemma. Conversely, if A is the union of a smooth chain of the type described, then A satisfies Chase's condition because any countable subgroup B of A is contained in A,+1 for some v < wl. This completes the proof of the lemma.

Referring to the notation of the lemma, let E be the set of all limit ordinals A < W1 such that A, is not Xi-pure in A. We shall prove that the "size" of E determines whether or not A is free. First we need some definitions.

A function f: w1 -> w1 is called normal if it is strictly increasing (i.e., ,u < v implies f(Au) < f(v)) and continuous (i.e., for any limit ordinal A, f(A) = supff(v)I v < A}). Notice that if f is normal then the image of f is unbounded in w, since the image is uncountable. A subset S of w, is called stationary if the image of every normal function has non-empty intersection with S. Examples of stationary subsets of w, are: S = w1; S = the set of all limit ordinals < w1; S = the set of all limit ordinals < w1 which are not of the form a + w; S = any set which contains the image of a normal function. (We use only the first two examples.)

Making use of the same notation as in Lemma 5.2 and the comments following the lemma, we can state the main theorem of this section.

5.3 THEOREM [2]. The group A is free if and only if E is not a stationary subset of w1.

Proof. Suppose E is not stationary in wi. Let f: w1 -> w1 be a normal function whose image does not intersect E, and let A. = Af(). Then since f is an unbounded continuous function, {AD I v < wil is a smooth chain whose union is A. Since the range of f does not intersect E, A. is Xi-pure in A for every v < wi. Hence A+1I/AD is free for every v < wi, by the definition of Xi-pure. Therefore Theorem 2.6 implies that A is free.

Conversely, suppose that A is free and let X be a basis of A. We assert that there is a smooth chain {XD I v < wl} of subsets of X, and a normal function f: w- > w, such that for every v < wi, X. is a basis of Af(g). If this is the case, then E is not stationary in wi. Indeed, for every v < wi, f(v) is not in E since A lAft() is isomorphic to the free group generated by X - XD and hence Af(,) is xi-pure in A.

We define X. and f(v) by induction on v. Let XO = 0 and f(0) = 0. Suppose X,. and f(,) have been defined for all ,u < v. If v is a limit ordinal, let X. = U<. <X, and f(v) = sup {f(, ()I , < v}. Then XD is a basis of Af(,) since Af(,) = U,,<,Af(!,). If v = ,u + 1, let YO be a countable subset of X

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 783

properly containing X,. and let o-o be an ordinal such that YO C A r, Let Y, be a countable subset of X such that A lo is contained in (Y,). By induction on n we obtain a chain of countable subsets of X

XiY, C Y1C" C C YnC n<w

and a sequence of ordinals

f (tt< o-o 0-nc n <Ct)

such that for every n < ,

Yn CA_, C(Y+).

If we let X. = Un<c, Yn and f(v)= sup{o-n n < w}, then XD is a basis of Af(). This completes the proof of the theorem.

6. The axiom of constructibility. We shall make use of the following rather amazing conse- quence of Godel's axiom V = L which was discovered by Jensen [9]. (Although we cannot give the proof here, this result is a relatively easy consequence of the formal definition and some elementary properties of L. We could have simply taken this consequence of V = L as our axiom - replacing V =L in 1.3 (i) - but we preferred to indicate its relation to the well-known Axiom of Constructibility.)

6.1 THEOREM. Assume V = L. Let C be a set which is the union of a strictly increasing smooth chain of countable sets {C. I v < wid, and let E be a stationary subset of w1. Then there is a sequence {S. I v E E} such that S. C C. for all v E E and such that for any subset X of C, the set of v in E with x n CD = SD is stationary in w1.

We actually make use of the following corollary of the theorem:

6.2 COROLLARY. Assume V = L. Let B be a set which is the union of a strictly increasing snmooth chain of countable sets {B. I v < w1} and let Y be any countable set. Let E be a stationary subset of w1. Then there is a sequence of functions {g :B R-> BR x Y v E E} such that for any function h: B -> B x Ysatisfying h (B.) C B. x Yfor all v, there is an v in E such that h restricted to B. is gv.

Proof. To prove the corollary from the theorem let C. = B. X (B. X Y) and C = B X (B X Y). Let {S, I v E E} be the sequence given by the conclusion of Theorem 6.1. Note that S. is a subset of BD x (B. x Y). If SD is a function from B. to B. x Y, let g. = S.. Otherwise let g. be any function from BD to B. x Y. If h is any function from B to B x Y we may regard h as a subset of B x (B x Y). By Theorem 6.1 there is an v (in fact a stationary set of them) such that h n CD = S.. Because of the hypothesis that h (BD,) C B. x Y, we see that h n CD equals h restricted to B., which, therefore, equals gv. This completes the proof of the corollary.

The following theorem is the key to the solution of Whitehead's Problem under the assumption V = L. It is proved by techniques like those used in the proof of Theorem 4.1. In the proof of 4.1 we did not need any special assumptions because every potential splitting homomorphism for r was already determined by its values on B. In the situation of the following theorem where B = U.<,, B. is uncountable, we need Corollary 6.2 in order to obtain a list of homomorphisms on the B.'s which includes a restriction of every potential splitting homomorphism for V.

6.3 THEOREM. Assume V = L. Let B be the union of a strictly increasing smooth chain {B. I v < W1} of countable free groups such that E = {v < w1 I BD+1/BD is not free} is stationary in wi. Then B is not a W-group.

Proof. Just as in the proof of Theorem 4.1 we define by induction a smooth chain of groups {C. I v < wil such that C. is a (B., Z)-group and the union C is a (B, Z)-group such that iT: C-> B

784 P. C. EKLOF [December

does not split. By Corollary 6.2 there is a set of functions

{IgD B. * B. X Z I V C E}

such that for every h: B -B x Z with 7rh = 1B, there is a ,u in E such that h I B,, = g. Let CO be any (Bo, Z)-group. Suppose that C, has been defined for all ,u < v. If v is a limit ordinal,

let C. = U,<D C,. If v is a successor ordinal, say v = ,u + 1, consider two cases. First, if , E E and gB!, -*B, > B,. x Z is a splitting homomorphism for 7r: C, -> B/, let C. be an extension of C, such that gl. does not extend to a splitting homomorphism for v: C. -* B, (Note that C. exists by Lemma 4.3, since B,.+1/B,, is not free and therefore, by Theorem 4.1, is not a W-group.) In the second case, if , 0 E or gl, is not a splitting homomorphism for v: C, -> B, let C. be any (BD, Z)-group extending C, (cf. proof of 4.1).

Let C be the union of the chain {C. I v < wl}. If there is a splitting homomorphism p: B -> C for v C-> B then there is a ,u E E such that p I B, = g,. We are clearly in the first case of the definition. But then, just as in the proof of 4.1, we obtain a contradiction of our construction of C, +,. Hence v does not split and the theorem is proved.

Proof of Theorem 1.3(i). Let A be a W-group of cardinality Xi. We shall prove'that A is free in two steps. First we shall prove that A satisfies Chase's condition (5.1) and then we shall use Theorem 5.3 to prove that A is free.

Suppose, first that A does not satisfy Chase's condition. Since Corollary 4.4 says A is Xi-free, this means: (*) there is a countable subgroup Bo of A such that for any countable subgroup C of A which contains Bo there is a countable subgroup C' containing C such that C'/C is not free. It follows that there is a strictly increasing smooth chain {B. I v < w1} of countable subgroups of A such that for each v < w1, BD+1IBD is not free. (The chain is constructed by transfinite induction using (*).) If B is the union of this chain then, by Theorem 6.3, B is not a W-group (since in this case E = w1). We have thus obtained a contradiction of Theorem 3.1. Hence A must satisfy Chase's condition.

By Lemma 5.2, A is the union of a smooth chain {AD I v < w1} such that for each v < w1, A,+1 is r1-pure in A. Let E = {A < w1 I A, is not Ht1-pure in A} and let E' = {A < w 1 I AA+1/A, is not free}. We

assert that E = E'. Certainly E' C E. Conversely, if A is not in E', then for every v > A, Au/A, is free by Corollary 2.5 since

(AD /AAx)/(AA +1/AAk )- A. /AA +1.

It follows that A/Ak is Xi-free since every countable subgroup of A/Ak is contained in AD/Ak for some v > A. Thus A is not in E and the assertion is proved.

By Theorem 6.3, E' is not a stationary subset of w1 because A is a W-group. Since E equals E', Theorem 5.3 implies that A is free. The proof of Theorem 1.3(i) is complete.

7. Martin's axiom. We shall state a special case of Martin's axiom in the form of a theorem. The reader, if he is familiar with the general form of MA (see Shoenfield [18]; also Martin-Solovay [12]) should find it easy to supply a proof of the theorem. Otherwise, he may take it as an axiom.

7.1 THEOREM. Assume MA. Let A and B be sets of cardinality < 2Mo and let P be a family of functions with the following properties:

(7.1.1) for every f in P, f is a function from a subset of A into B; (7.1.2) for every a in A and every f in P, there exists g in P such thatf C g and a is in the domain of g;

and (7.1.3) for every uncountable subset P' of P, there exist fl, f2 in P' and f, in P such that f' X f2 and f,

extends both f' and f2. Then there exists a function g : A -* B such that for every finite subset F of A there exists f in P with F

contained in the domain of f and g I F = f IF.

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 785

(An example which shows that condition (7.1.3) - called the countable antichain condition - is necessary is: P = the set of all one-one functions from a finite subset of an uncountable set A into a countable set B.)

We shall use Theorem 7.1 to prove the following:

7.2 THEOREM. Assume MA + 2Mo > X4. Let A be any group of cardinality I4 which satisfies Chase's condition. Then A is a W-group.

Before proving 7.2 let us see that it implies Theorem 1.3 (ii). The following theorem is proved in ZFC.

7.3 THEOREM [6]. There is a group A of cardinality I4 which satisfies Chase's condition but is not free.

Proof. We shall define by induction a smooth chain {A. I v < wd of countable groups satisfying the following three properties:

(i) for every v < w, A. is free; (ii) for every A < v < w1, AI/A,,,+1 is free; and (iii) for every limit ordinal A < w1, AA+1/AA is not free. If A is the union of such a chain, A satisfies the requirements of the theorem. Indeed, A satisfies

Chase's condition by Lemma 5.2 and properties (i) and (ii). (Notice that for every A < W1,A,,+1 is N1-pure in A because every countable subgroup of A/A,,+1 is contained in A,/A,+1 for some v < w,.) By (iii) the set of limit ordinals A such that AA is not N1-pure in A equals the set of all limit ordinals, which is clearly a stationary set. Therefore by Theorem 5.3, A is not free.

Now let us define the chain. Let Ao = 0 and suppose that for some 8 < wi, a smooth chain {A. I v < 8} has been defined which satisfies (i), (ii) and (iii). We must define AS and verify the three properties for the chain {A, : v < 8 + 1}. There are three cases.

Case 1: 8 = v + 1, where v is not a limit ordinal. Let A, = A. E Z. The three properties are easily verified. (Property (ii) uses Corollary 2.5.)

Case 2: 8 = A, a limit ordinal. Let AA = UV<A.A. Choose a strictly increasing sequence {on I n < w} whose limit is A such that o-n is a successor ordinal for every n. Then AA = Un<wA<n and by (ii), AO,n+I/AO,n is free for every n < w. By Theorem 2.6, AA is free, and A /AI, is free for every n < w. Hence (i) holds; and (ii) follows from Corollary 2.5 since if A < A, A,+1 is contained in Ac, for some n < w. Property (iii) is obvious in this case.

Case 3: 8 = A + 1, where A is a limit ordinal. This is the most difficult case. Let {on I n < } be as in Case 2, except that for convenience we require aO = 0. By the proof of Theorem 2.6 we know that there exists a smooth chain of sets {Xn I n < } such that Xn is a basis of A,. For each n > 1, choose xn E Xn - Xn_1. Let Yn = Xn - {xnl for each n > 1 and let B be the subgroup of AA generated by UnYn. Let P= ln=l(xn). Define AA+, to be the subgroup of BE[ P generated by AA and {Zm : 1 -< m < w}, where Zm is the element of P represented (in the obvious sense) by the formal sum

Zm = > (n!/m!)xn. n m

It is an easy exercise to verify that Un Yn U {Zm :1 C m < } is a basis of AA+,. Thus (i) holds. For each k < w, AA+l/A,k is isomorphic to the subgroup of AA+l generated by

U (Yn Yk)U{ zm: k + 1m <}. n>k

Thus AA+11A,k is free; property (ii) follows from Corollary 2.5 (cf. Case 2). As for property (iii), notice that m !zm - z1 E AA for every m -' 1. Thus z1 + AA is a non-zero element of AA+,/AA which is divisible by n for every n > 0. Since a free group does not have any elements of this type, property (iii) and the theorem are proved.

786 P. C. EKLOF [December

Proof of 7.2. Let A be a group of cardinality N, satisfying Chase's condition and let 7r :B -* A be a surjective homomorphism with kernel Z. (All we really need is that the kernel is countable.) We shall prove that 7r splits by applying Theorem 7.1 to the set P of all homomorphisms ep: S -* B satisfying:

(7.2.1) 7r(p = ls; and S is a finitely-generated pure subgroup of A.

If we prove that P satisfies (7.1.1), (7.1.2) and (7.1.3), then by Theorem 7.1 there is a function g: A -* B which agrees with a member of P on any given finite subset of A. It follows that g is a homomorphism and that irg = 1A. Hence 7r splits.

Since (7.1.1) is obvious for P, it remains to verify (7.1.2) and (7.1.3). Property (7.1.2) is a consequence of the following lemma.

7.4 LEMMA. If ep is in P and F is a finite subset of A, then there is a function ep' in P such that ep' extends ep and F is contained in the domain of so'.

Proof. Let S be the domain of q'. Since A is K,-free, there is a finitely-generated pure subgroup S' of A which contains S U F (cf. the remarks preceding Theorem 4.2). Now S'/S is finitely-generated and torsion-free (since S is pure in A) and therefore', by Theorem 2.2, S'/S is free. By Corollary 2.5 there is a basis of S'of the form X U Y where X is a basis of S. If x E X, define ( '(x) = qp (x). If y E Y define ep'(y) = by, where by is some element of B such that 1r(by) = y. This defines a homomorphism (P': S'--* B which has the desired properties; thus the lemma is proved.

The proof of property (7.1.3) requires a little more work. Let us first prove it under the assumption that P' is an uncountable subset of P such that there is a pure subgroup A' of A which is free and which contains the domains of all the elements of P'. Choose a. basis X = {x, I v < W1} of A'. Because of Lemma 7.4 we may assume that the domain of each ep in P' is generated by a finite subset of X. Moreover, since a countable union of countable sets is countable, we may assume (replacing P' by an uncountable subset if necessary) there is an m such that for every ep in P', the domain of ep is generated by exactly m elements of X. Let P' = {ep, I v < w1} and let Y, C X be a basis of the domain of qp,. Since Y, has cardinality m for v, there is a subset T of X which is maximal with respect to the property that T is contained in Y, for uncountably many v. (Possibly T = 0.) Notice that since the kernel of 7r is countable there are only countably many functions on T which belong to P. Hence we may assume without loss of generality that ep, and ep, agree on T whenever T C Y_ and T C Yr. Renumbering, we may assume that T C YO. For each y E YO - T, there are only countably many v such that y E Y, (by the maximality of T). Hence there exists v7 0 such that Y, f Yo = T. Since epo and ep, agree on T they have a common extension 0 : (Yo U Yr,) -* B. Now (Y0 U Y,) is a pure subgroup of A' since it is generated by a subset of a basis of A'. Therefore (YO U Y,) is a pure subgroup of A since

(A/(Yo U Y,))/(A'/(Yo U Y,)) A A'.

(We make use of the easily proved fact that if AIB and B are torsion-free, then A is torsion-free.) Hence df is an element of P and property (7.1.3) is proved under the special assumption. The general case of (7.1.3) follows from the next lemma. (Recall that we are assuming that A satisfies Chase's condition. Here we finally use that hypothesis!)

7.5 LEMMA. For any uncountable subset P' of P there is a free subgroup A' which is pure in A and an uncountable subset P" of P' such that dom (so) C A' for every ep in P".

Proof. Suppose P' = {Sp, I v < wo} where p Sv, -* B. Replacing P' by an uncountable subset if necessary, we may assume there is an m such that S, has a basis of cardinality m for all v < w0. There is a pure subgroup T of A maximal with respect to the property that T is contained in uncountably many S,. We may assume that T is contained in S, for all P. By Theorem 2.2 and Corollary 2.5 we may assume there is a (finite) basis of S, of the form X U Y, where X is a basis of T.

1976] WHITEHEAD'S PROBLEM IS UNDECIDABLE 787

We shall construct A' as the union of a smooth chain {A, I v < w} such that for each v < w1, A, is a pure subgroup of A and A,+,/A, is free. Then A' will be free by Theorem 2.6; moreover, A' will be pure in A since it is a union of pure subgroups of A.

Let Ao = T Suppose we have defined {A,, I A < } and a strictly increasing sequence of ordinals fo-, IA,u < } such that Y,+l CA,+,. If v is a limit ordinal let A, = U, <,A,. If v is a successor ordinal, v = 8 + 1, let C, be a countable Mi-pure subgroup of A which contains A,. (Here we use the hypothesis that A satisfies Chase's condition.) There exists o, > O>+ for all A <v such that (YeV) n c8 = 0. (Otherwise, since C, is countable there would be an element c E C8 and uncountably many T < wl such that c E (YT); but then the pure closure of T + (c) would contradict the maximality of T.) Let A, be the pure closure of A, + (Y0). Since (Y,) n c, = 0, it follows that A, n C. = A8. Hence A,/A, is isomorphic to a countable subgroup of A/C, and is therefore free - since C, is N1-pure in A. If we let P" = A < wo} we see that the conclusions of the lemma are satisfied and hence Theorem 7.2 is proved.

8. Generalizations. Theorem 1.3 (ii) may be generalized as follows. Assuming MA + 2Mo > X1, for every uncountable cardinal K, there is a non-free W-group of cardinality K. This follows because if A is a non-free W-group of cardinality Ml then the direct sum of K copies of A is a W-group, because in general

Ext ( Ai, C) =HExt (Ai, C).

Shelah (Is. J. Math., 21 (1975) 319-349) has shown that Theorem 1.3 (i) generalizes also, i.e., assuming V = L, every W-group is free. The proof is by induction on the cardinality of the W-group. At singular cardinalities the proof requires the following result of Shelah's: if A is a group of singular cardinality K such that every subgroup of A of cardinality < K is free, then A is free.

Also, recently Shelah has proved that a negative answer to Whitehead's problem for groups of cardinality Ml is consistent with ZFC + 2o = _l. This together with Theorems 1.3 (i) and 1.1 (ii) implies that Whitehead's problem is undecidable in ZFC + 20o = N1. (An assertion to the contrary in Math. Reviews, vol. 50 # 2362 is in error.)

The proof of 7.3 can be generalized to prove that there are 2', pairwise non-isomorphic groups of cardinality NI which satisfy Chase's condition.

The proof of Theorem 1.3 which we have given is somewhat different in detail from that in [17], although the key ideas are the same. The advantage of the proof given here is that it may easily be generalized to obtain other undecidability results. See [3] for these as well as related results.

Work on this paper was partially supported by NSF grant GP-43910. I wish to thank Professors S. Feferman, U. Felgner and R. Solovay for their helpful comments.

References

1. S. U. Chase, On group extensions and a problem of J. H. C. Whitehead, pp. 173-193 in Topics in Abelian Groups, Scott-Foresman, Glenview, Ill., 1963.

2. P. C. Eklof, On the existence of K-free abelian groups, Proc. Amer. Math. Soc., 47 (1975) 65-72. 3. , Homological algebra and set theory, to appear in Trans. Amer. Math. Soc. 4. L. Fuchs, Infinite Abelian Groups, Vol. I, Academic Press, New York, 1970. 5. K. Godel, The Consistency of the Continuum Hypothesis, Princeton University Press, 1940. 6. P. Griffith. A note on a theorem of Hill, Pac. J. Math., 29 (1969) 279-284. 7. P. Hill, New criteria for freeness in abelian groups II, Trans. Amer. Math. Soc., 196 (1974) 191-201. 8. J. P. Jans, Rings and Homology, Holt-Rinehart and Winston, New York, 1964. 9. R. B. Jensen, The fine structure of the constructible hierarchy, Ann. of Math. Logic, 4 (1972) 229-308. 10. I. Kaplansky, Infinite Abelian Groups (rev. ed.), Univ. of Michigan Press, 1969. 11. S. Lang, Algebra, Addison-Wesley, Reading, Mass., 1971. 12. D. A. Martin and R. M. Solovay, Internal Cohen extensions, Ann. of Math. Logic, 2 (1970) 143-178.

788 MORRIS MARDEN [December

13. J. D. Monk, On the foundations of set theory, this MONTHLY, 77 (1970) 703-711. 14. D. G. Northcott, An Introduction to Homological Algebra, Cambridge University Press, 1960. 15. L. S. Pontryagin, The theory of topological commutative groups, Ann. of Math., 35 (1934) 361-388. 16. M. E. Rudin, Souslin's conjecture, this MONTHLY, 76 (1969) 113-119. 17. S. Shelah, Infinite abelian groups - Whitehead problem and some constructions, Is. J. Math., 18 (1974)

243-256. 18. J. R. Shoenfield, Martin's Axiom, this MONTHLY, 82 (1975) 610-617. 19. R. M. Solovay and S. Tennenbaum, Cohen Extensions and Souslin's problem, Ann. of Math., 94 (1971)

201-245. 20. K. Stein, Analytische Funktionen mehrerer komplexer Verainderlichen zu vorgegebenen

Periodizitaitsmoduln und das zweite Cousinsche Problem, Math. Ann., 123 (195.1) 201-222.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, CA 92717

MUCH ADO ABOUT NOTHING

MORRIS MARDEN

Let me begin by explaining my use of the Shakespearean title for this talk. Ever since I have been studying the zeros of functions, some of my non-mathematical friends have poked fun at me as being the world's expert on nothing. As I have been the innocent victim of this pun for almost fifty years during which I have tried my best to do much about zeros, the appropriateness of the title is obvious. Of course, Shakespeare applied his title to a comedy, so I hope that I have not misled anyone to expect anything but a very serious mathematical talk today.

By request of the committee on arrangements*, this talk is to be a retrospect on some of my past research. In order to give the talk some unity, I shall limit it to a single topic: the extension of Rolle's theorem to the complex plane. This is, of course, the theorem which states that between any pair of real zeros of a real differentiable function f lies at least one zero of its derivative; that is, at least one critical point of f. It is a theorem which one meets in any introductory course in the calculus. Yet its extension to the complex plane is by no means trivial. In fact Rolle's theorem does not hold for arbitrary analytic functions of a complex variable as is shown by the example f(z) = ezi - 1. This function has zeros at z = 0 and z = 2- but its derivative f'(z) = jezi has no zeros whatsoever.

However, for certain classes of analytic functions of a complex variable some analogues to Rolle's theorem have been found; namely, for polynomial, rational, entire and meromorphic functions. I shall describe these analogues including my own contributions. In general I shall omit proofs but give physical interpretations where possible. Now, these analogues of Rolle's theorem fall into two types. The first type pertains to functions for which the location of all the zeros and poles is prescribed. The second type pertains to functions for which the location of only some zeros and poles is prescribed, the remaining ones being unspecified.

The theorems of the first type aim to generalize the corollary to Rolle's theorem that any interval of the real axis containing all the zeros of a given polynomial P also contains all the critical points of P. The initial complex variable discovery along these lines goes back to the earliest days of coniplex variables and in fact to the man who was most influential in getting the geometric representation of complex variables accepted. I refer to Gauss who in 1836 stated that the critical points of a polynomial P are the multiple zeros of P and the equilibrium points in a field of force due to particles at the zeros of P, attracting according to the inverse distance law. Gauss' mechanical interpretation may be easily

* Lecture given on May 9, 1975 as part of the Colloquium at the University of Wisconsin-Milwaukee honoring the author's retirement from the UWM.