FREE TOPOLOGICAL GROUPS AND FREE PRODUCTS OF TOPOLOGICAL GROUPS
by
PETER NICKOLAS
A thesis submitted for the degree
of Doctor of Philosophy at
the University of New South Wales.
School of Mathematics, University of New South Wales, Kensington, N.S.W., 2033. July, 1976...... I —-> UNIVERoiTV ur N.S.W. !
C3525 -O.ftM.77 LIBRARY (ii) CONTENTS
ABSTRACT ...... (iv)
INTRODUCTION (vi)
ACKNOWLEDGEMENTS (x)
CHAPTER 1
Free Topological Groups and Free Products of Topological Groups:
Their Basic Properties.
§1. Free topological groups ...... 1
§2. The Graev pseudometric topology ...... 7
§3. Free products of topological groups ...... 13
CHAPTER 2
Local Compactness and Free Products
§1. Introduction and preliminaries ...... 17
§2. Homomorphisms from locally compact groups into free products .. 20
CHAPTER 3
Subgroups of Free Topological Groups
§1. The role of k^-spaces ...... 28
§2. The freeness of subgroups ...... 31
§3, Subgroups of the free topological group on [0,1] ...... 40 (iii)
CHAPTER 4
Subgroups of Free Products of Topological Groups
PART 1 : Preliminary Material ...... 51
§1. Topological groupoids and universal morphisms ...... 52
§2. Some specific universal morphisms ...... 56
§3. Equivalence under the functor FM...... 66
PART 2 : The Topological Kurosh Theorem ...... 68
§4. The statement of the Kurosh theorem ...... 68
§5. The proof of the Kurosh theorem...... 72
§6. Some consequences of the Kurosh theorem ...... 85
§7. More on FM([0,1]) ...... 92
CHAPTER 5
Universal Constructions and Function Spaces
§1. The universal topological group and spaces of morphisms ...... 96
§2. An exponential law for k-categories ...... 103
§3. Reflexivity and free Abelian topological groups ...... 116
REFERENCES 124 (iv)
ABSTRACT
This thesis makes several contributions towards an understanding of the topologies, and of the subgroups, of free topological groups and free products of topological groups.
The central result is an analogue for free products of topological groups of the Kurosh subgroup theorem, giving conditions under which a subgroup of a free product of countably many k^-groups can in turn be expressed as a free product of certain of its subgroups. The theorem given here generalises most of the known results of its kind. Its proof makes use of the theory of topological groupoids.
An examination is also made of the subgroups of free topological groups, and a detailed analysis is made in particular of the subgroups of FM([0,1]) , the (Markov) free topological group on the interval
[0,1] . It is shown, for example, that the free topological group on a compact space C occurs as a subgroup of FM([0,1]) if and only if C is also metrizable and of finite dimension. One step in the proof of this fact involves showing that the free topological group on any k^-space
X contains as subgroups the free topological groups on all the finite product spaces Xn .
The principal result on the topologies of the groups under consider ation here is that the only locally compact group topology possible on an algebraic free product of groups (and on a free group in particular) is the discrete topology. An explicit description is also given of the topology of the (Graev) free Abelian topological group on a completely (v)
regular Hausdorff space X , in terms of certain extensions of the
continuous pseudometrics on X .
The last chapter of the thesis links together some results of a
different nature again, the main one concerning spaces of topological
groupoid morphisms. If C , D are topological groupoids, the set of morphisms M(C,D) from C to D is given the compact-open
topology; then if G is a k^-groupoid and U(G) its universal
topological group, with i : G U(G) the universal morphism, the induced map
i* : M(U(G),H) M(G,H) is a homeomorphism for every topological group H . Applications to free topological groups and free products follow. (vi)
INTRODUCTION
Markov [1] and Graev [1] introduced their respective definitions of the free topological group on a topological space in the 1940’s, and the notion of a free product of topological groups was first discussed by Graev [2] a little later. The theory of both these constructions has been developed quite extensively in recent years.
Work here has proceeded in three main directions: that of analysing the topology of free topological groups and free products of topological groups; of classifying their subgroups; and of describing their algebraic topology.
Our principal concern in this thesis is with the first two of the above problems: Chapters 1 and 2 deal largely with the first of them and Chapters 3 and 4 with the second. Of course, some overlap is inevitable here, so that some subgroup material is included in Chapter 1, while some material on the topology of free topological groups finds its most natural place in Chapter 3. A few ideas of a different nature again are drawn together in Chapter 5.
Chapter 1 opens with a discussion of the definitions and basic properties of the Markov and Graev free topological groups, FM(X) and
FG(X) , on a space X , and we analyse and classify the numerous proofs of the fact that these groups are Hausdorff when X is completely regular and Hausdorff. Graev's proof of this involves the construction of certain pseudometrics on the free group, and we examine in some detail the topology they define: we show that this topology on FG(X) rarely (vii)
coincides with the free topology, while on the free Abelian topological
group AG(X) the topology is always equal to the free topology. This latter result leads to a proof that certain subgroups of AG(X) are again free Abelian topological groups. The chapter ends with a brief discussion of free products of topological groups.
A very natural question to ask about free topological groups and free products is whether their topologies are ever locally compact.
For free (and free Abelian) topological groups this question was settled by a result of Dudley [1], showing that the discrete topology was the only locally compact group topology possible on an abstract free (or free Abelian) group. Also, Morris, Ordman and Thompson [1] have used Dudley's result to show that a free product of topological groups is locally compact only if it is discrete. In Chapter 2 we generalize both of the above results by showing that an abstract free product of groups admits no locally compact topology other than the discrete topology. (The result is actually somewhat more general than this.) As might be expected, the proof uses known facts on the structure both of free products and of locally compact groups.
Our most important single result on the question of subgroups is a topological version of the Kurosh subgroup theorem, which is proved in
Chapter 4. Brown and Hardy [1] have already shown that an open subgroup of a free product of countably many k^-groups is again a free product, while Hardy and Morris [2] have shown that the Cartesian subgroup of a finite free product G. * G~ * . . . * G of k -groups is a free 12 n GO topological group. The proofs in both cases make use of the theory of topological groupoids as developed by Brown and Hardy Cl] and Hardy [1] (viii)
(cf. also Higgins [1]). Our version of the Kurosh theorem yields both of the above results as corollaries, as well as several other results of interest, including some applications to free topological groups; in its proof we also use topological groupoids, along with some ideas of Weir [1] and MacLane [2].
In Chapter 3 subgroup questions of a different kind are considered.
In particular, we examine and classify a large class of subgroups of
FM([0,1]) , the Markov free topological group on [0,1] . The principal result is that if X is a compact space, its free topological group is a subgroup of FM([0,1]) if and only if X is compact and finite dimensional. This work was prompted by results of Hardy, Morris and
Thompson [1] and Thomas [1] showing that for any completely regular
Hausdorff space X , FM(X) contains a closed subspace homeomorphic to the product Xn , for each positive integer n . One of the steps in the proof of our result on FM([0,1]) consists of showing that FM(X) contains the subgroup FM(Xn) for each n , provided X is a k^-space; we also show, however, that the embedding of Xn in FM(X) required for this result must be different from that given by Thomas and Hardy, Morris and Thompson.
In the fifth and final chapter of the thesis we turn to some questions of a nature different from those considered in the earlier chapters. If
C , D are topological groupoids, we give the set M(C,D) of morphisms from C to D the compact-open topology, and then the first main result states that if the topological groupoid G is a k^-space and U(G) is its universal topological group, with i : G U(G) the universal (ix) morphism, then the induced map
i* : M(U(G),H) “► M(G,H) is a homeomorphism, for every topological group H . The proof utilises the construction of U(G) given by Brown and Hardy [1]. Since free topological groups and free products of topological groups can both be expressed as universal groups on appropriate groupoids, applications to both these constructions follow, and from these in turn we deduce results on free Abelian topological groups and on coproducts in the category of
Abelian topological groups. R. Brown has pointed out that the correct general setting for results such as these is as consequences of an exponential law in a suitable category of topological groupoids, and §2 is devoted to the derivation of such a law.
The final section of Chapter 5 arose through an attempt to answer the question: Is the free Abelian topological on [0,1] reflexive?
(that is, does it satisfy the Pontrjagin Duality Theorem?). In the process of finding a negative answer to this question, we derive a fairly general necessary condition for reflexivity, which has several corollaries of independent interest. 1. CHAPTER 1
Free Topological Groups and Free Products of Topological Groups:
Their Basic Properties
§1. Free topological groups.
A.A. Markov [1] and M.I. Graev [1] were the first authors to define and construct free topological groups. We begin by introducing and discussing their respective definitions.
Definition 1.1.1. Let fx be a conttnuouA map fi/com a topological
Apace, x Into a topological gtioup FM(X) . Then FM(X) [oft, Atfttctly, the pcuA (f^, FM(X)) ) lA a Markov free topological group on X l^ {^Oft any topological gftoup G and any conttnuouA map (p : X G , thefte tA a unique conttnuouA homomoftphlAm $ : FM(X) G Auck that
The existence of FM(X) for all spaces X is established easily, as one might expect, by categorical arguments. Specifically, the Freyd adjoint functor theorem (MacLane [1]) constructs a functor FM as a left adjoint to the forgetful functor from the category of topological groups to that of topological spaces, and the map f : X FM(X) is then given by the unit of this adjunction.
In the same way, a left adjoint FG to the forgetful functor from topological groups to pointed topological spaces can be constructed, and this gives rise to the Graev free topological group. 2.
Definition 1.1.2. Let g„ be a continuous map faom a pointed 4pa.ce A
(X, e) Into a topological group FG(X), mapping e to the Identity.
Thun FG(X) Is a Graev free topological group on X Ifi far any topological group G and any continuous map
the Identity oft G , there Is a unique continuous homomorphism
$ : FG(X) -► G Such that
It is clear from their definitions that FM(X) and FG(X) are
unique up to topological isomorphism. Also, by considering maps from
X into the appropriate abstract free groups with the indiscrete topology,
it is easy to see that the maps f : X FM(X) and g : X FG(X) are
always injections, and that FM(X) is algebraically the free group on
the subset f (X), while FG(X) is free on g (X\{e}) . A A
The most basic non-trivial result, proved by Markov and Graev, and
by many authors since, is that when X is completely regular and
Hausdorff, the maps f and g are closed (topological) embeddings,
and FM(X) and FG(X) are Hausdorff. Under these conditions, therefore,
we will usually look upon X as a closed subspace of FM(X) and FG(X),
and dispense with the maps f and g . (We should point out that for A A the early writers on the subject, as well as some recent ones such as
Abels [1], the objects under study were really free Hausdorfa topological
groups, so that what for us is the problem of proving the Hausdorffness
of FM(X) and FG(X) was for them the problem of showing that these
groups have the expected (free) algebraic structure.) The following
proposition completes the story for arbitrary spaces (see Thomas [1],
for example). 3.
Proposition 1.1.3. VoK, any -6pfl.ee X
(i) f : x -► FM(X) and gx : x FG(X) a/ie embedding* If and
only If x Is completely KeguJUui ;
avid (ii) FM(X) and FG(X) a/ie Hausdotff If and only If X It
functionally *epaAable [that Is, X ha* enough continuous tieal-valued
functions to *epa/iate points) .
Note that for completely regular spaces (and, in particular, for
topological groups) the concepts of functional separability and
Hausdorffness are equivalent.
1.1.4. Notation. If G is any group and Y is a subset of G, we
shall denote by gp(Y) the subgroup of G generated by Y . For an
integer n we denote by gp^(Y) the set of those elements of gp(Y) which can be written as words of lengths less than or equal to n with
respect to Y . In particular, the subset gp^(X) of either FM(X) or FG(X) will be written as F (X) . n
It has been proved that when X is completely regular and Hausdorff
the subspaces F^(X) are closed for each n , in FM(X) and FG(X) .
(See, for example, Hardy, Morris and Thompson [1].) Observe also that
in FM(X) , F^(X) is the disjoint union of X , X ^ and {e} , whilst
in FG(X) it is the wedge (one-point union) of X and X ^ at their
common point e. 4.
1.1.5. The Hausdorffness of the free topological groups.
The fact that FM(X) and FG(X) are Hausdorff when X is completely regular and Hausdorff has been established in several different ways. The proofs of Markov Cl] and Graev [1], and the later one of Swierczkowski [1] (which holds for general free topological algebras), use the continuous real-valued functions on X to construct pseudometrics (or other equivalent structures) on the appropriate abstract free group. It is then shown that the topology defined by these pseudometrics makes the group operations continuous, and induces the original topology on X . The free topology is clearly the finest topology of this kind, and will thus be Hausdorff if the pseudometric topology is. But since X has enough real functions to separate points, enough pseudometrics can be constructed on the free group to do the same, and Hausdorffness follows. Graev’s proof will be discussed further in §2.
Another method, used in different ways by several authors, is to show that points of the free topological group on X can be separated by means of continuous homomorphisms into various Hausdorff topological groups. Given distinct points x^, x^, ..., x € X , such an argument typically makes use of some topological group G which is Hausdorff and path-connected, and which has a subgroup freely generated by n elements g^, g^, ..., g^ . The complete regularity of X and the path-connectedness of G allow the map x^ H* g^ , i = 1, 2, ...» n, to be extended to a continuous function from X to G , and this function then extends to a continuous homomorphism from the free topological group into G , which separates each non-trivial element of gp{x^, x^, . .., x^} from the identity, because of the freeness of gp{g^, g£> •••> gn) • (Of course, a 5.
group G with a free subgroup of rank 2 will always suffice, since
such a subgroup contains free subgroups of every finite rank.)
Thomas [1] and Brown and Morris [1] both use such a technique, although
the proof in the second paper is simpler, making use of a general
construction which embeds any Hausdorff topological group in a path-
connected one (Brown and Morris [1], Hartman and Mycielski [1],
Hewitt and Ross [1, 7.20]). Hewitt and Ross [1, Theorem 8.8] use homomorphisms into the unitary groups for a similar purpose, completing
the rather sketchy proof of Kakutani [1], which otherwise proceeds by what is now a standard categorical argument. Abels [1] not only proves
the Hausdorffness of the free topological group, but also obtains
interesting information on some special pseudometrics associated with
it, by considering homomorphisms into a certain Lie group.
A third type of proof is given by Ordman [3]. He begins by deriving an explicit description of the (Hausdorff) topology of the free topological
group on a compact Hausdorff space (cf. Chapter 3). Using the embedding of a completely regular Hausdorff space X in its Stone-Cech compacti-
fication $X he then defines a continuous monomorphism from the free
topological group on X into that on 3X . This induces a Hausdorff
group topology on the free group yielding the original topology on X , and so once again the free topology is Hausdorff. Hardy and Morris [1] use the same idea, in combination with work of Brown and Hardy [1], to
show that the free topological groupoid F(T) on a completely regular
Hausdorff topological graph T is Hausdorff, and that the canonical
graph map i : T F(F) is a closed embedding. 6.
1.1.6. Free Abelian topological groups.
Markov and Graev also introduced the Markov and Graev free Abelian
topological groups, AM(X) and AG(X) respectively, on a space X ,
the definitions being completely analogous to Definitions 1.1.1 and
1.1.2 , replacing "group" by "Abelian group" at each occurrence. All
the comments made above about the basic theory of FM(X) and FG(X)
carry over (with the obvious changes) to the free Abelian topological
groups.
Note that each free Abelian topological group can be obtained from
the corresponding free topological group by factoring out the commutator
subgroup.
Finally, we record for our future use a result of Joiner [1] (see
also Hardy, Morris and Thompson [1H).
Theorem 1.1.7. Let X be a completely regular HcuudosiH 4pace, and £1 C2 en AuppoAc that x2 ... xn Ia an element oft FM(X) In deduced ionm.
[That lA, x^, x2» ..., xn e X , , e2, ...» en = ± 1 and ior
i = 1, 2, ..., n-1 , x^ = xi+1 ImplleA e± = ei+1 .) Then a baAe ofi
C1 e2 £n - neighbourhoods oi x^ x2 __x^ In the AubApace Fn(x) ^ given by e e e the collection oi all AetA oi the ionm u2 ... Unn , where ion,
each i u. Is a neighbourhood oi xi In x .
Analogous results hold ion, FG(x), AM(X) and AG(x) . 7.
§2. The Graev pseudometric topology.
In this section we discuss in some detail the pseudometric topology which Graev [1] defines on FG(X) and AG(X) , in his proof that these
groups are Hausdorff when X is completely regular and Hausdorff. We
show that for a large class of such spaces Graev’s topology on FG(X)
is strictly coarser than the free topology, complementing a result of
this type already known, while his topology on AG(X) does in fact
coincide with the free topology. Using the latter result, we show that a certain class of subgroups of AG(X) are again free Abelian topological
groups.
For the remainder of §2, let X be a completely regular Hausdorff space with basepoint e . As is well-known, the topology of X is defined by the family of continuous pseudometrics on X : more precisely,
the collection of all sets {y e X : p(y,x) < e} , where e > 0 and p varies over all continuous pseudometrics on X , forms an open basis at any point x e X . For each such pseudometric p , Graev constructs on
the abstract free group F(X\{e}) (where e e X is regarded as the identity of the group) a two-sided invariant pseudometric p , whose restriction to X is p . (We say that p is two-sided invariant if, for any elements a, b, u, v, p(aub, avb) = p(u,v). ) Then the class of all such p defines a group topology on F(X\{e}) (cf. Hewitt and Ross
[1, Theorem 4.5]), inducing the original topology on X and making the group Hausdorff. A topology is defined in a similar way in the Abelian case. It is natural to ask whether in either case Graev’s topology is equal to the free topology. 8.
Morris and Thompson [1] have shown that when X is not totally disconnected, Graev's pseudometric topology on FG(X) is strictly coarser than the free topology. The following result records a similar fact about another class of spaces.
Proposition 1.2.1. Suppose that the. completely hegula/i HauAdohH
Apace X contalnA a (non-t/Uvlal) Aequence {x^} converging to Aome potnt x . Then the Ghaev pa eudo metric topology on FG(X) Ia not the iree topology .
Proof. For any continuous pseudometric p on X , we have p(x^,x) 0.
Using the invariance of the extension p , we have
p(xn x x ^n+^, e) = p(xn x x ^ x n, e) n n
= p(xn x 1, e)
= P(xn> x) 0 .
That is, the sequence {x11 x^ x } converges to e in the pseudometric topology. But it is well-known (see Abels [1], for example) that a convergent sequence in a free topological group cannot consist of words of increasing length, and so the result follows.
Corollary 1.2.2. Ii X Ia a non-dlAcrete metric Apace, then the Graev pAeudometrlc topology on FG(x) Ia not the ih.ee topology.
Corollary 1.2.3. li the completely regular HauAdorU Apace x Ia not totally path-citAconnected, then the Graev pAeudometrlc topology on FG(x)
Ia not the iree topology. 9.
Proof. Theorems 3-15 and 3-24 of Hocking and Young [1] show that X contains a homeomorphic copy of the interval [0,1], and thus contains a convergent sequence.
When X is discrete it is clear that Graev's topology is discrete and hence equal to the free topology, showing that some restriction on
X is needed for the conclusion of Proposition 1.2.1 to hold. Non discreteness may well be all that is necessary.
We now turn to the case of the free Abelian topological group
AG(X) , and study Graev's pseudometric extension procedure in some detail. (Here we write the operation of the group additively, and represent the identity of the group, and the basepoint of X, by 0.)
For a continuous pseudometric p on X , extend p to X U (-X) by setting p(-x,-y) = p(x,y) and p(-x,y) = p(x,-y) = p(x,0) + p(y,0), for x,y £ X . Now suppose that
1.2.4
are elements of AG(X) written in reduced form. That is, 6^ = ± 1 and x^ e X\{0} for i = 1, 2 • • • ^ n for each i and j ; and similarly for v . We now consider all possible representations of u and v as (not necessarily reduced) words
u = c^ + c^ + .••. + c s
1.2.5
v = d^ + + •••• + d s of equal length, and with c_^,d^ e X U (-X), i = 1, 2 . . , s. Then, by 10.
s definition, p(u,v) = inf { £ p(c.,d.) } , where the infimum is taken i=l 1 1 over all representations of u and v of the type mentioned. Graev
produces an algorithm which, beginning with the expressions 1.2.5 for
u and v , constructs new representations
u = a, + a0 + .... + a 12 s 1.2.6 (i)
v=b1+b0+....+b , 12 s with the following properties: s s 1.2.6 (ii) l p(a ,b ) < £ p(c ,d ) ; i=l i=l
1.2.6 (iii) At most n + m of the terms p(a^,b^) are non-zero; and
1.2.6 (iv) {a1#a2,...,ag,b1,b2,...,b } C
{x1,-x1)x2>-x2>...>xn>-xn,y1,-y1,y2,-y2>...,V-ym,0} .
It follows that the value p(u,v) is actually attained for some
representation 1.2.6 (i), satisfying (iii) and (iv). We can now prove
Theorem 1.2.7. Let X be a completeZy n.egudLan. HauAdosififi 4pace. Then the Aet ofi pi eadometAlcA p on AG(x) , obtained
contlnuouA pAeudometfUeA on x , defalneA the fisiee topology on AG(x) .
That aJ>, the, AetA {w e AG(X) : p(w,0) < e} , ^ok p oa above., and
e > 0 , a/ie an open baAlA at 0 In AG(X) .
Proof. The topology of any Abelian topological group is defined by the
family of continuous two-sided invariant pseudometrics on the group
(cf. Hewitt and Ross [1, Theorem 8.2]). Let the family of such pseudometrics on AG(X) be {a.} , and let the restriction of O to J J X be Pj * Then we claim that for any u,v e AG(X), G.(u,v) < p.(u,v) . 11.
For if x,y e X , we have
Pj (x,y) = CT (x,y) = Oj(-x,-y) = p^(-x,-y) , and
Pj (_x,y) = pj(x,0) + p^(y,0) = c^(x,0) + a^(y,0) > CK(-x,y) , so that for a,b e X U (-X) , Q_. (a,b) < p_.(a,b) . If now u and v are reduced words as in 1.2.4 , we can choose representations 1.2.6 (i) of the form u=a1 + a_ + ... + a , v=b, + b» + .. . + b realising ■L b S -L d. S the distance p_.(u,v) , and satisfying 1.2.6 (iv) . Hence, using the invariance of G. , J
& & r (u,v) < l a (a ,b ) < l p (a ,b ) = p (u,v) . J i=l 3 i=l 3 3
Therefore {w e AG(X) : Pj(*,0) < e} C {w e AG(X) : a.(„,0) < £} , and the result follows.
To obtain Graev’s topology on AG(X) it is generally necessary to extend all the continuous pseudometrics on X, rather than just the members of some subclass which still defines the topology of X. For otherwise, AG(X) would be metrizable whenever X is, but it is known that AG(X) is metrizable only when both X and AG(X) are discrete
(Abels [1, Korollar 3.6] for example).
Definition 1.2.8 (Bing [1], Michael [1]). A Apace. X Is collectionwise normal Ifi ^on. any locally finite collection {A^} o£ pairwise disjoint cloAed AubAets ofi x , there Ia a disjoint collection {u^} oi open
Aets Auch that A a c“ u a Ionu each, a .
(A collection {A^} Is locally finite l{\ each, point ojj x has a neighbourhood Intersecting only finitely many oft the Aets {A^} .) 12.
Dowker [1] and Shapiro Cl] have shown that a continuous pseudometric
on a closed subspace of a collectionwise normal space extends to a
continuous pseudometric on the whole space (and, indeed, that this
property characterises the collectionwise normal spaces).
Theorem 1.2.9. Let X be a ao Hecticnwl6e normal HauAdon^ 4pa.ce with
baAepolnt 0 , and Y c x a closed 4ab4pa.ce containing 0 . Then the
Aubgdoup gp (Y) o£ AG(X) generated by Y Ia to polo giddily Isomodphlc
[In the natu/ial mjy) to AG(Y) .
Proof. Note that X is normal, and hence, in particular, completely
regular.
By the result of Dowker and Shapiro, a continuous pseudometric G
on Y extends to a continuous pseudometric p on X . Thus G gives
rise in two different ways to a two-sided invariant pseudometric on
gp(Y) . The first such pseudometric, G , is given by extending G to
gp(Y) using Graev’s construction and the fact that gp(Y) is algebraically
the free Abelian group on Y\{0} . The second pseudometric, say G , is
the restriction to gp(Y) of p , Graev’s extension of p to AG(X) .
We claim that 0=0.
It is clear that G(u,v) < G(u,v) for u,v e gp(Y) , since the
greatest lower bound defining g(u,v) is taken over a set of values
including all those used to define G(u,v) . On the other hand, if we
choose representations 1.2.6 (i) of the form u = a^ + a^ + ••• + as »
v = b^ + b^ + ... + bg , satisfying 1.2.6 (iv) and for which
s G(u,v) = p(u,v) = £ p (a.,b.) , 1.2.6 (iv) shows that in fact 1=1 1 1 _ _ a^a^, • . . ^^jb^jb^, . . . ,bg e Y U (-Y) , and thus G(u,v) < g(u,v). Hence 0=0 . 13.
We have therefore shown that the relative topology on gp(Y) is at least as strong as that induced by Graev’s extensions to gp(Y) of the continuous pseudometrics on Y , and so, by Theorem 1.2.7 , gp(Y) = AG(Y) , completing the proof.
If the subspace Y does not contain 0 , gp(Y) is clearly
AG(Y U {0}) , which, as we shall see in §3 , is just AM(Y) .
We also remark that Flood [1] has results similar to Theorems 1.2.7 and 1.2.9 for free locally convex topological vector spaces, although the results here were obtained independently.
Among the familiar classes of spaces which are collectionwise normal is the class of paracompact spaces (Bing [1], Shapiro [1]), which in turn includes all metric spaces and all (Hausdorff) k^-spaces (a class of spaces discussed in detail later). The k^-space case of the following corollary is well-known (see Mack, Morris, Ordman [1], Graev [1, Theorem 10], and our Chapter 3).
Corollary 1.2.10. Let X be a paAacompact HauA do sijSjj 4pace, with baicpoint 0 , and Y a cJLo&ed AubAet containing 0 . Then the 6ubgA,oup gp (Y) Ojj AG(X) it> topologicaiiy ibomotiphic to AG (Y) .
§3. Free products of topological groups.
In this section we introduce the notion of a free product of topological groups, first defined by Graev [2], and discuss its basic properties. 14.
Definition 1.3.1. Let {Ga } be a {\amity o{\ topological groups, A XeA and let i^ : G^ -> G be [ion. eacli X) a continuous komomon.pklsm Into
Some topo-tog-teat gsioup G . Than G li> the free product of the groups {G,} , denoted n G, , li {>oti any topological gn.oup H and any A XeA A collection ojj continuous komomon.pklsms f^ : G^ -► H , thene is a
unique continuous komomon.pfusm f : G H Suck that f ^ = f ° i^ , ion. eack XeA.
The free product of a finite collection {G., G0, ..., G } of 12 n topological groups is written as G. * G0 * . . . * G 12 n
* Of course II G-^ is just the coproduct of the groups {G-^j in the
category of topological groups, and its existence and uniqueness (up to
topological isomorphism) follow by categorical arguments. Also, the
•k underlying algebraic structure of IT G^ is just that of the algebraic
free product of the abstract groups {G^} with the {i^} being the
usual injections. In fact, the {i-^} are topological isomorphisms, so
k that we may identify G^ with the subgroup i-^(G^) of IT G^
(Ordman [1]); if further IT G-^ is Hausdorff, then G^ is a closed
subgroup (Ordman [2]) .
This, of course, leaves open the question of when IT G-^ is
* Hausdorff, which was answered in Graev s original paper: IT G-^ is
Hausdorff if and only if each G^ is Hausdorff. Graev notes firstly
that it suffices to prove the result for a free product G * H of two
Hausdorff groups G and H . For subsets A and B of any group,
denote by [A, B] the set of commutators [a, b] = a'*'b^ab for
a e A, b e B . Then it is easily seen that the kernel K of the
natural homomorphism from the abstract group G * H onto G x H , 15. known as the Cartesian subgroup of G * H, is algebraically freely generated by [G, H]\{e} , and that every element of G * H can be expressed uniquely in the form ghk , for g e G , h e H and k £ K .
(For further information see Chapter 4.) Graev then uses the continuous left-invariant pseudometrics on G and H to define pseudometrics on the set [G, H] , and uses his pseudometric extension process for free groups (§2) to extend these to K = gp([G, H]) , giving a Hausdorff topologization of K . This leads in turn to a Hausdorff topologization of G * H inducing the original topologies on G and H , and Graev shows (after considerable calculation) that the group operations of
G * H are continuous in this topology. The free topology on G * H is, of course, at least as fine as Graev's, and is therefore Hausdorff.
It is noteworthy that Graev's proof of the above result, in contrast with the analogous result for free topological groups, remains the only one known. (That given in Morris [1] is invalid.) Several other authors, however, have proved the result in special cases: Hulanicki [1] for products of compact groups, Ordman [1] for products of locally invariant groups, and Katz [1] and Brown and Hardy [1] for countable products of groups which are (Hausdorff) k^-spaces.
1.3.2. The relation between FM(X) and FG(X).
The functors FM and FG are both left adjoints, and consequently preserve coproducts in their respective domains. Thus, for any class of spaces {X^} , we have FM( U X-^) = II FM(X^) and FG( V X-^) = II FG(X^) , since the disjoint union LI is the coproduct in the category of spaces, and the wedge or one-point union V is the coproduct of pointed spaces.
It is easy to see that for a completely regular Hausdorff space X ,
FM(X) is topologically isomorphic to FG(X U {e}) , where e is the 16. identity of FM(X) . Since X U {e} can be written as X V {xQ,e} for any point Xq e X , the above remarks therefore give us topological isomorphisms
FM(X) = FG(X V {xQ,e}) = FG(X) * FG({xQ,e}) = FG(X) * Z , where Z is the discrete infinite cyclic group.
More explicitly, it is easy to check that the subgroup gp(x^ ^ X) of FM(X) is topologically isomorphic to FG(X) (with respect to the map gx : X gp(xQ * X) sending x to x^ ^ x) , and that there are then isomorphisms FM(X) = gp(x^ * X) * gp({xQ}) = FG(X) * Z . Of course, for the free Abelian topological groups, we have the analogous result AM(X) = AG(X) $ Z .
Implicit in the comments above is the fact that FG(X) is independent of the basepoint chosen in X . This fact has been used to provide examples of non-homeomorphic spaces with isomorphic free topological groups. For example, if X and Y are copies of the interval [0,1], we know FG(X V Y) is isomorphic to FG(X) * FG(Y) , regardless of the choice of the points at which X and Y are wedged. Then, by choosing these points differently we can obtain three non-homeomorphic spaces, an interval, a T-shaped space and a cross-shaped space, with isomorphic free topological groups.
Note that if FG(X) and FG(Y) are isomorphic, for any X and Y , we have FM(X) = FG(X) * Z = FG(Y) * Z = FM(Y) , so that FM(X) and
FM(Y) are isomorphic. Whether this implication can be reversed is unknown.
The contents of 1.3.2 are taken essentially from Morris [2] . 17 . CHAPTER 2
Local Compactness and Free Products
§1. Introduction and preliminaries.
If {G^} is a collection of locally compact Hausdorff topological
k groups, it is natural to ask when the free product II G-^ is locally compact. Morris [3, 4] and Ordman [2, 5] obtained partial answers to this question, and a full solution was given by Morris, Ordman and & Thompson [1] : IT is locally compact only when each G^ is
k discrete (in which case II G^ is, of course, discrete itself). Observe
k k that II G, can be decomposed as a free product G, * II G, of two A A0 A*A0 A of its closed subgroups, so that it suffices to prove the result for a product G * H of just two factors G and H . To do this, Morris,
Ordman and Thompson note firstly that the (algebraically free) Cartesian subgroup gp([G, H]) of G * H (see §3 of Chapter 1) is closed, and thus has a locally compact topology. They then invoke a result of
Dudley [1], that the only locally compact group topology possible on a free group is the discrete one, so that gp([G, H]) is discrete. If now {g^} is a convergent net in G , then {[g^, h]} is also convergent in gp([G, H]) (taking for H any non-trivial element of H) and is therefore eventually constant, whence {g^} is also eventually constant, and G is discrete. Similarly, H is discrete, and so is
G * H . 18.
Dudley's result is actually more general than the case just quoted.
He defines a "norm" on a group to be an integer-valued function with properties imitating those of the "reduced word length" function on a
free (or free Abelian) group - the latter function constituting a basic example. He then shows that any homomorphism from a locally compact group into a normed group must be continuous, if the normed group is given the discrete (and hence any other) topology. The above application
to free groups follows easily.
In this chapter, we prove a theorem which generalises both the work of Dudley in the case of free groups, and the work of Morris, Ordman and
Thompson. We show that any homomorphism from a locally compact group H into an algebraic free product of groups with the discrete topology is either continuous, or trivial, in the sense that the image of H lies wholly in a conjugate of one of the factors of the free product (Morris and Nickolas [1]). In particular, we see that a (non-trivial) algebraic free product admits no locally compact group topology other than the discrete topology. The same is true for a free group, which we may regard as a free product of infinite cyclic groups.
Our results may be thought of as saying that algebraic freeness is, in a strong sense, incompatible with local compactness, and one might therefore expect that our proof would require fairly detailed information on the structure both of free products and of locally compact groups.
This is the case; the main tools that we use, in fact, are the Kurosh subgroup theorem for free products, and a theorem of Iwasawa [1] on the structure of connected locally compact groups. Dudley's methods are not immediately applicable to our problem - a free product is normable only if each factor is normable, and, while the class of normed groups is not very clearly delineated, it is certainly still quite restricted. (Any 19. normed group must, for example, be torsion-free.)
* In this chapter, II may denote either the algebraic or topological
free product of the (abstract or topological) groups {G-^} (but mostly
the former). Where confusion is possible we shall indicate which is meant. All topologies are assumed Hausdorff.
We now state a version of the Kurosh subgroup theorem (Kurosh [1],
Higgins [1], Magnus, Karrass and Solitar [1] etc.), and note a few
consequences.
Theorem 2.1.1. l{\ G = II G^ iA an aZgebhatc {\h.ee product o{\ gh.ou.p6 XeA ^oh. 6ome tndex 6et A , and s t6 a 6ubghoup ofi G , then S t6 itAelfi a fih.ee product loaXA do.compo6AjU.on
* s p * n S 5 yeM y wkoAc M iA 6ome Index 6et, F iA a fih.ee gsioup, each w^ iA an element ofi G , and each iA a 6u.bgh.oup ofi one ofi the , X e A .
Considerably more detailed versions of this theorem are known (see
the references above) - one of these will be discussed in Chapter 4 - but
the result as just stated is sufficient for our purposes here. We also need
Lemma 2.1.2. LoJ S be a 6 ubghoup ofi the algebhatc fihee product
Then tfi S [l] iA dlviAlble, oh. [it] iA Aboltan and iA not iAomohphtc to the tnfitnite
cyclic ghoup z, 20.
on. [Hi] has bounded u)on.d length fieZatlve to the groups {G^} , then S li> cl conjugate In G ofi a subgroup ofi G^ , jjo/1 some X .
It is obvious that a group which is either divisible or Abelian cannot be decomposed non-trivially as a free product, and (i) and (ii) therefore follow from Theorem 2.1.1. A slightly longer argument, which we omit, proves (iii).
§2. Homomorphisms from locally compact groups into free products.
The following lemma is actually a special case of Dudley's results: we include a proof here, however, which depends on the known algebraic structure of compact Abelian groups, and is thus more in the spirit of our other proofs.
All the results of §2 appear in Morris and Nickolas [1].
Lemma 2.2.1.
Proof. Suppose that Z , and so we can assume that If H is Abelian, then since Z is projective, H is algebraically isomorphic to K x Z , where K is the kernel of Now let H be any compact group. For each x e H , the closure of the group generated by x is a compact Abelian group, and its image under (j) is therefore trivial. In particular cf)(x) = 0 , and since this holds for each x e H , (j) is trivial as required. We now begin our proof of the main result. The argument splits naturally into three steps, which we record separately. Theorem 2.2.2. 1^ (J) tt> any komomofipktbm fatiom a compact gsioup H 'k — 1 tnto an atgcbnatc {nee product G = n G, , then cj)(H) c x G, x , AeA A A {on borne x e G and AeA. Proof. Let us suppose firstly that (J) is onto and that there are two non-trivial groups G^ and G^ in {G^} . Choose non-trivial elements g^ e G^ and e G-^ and an element h e H such that 4>(h) = g2 • If we now set A = gp({h}) , the closure in H of the subgroup generated by h , we see that A is a compact Abelian group, and hence that gp({g^ 82^) • By Lemma 2.1.2 , we therefore have either (1) (j)(A) is algebraically isomorphic to Z , or (2) cf)(A) ^ x G-^ x ^ for some x e G and AeA. Case (1) is shown to be impossible by Lemma 2.2.1 , while if Case (2) occurs, the elements of since gp((g^ g2^) S. ^(A) • Thus if (J) is onto, there can in fact be at most one non-trivial group in {G, : A e A} , and the statement of the theorem clearly holds. 22. If (f) is not onto, the Kurosh subgroup theorem (2.1.1) nonetheless & assures us that (j)(H) , being a subgroup of II , is a free product * -1 F * II (w S w ) , where F is a gree group, M is some index yeM h y y set, each w is an element of G , and each S is a subgroup of some y y G^ . By the argument of the first paragraph, this product can have only one factor. That is, either (a) (H) is isomorphic to F , or (b) cb(H) C w G, w ^ , for some A e A and w € G . — y A y * y If (a) holds, F is itself a free product of copies of Z , so that a further application of the argument of the first paragraph shows that (j)(H) is isomorphic to Z . This possibility is again ruled out by Lemma 2.2.1 , and so (b) holds, which proves the theorem. Theorem 2.2.3. 1^ (j) iA any homomosipfuAm fisiom a connected loca&Ly compact gsioup -into an aZgebsuitc ^nee product G = II G, , then _1 AeA A 4>(H) C x G^ x , {±0*1 t>ome x e H and AeA. Proof. Since H is connected and locally compact, the structure theorem of Iwasawa [1] (see also Montgomery and Zippin [1, §4.13]) says that H has a compact connected subgroup K and subgroups R^, R2, ...» R^ , for some n , where each R^ is topologically isomorphic to the additive group of real numbers with the usual topology, such that each h e H can be decomposed in the form h = r^ r^ ... r^ k , with k e K and r. e R. , for each i . Each R. is divisible and 1 1 1 so, by Lemma 2.1.2 , <})(R^) is contained in some conjugate -1 * w^ G^ w^ in n Gx , i = 1, 2, ..., n . Also, by the previous i i i theorem, (J>(K) ^ w, G, w. ^ , for some An e A . But since any h e H “ A0 Ao Ao ° has the representation h = r, r„ ... r k , we see that the lengths of 12 n 23. the elements of (j)(G) are bounded. Thus, by Lemma 2.1.2 , (J)(G) C x G^ x ^ , for some x e H and A e A . We now present out main theorem. Theorem 2.2.4. let Q be any komomonphlsm ^nom a locally compact gnoup H Into an algebraic fatiee product G = II G, . I| G given the AeA A dlsc/iete topology then one [at least) ofa the following holds: [1) (2) cj)(H) ^ x G^ x ^ , fcoh some x e G and AeA. Proof. Let C be the connected component of the identity in H . Then C is a closed, hence locally compact, normal subgroup of H . Our proof is divided into two cases, depending on whether or not 4>(C) is trivial. Suppose firstly that cf>(C) is not equal to {e} , where e is the identity in G . The previous theorem tells us that (j)(C) <2 x G^ x ^ , for some x e H and AeA. More precisely, 4)(C) = x S x ^ for some non-trivial subgroup S of G^ , and since C is normal in H , we see that x ^ (f>(C)x = S is a normal subgroup of x ^ (|)(H)x . So for any w e x ^ (f)(H)x , we have w^Sw=SCg^ , and this implies that w e G^ . That is x ^ cj)(H)x G^ , so that cf)(H) ^ x G^ x ^ , as required. Suppose secondly that (C) = {e} . It is clear that D = H/C is a locally compact (Hausdorff) totally disconnected group and that (f) induces a homomorphism iJj : D G such that (f) = ijj o p , where p is the natural quotient homomorphism from H onto D . If ip is continuous, the continuity of p ensures that (j) is als° continuous, 24. while if ip(D) C x x ^ , the identity (j) = ip ° p ensures that 4>(H) C x x ^ : consequently, we need only show that the theorem * holds for the map ip : D “*• II G-^ . By Theorem 7.7 of Hewitt and Ross [1] every neighbourhood of the identity in D contains a compact open subgroup K . If for any such K , ip(K) = {e} , it is clear that the kernel of ip is open in D , so that ip is continuous and the theorem is proved. Suppose on the other hand that no compact open subgroup of D maps to the identity under ip . If and are two such subgroups, Theorem 2.2.2 shows that *KK^) C G-^ x^ and ip(K^) C x^ , for some x ,x2 e G and X1,X2 e A . Now H K2 is again a compact open subgroup of D , and by assumption H K2) / {e} , but ^(Kx n k2) c n ip(K2) c Xl Gx x"1 n x2 Gx x”1 . Therefore x, G, x. ^ H x0 G-, x^ / {e} , and thus G-. = G, and 1 Ai I l a2 l ^1^2 X1 ^X x]/ = X2 x2^ * Hence ^(K) C x^ G^ x^ for every compact open subgroup K of D . Fixing such a subgroup K , we have x^ ip(K) x^ = S , for some subgroup S of G^ Let w be any -1 element of ip(B) and choose d e D such that w = ip(d) . Then d K d is another compact open subgroup of D , and so ip(d K d C x G, xA 1 A ^ I But now we have Gx - xiX *Kd K d_1)x1 = x^ w i|i(K)w ^ x^ = X^-1 W Xj Sc x^-1 w "I x^ , which, since S C G-^ , is possible only if x^ w x^ e G^ . This, however, implies that w e xn G, xn^ , and so ip(B) ^ x. G, x.^ , 1 A^ 1 — 1 A^ 1 which finishes the proof. 25. Corollary 2.2.5. A fan.ee product II ofa at teast two non-tniviat gnoups admits no to catty compact gnoup topotogy otken. than the disenete one. k Proof. In the statement of Theorem 2.2.4 put H = G = II , giving H a locally compact topology and G the discrete topology, and let (f> be the identity map. Clearly (2) is false, and so (j) is continuous and H is discrete. Corollary 2.2.6. I fa (p any komomoApkism faaom a tocatty compact gnoup H tnto a fan.ee gnoup with the disenete topotogy, tken <|> is continuous. In panticutan, a fanee gnoup admits no tocatty compact gnoup topotogy otken than the discrete one. Proof. We can regard the free group as a free product of copies of Z, so that by Theorem 2.2.4 , (J> is either continuous or its image is a subgroup of a conjugate of one of the copies of Z . The problem is therefore reduced to that of showing that a homomorphism (which we again call (j) ) from H onto Z must be continuous. Iwasawa’s structure theorem (cf. the proof of Theorem 2.2.3) shows that the component C of the identity in H has subgroups R^, R^, ...» R^ topologically isomorphic to the reals, and a compact connected subgroup K , such that each element c e C can be written as a product c = r^ r^ ... r^ k , with e R^ for i = 1, 2, ..., n , and k e K . Since each R^ is divisible we have (j)(R^) = {0} > and Lemma 2.2.1 shows that <{)(K) = {0} . Hence cj)(C) = {0} . As in the proof of Theorem 2.2.4 , (J) can therefore be factored 26. through the quotient p : H H/C , giving a homomorphism \jj : H/C Z . But H/C is locally compact and totally disconnected, and so has a basis at the identity of compact open subgroups. By Lemma 2.2.1 all of these subgroups are mapped to 0 by f , so that ip is continuous. It follows that (J) is continuous, as required. 2.2.7. Remarks. (1) Morris, Ordman and Thompson [1] asked: What are the locally compact subgroups of T * T , the free topological product of two copies of the circle group T ? It is now clear that a locally compact subgroup H of A * B , for any topological groups A and B , is either discrete or topologically isomorphic to a closed subgroup of A or B . In particular, the locally compact subgroups of T * T are either discrete or topologically isomorphic to T (and therefore compact). (2) It should be noted that Lemma 2.2.1 is still valid if we replace Z by an arbitrary free Abelian group. An argument such as that in the proof of Corollary 2.2.6 then shows that any homomorphism from a locally compact group into a free Abelian group with the discrete topology is continuous. This fact is also given by Dudley’s work. (3) A free Abelian group is a "free Abelian product" - that is, a (restricted) direct product - of copies of Z . In contrast with both the preceding remark and Theorem 2.2.4 , a direct product of arbitrary Abelian groups may admit non-discrete locally compact group topologies in profusion. It is well-known, for example, that the group R of real numbers is (algebraically) a direct product of c copies of the group 27. Q of rational numbers (Fuchs [1, §23]), while the usual topology for R is locally compact. Indeed there exist many compact topologizations of R (Hawley [1]). (4) Dudley [1] has shown that a homomorphism from a complete metric group into a free (or normed) group with the discrete topology must be continuous, and it is tempting to suggest that the complete metric analogue of Theorem 2.2.4 is true. A proof of the kind we have given here for the locally compact case, however, would require some knowledge of the structure of complete metric groups, and this is not available. Morris, Ordman and Thompson [1] have nevertheless used Dudley’s result to show that a free product of topological groups is complete metric only when it is discrete. The argument is exactly analogous to the one they used in the locally compact case (§1). 28. CHAPTER 3 Subgroups of Free Topological Groups §1. The role of ky spaces. In this chapter we shall study the free topological groups on ky spaces, and their subgroups. The restriction to ky spaces allows us to describe the topology on the free group quite explicitly, in a way that is not possible in the general case. This in turn gives us a means of examining the topology on a subgroup of a free topological group and finding when such a subgroup is again free. Definition 3.1.1. A HauA do Apace. X lA a ky space Ifi It lA the union o^ an IncAexiAlng Acqucncc x.^ c x2 c ... ofi compact AubAelA, Auck that a Aet A C x 1a cloAcd 1^ and only Ifi A n Xn 1a cloAcd In x^ , fion. each, n . The inclusion of Hausdorffness in the definition is not essential but it will be our standard assumption here. It clearly allows the condition in the definition (that X has the weak topology with respect to the sets {Xn}) to be replaced by: A set A C x is closed if and only if A H is compact for each n . We say that X has kydecomposition U X^ , or that U X^ is a decomposition of X as a kyspace. The assumption of Hausdorffness also enables us to conclude that ky spaces are paracompact (cf. VIII, 6.5 of Dugundji [1]), and hence are completely regular Hausdorff spaces. 29. On the other hand, the class of k^-spaces includes all compact Hausdorff spaces, countable CW-complexes and a-compact locally compact Hausdorff spaces. Madison [1] shows that the k^-spaces are precisely the Hausdorff quotients of the a-compact locally compact Hausdorff spaces. We now list the standard facts about k^-spaces that we shall need. Proposition 3.1.2. Let x be a k -Apace with decomposition u . Then (1) A closed subspace w ofi x is also a k -4pace, with decomposition u (w n x ) ; (2) Ifi c is a compact subspace ofi X , tkcJie is an n such that C C x ; — n (3) Ifi {X^} is an incheasing sequence ofi compact subsets ofi X uiith union x , then X haA k - decomposition u x^ ifi and only ifi fioh each n thehe is an m such that X c x' ; (4) Ifi Y is a k^-Apace with decomposition u y^ , then X x y is a k^-space with decomposition u (x x Yn) ; and (5) Ifi X is also a topological ghoup, then it is complete (in the left on night unifiohmity). Hence it is closed in any Hausdohfifi topological ghoup ofi which it is a subgroup. It is simple to prove (1) and (3); (2) is Lemma 9.3 of Steenrod [1]; (4) is Proposition 4.26 of Madison [1] (see also Michael [2] and Milnor [1]); and the first part of (5) is Theorem 2 of Hunt and Morris [1], while the second part follows from Proposition 8 of Chapter II, §3, no. 4 of Bourbaki [1]. Let (X,e) be a pointed k^-space with decomposition U . Suppose for convenience that the basepoint e lies in X^ . (If e e Xn say, 30. we may simply forget X-, XOJ ...» X , and renumber the sets {X } 1 l n^—1 n beginning at X .) We have noted in §1 of Chapter 1 that the subspace no F^(X) of FG(X) , consisting of the words of lengths less than or equal to 1 , is the wedge X V X ^ of X and X ^ at e . For any n we can therefore identify X^ V X^ with an obvious subspace of F^(X) . Then, denoting gp^CX^) by F^(Xr) , there is a surjection p : (X V X ^)n “■* F (X ) sending any n-tuple of elements of X V X ^ to their product in FG(X) . Also, continuity of the group operations shows that p is continuous, and, since (X V X ^)n is compact and n n n FG(X) is Hausdorff, we see that p^ is an identification map. In other words, the topologies on the sets F (X ) are fixed once n n we know the topology of X , and, in fact, the above paragraph shows that the same is true for any Hausdorff group topology on FG(X) inducing the original topology on X . Since the free topology is the finest topology of this kind, it is therefore natural to hope that the free topology might be given as the weak topology relative to the compact subspaces Fn(Xn) . This is indeed the case: Graev [1] proved the result when X is compact, and the general result was proved by Mack, Morris and Ordman [1], Theorem 3.1.3. (X,e) a pointed k^-4pac.£ with dzcompoA'OUon U x^ , tkzn FG(X) a k^-4pace, uuXk dzc,ompoi>djtiovL U F^CX^) . AnaZogouA JieAuZtA hold £oh. FM(X), AG(X) and AM(X) . The proof that the group multiplication of FG(X) is continuous with respect to the weak topology on FG(X) rests upon the vital fact that, for maps between k^-spaces, products of identification maps are 31. identifications. That is, if -+ and : X2 Y2 are identifications, then so is x ^2 : x X2 ^ Y1 X Y2 ’ Provided that the spaces involved are all k^-spaces. This is an easy consequence of Proposition 3.1.2 (4), which shows that x and x are both k^-spaces. It is well-known that, between arbitrary spaces, a product of identification maps may fail to be an identification; this may even happen for maps between (Hausdorff) k-spaces (Dugundji [1], VI, 7, Ex.l). By restricting our attention to k^-spaces, we are in effect moving into a (full) subcategory of the category of topological spaces in which this difficulty vanishes. Other authors (Ordman [3 and 4], Hardy [1]) have successfully used another category, the category of k-Hausdorff k-spaces and continuous maps, to overcome the same problem. In the next chapter we shall discuss a construction of Brown and Hardy [1] which includes both the free topological group and free products as special cases, and we shall see that in this more general context as well, explicit descriptions of the appropriate topologies are made possible by restriction to k^-spaces. §2, The freeness of subgroups. Several authors have given examples of subgroups of free (or free Abelian) topological groups which are not free (Graev [1], Brown [2], Hunt and Morris [1]). The precise information we now have about the topology of FG(X) when X is a k^-space makes available straightforward criteria for the freeness of its subgroups. Such criteria are given in Theorem 3 of Mack, Morris and Ordman [1]. We prove here a sharpened version of this result (Nickolas [1]). 32. Theorem 3.2.1. Let (X,e) be a pointed k -Space with decomposition u x^ , and let Y be a subspace FG(X) -6ucA ;£fiaX Y\{e} ^Aeely genenateb gp(Y) . Then the fallowing aAe equivalent: (1) Y is the union ofi an increasing sequence ofi compact sets {Yn> such that fa A each n there exists an m fa/1 which gp(Y) n Fn(Xn) C gpm(Ym) ; (2) there Is a decomposition u y^ o£ y as a k^-space suck that far each n thlere extsts an m fa/i which gp(Y) n Fn(xn) C gpm(Ym) ; (3) gp(Y) Is FG(Y) , and gp(Y) and Y are closed in FG(X) . Proof. That (2) implies (3) is Theorem 3 of Mack, Morris and Ordman [1], and that (2) implies (1) is trivial. Suppose (1) holds: we shall show that the given union Y = Uy n is a decomposition of Y as a k^-space (cf. Theorem 4 op.cit.). Now y n f (x ) = y n gp(Y) n f (x ) n n or n n C Y H gp (Y ) for some m , by (1) — m m = Y . m Therefore Y H F (X ) C y H F(X), and since Y C y we have n n — m n n m — YH F (X ) = Y H F(X). But both Y and F (X ) are compact, nnmnn m nn and so Y T) F^CX^) is compact. Since this holds for each n , and FG(X) = U Fn(Xn) is a k^-space, Y is closed in FG(X) , and is thus a k^-space with decomposition U(y D F^CX^)) (Proposition 3.1.2 (1)). To prove that U is another such decomposition, Proposition 3.1.2 (3) shows that we must find for each n an m for which Y H F (X ) C Y ; and this we have already done. Thus (2) holds, n n — m 33. Suppose now that (3) holds. If we set Y = Y ^ F (X ) , the n n n fact that Y is closed implies that U Y^ is a decomposition of Y as a k^-space. By Theorem 3.1.3 (Theorem 1 ibid.), gp(Y) = FG(Y) is therefore a k^-space with decomposition U gp^(Y^) . But gp(Y) is closed in FG(X) and so gp(Y) ^ F (X ) is compact for each n , and there is an m such that gp(Y) ^ F (X ) ^ gp (Y ) (Proposition 3.1.2 (2)). Therefore (2) holds, completing the proof. Of course, there is an analogous result for FM(X) , AG(X) and AM(X) . In the Abelian case we write gp (X ) as A (X ) rather than n n n n as F (X ) . n n In the following example we investigate the freeness of two subgroups of a free Abelian topological group, and illustrate two ways in which Theorem 3.2.1. may be applied. 3.2.2. Example. Let Xq, x^, x^, ... be a sequence of distinct elements in some compact Hausdorff space X . In AM(X) consider the set Y of elements y = nx~ + x , n = 1, 2, ... (using additive notation). We claim n 0 n that Y is a closed, discrete subspace of AM(X) and that gp(Y) is algebraically the free Abelian group on Y , but is not equal to AM(Y) . We know that AM(X) is a k^-space with decomposition U A^(X) , and it is clear that if S is any subset of Y , each intersection S H A^(X) is finite and hence compact, showing that S is closed in AM(X) . This proves that Y is closed in AM(X) and is discrete. In particular, Y has a k^-decomposition U Y^ , where Y = Y n A (X) = {y., y0, ..., y .} . n n •/l y2 n-1 34. Now suppose that w = a^y + a y + ... + a,y is an element of i n1 z n« K. n- 12 k gp(Y) written in reduced form, so that {n^} are distinct positive integers, {a_^} are non-zero integers, and k > 1 . Then w cannot be equal to 0 , since w = (a^, 4- a0n0 + ... + a. n. )x-. + a.x + a0x + ... + a, x which 11 22 k k 0 In, 2 n0 k n. 12 k has length £ a,n, + amn + ... + a, n, + a. + art + ... + a, 11 2 2 k k with respect to X , and £ = 0 implies that a^ = a^ \ - 0 ’ contradicting our assumption. Thus gp(Y) is algebraically the free Abelian group on Y . Theorem 3.2.1. (adapted to the case of AM(X)) shows that gp(Y) is AM(Y) if and only if for each n there exists an m such that gp(Y) H A^(X) C gpm(Y^) . But the sequence of all elements yn+l “yn = X0 + Xn+1 “Xn clearlY lies in gpOO n A3(x) Yet is contained in no set gpm(Y^) , and so gp(Y) is not AM(Y) . It is still possible, of course, that gp(Y) is the free Abelian topological group on some other space. The work above does show, however, that such a space (if it exists) is not discrete, for if it were, gp(Y) would also be discrete and hence equal to AM(Y). On the other hand, consider the subset Z of Y , consisting of the elements z = y . = n!x~ + x . . We claim that gp(Z) is AM(Z) . n nl 0 nl For convenience we leave out the factorial sign in the subscript, writing z^ = nlxQ + x^ . (The sequence "txn^ was i-n anY case chosen arbitrarily, so that (xn,} can be replaced by any other desired sequence, and, in particular, by itself.) By the work above, we already know that gp(Z) is the free Abelian group on Z , that Z is discrete, and that Z is closed in AM(X) and has k^-decomposition U , with Z = Z H A (X) . Thus we have only to show that for each n there is n n J 35. an m for which gp(Z) H A (X) C gp (z ) . n — m As before, let w = a.z + a0z + . . . + a, z be an element of In, 2 n_ k n, 12 k gp(Z) H A^(X) written in reduced form relative to Z , so that {n^} are distinct positive integers, {a^} are non-zero integers, and k > 1 Also suppose without loss of generality that 1 < n^ < n^ < ... < . Then w = (a.n, I + a^n,.! + ... + a. n, \) x„ + a.x + a.x + ... + a, x 11 22 kk 0 1 n^ 2 k n^ and so the length of w with respect to X is |a^n^! + a2n2* + ••• + ajcnjc* I + || + | a21 + . • • + |a^ | , which by hypothesis is less than n . Therefore we have |a | < n for i= 1, 2, ..., k , |a^n^I + a2n2* + ••• + a|ctl]c* I - n > anc* k < n Furthermore, writing p = n^ , we also have a,n,! + a~n„! + ... + a < la.ln.! 4- |a0|n0! + ... + la 11 2 2 k-1 k-1* 11 1 2 2 k-11 k-1 < nCn^^! + n2! + ... + < n(1! + 21 + ... + (p - 1)!) < 2n(p - 1)! Hence n > a.n,! + a~n0! + ... + a. n. ! 'll 22 k k > |aknk:| - la^! + .ft! + . + ak-lnk-i: " IakI"k: _ 2n(-p " 1-)! > p'. — 2n(p -1) ! , and so p! < n + 2n(p - 1)! = n(l + 2(p - 1)1) ^ 3n(p - 1)! Thus, p = max {n^, n2, ..., n^} < 3n , and since we have already shown that k < n , we see that w e gp (Z ) for m = (3n)! + 1 . Therefore m m gp(Z) H Ar(X) C gpm(Zm) , and so gp(Z) = AM(Z) . 36. 3.2.3. The Schreier theorem for free topological groups. Let F be the free group on a set of generators X and let H be a subgroup of F . We recall that a Schreier transversal for H in F is a transversal (or complete set of coset representatives) for H in F satisfying (i) the representative of H is the identity, ^2 £ £ and (ii) if x. x„ ... x , x is the representative of its coset, 12 n-1 n written in reduced form with respect to X (so that for each i x^ e X and = ± 1 , and x^ = x^+^ implies £^ = , then ^1 e2 en-l the initial segment x^ X2 ... xn_^ is also the representative of its coset. The well-known Nielsen-Schreier theorem of abstract group theory states that a subgroup of a free group is free, and one of the standard proofs of this fact makes use of a Schreier transversal to construct a set of free generators for the subgroup (Hall [1], Kurosh Cl] for example). Given a subgroup H of the free topological group FG(X) on a k^-space X , it is clear that a transversal for H in FG(X) can be regarded as a section s : FG(X)/H FG(X) of the natural projection p : FG(X) FG(X)/H . (We say that s is a section of p if the composite p 0 s is the identity on FG(X)/H.) Brown and Hardy [1] have shown in these circumstances that if s is continuous (relative to the quotient topology on FG(X)/H) and defines a Schreier transversal, then the subgroup H is a free topological group. Their proof employs the theory of topological groupoids (cf. Chapter 4) and is a topologized version of the proof of the usual Nielsen-Schreier theorem given by Higgins [1]. Here we use Theorem 3.2.1 to provide a proof of their result without recourse to the theory of topological groupoids. 37. Theorem 3.2.4. Lit (X,e) bi a pointed, k^-Apaci utt^i dicompoAltlon U , and lit H be a Aubgsioup ofi G = FG(X) . Suppose, that the. pfiojictlon p : G -* G/H (wheAi G/H -tfie 4 pace night iot>itA ofi H) ha6 a aontlnuouA fiction s : G/H -* G 4ucb -Cfiat s(G/H) aj> a SihAiloA tAanAveAAal faon. H In G . Thin H li> cloAid and aa a [Gsiaiv] hh.ii ^opo^og-ccaZ gsioup. Proof. If we set B = {s(Hg)xs(Hgx) ^ : g e G , x e X} , the usual proof of the Nielsen-Schreier theorem (e.g. Hall [1]) shows that B\{e} is algebraically a free basis for H . Define (j) : G x X G by (f) : (g,x) ^ s(Hg)xs(Hgx) ^ , for g e G and x e X . Clearly B = (f)(G * X) . Now each of the following functions is continuous on G x X : (g,x) g »"► p(g) = Hg ^ s(Hg) , (g,x) ^ x , (g,x) hgxH p(gx) = Hgx H- s(Hgx) ; and thus (p is itself continuous, by the continuity of multiplication and inversion in G . Defining B to be 6(F (X ) x x ) we therefore n n n n see that each B is compact, that B C B C ,,, and that n 1 — 2 — OO B = U B . According to Theorem 3.2.1, to show that H is FG(B) n=l n and is closed in G we need only find for each n an m for which H n F (X ) c gp (B ) . n n — m m To this end, let h e H H F (X ) . Then h can be written in n n £1 £2 reduced form as x^ x^ ... (so that = £_^+^ whenever x^ = x^^) where x. e X for i = 1, 2, ...,k, and k < n . Since h e H the in proof of the Nielsen-Schreier theorem shows that we can also write 38. = c. c. ... c. , where 12 k e. e0 e. . e. e. e0 e. . e^-1 it 1 2 i-I i „ 1 2 i-I i = s Hx. x„ ... x. n X. s Hxi x0 ... x. n x. 12 i-I 1 12 i-I i for i = 1, 2, ..., k . It is also clear that, for each i < k , £1 e2 £i-l £1 £2 £i xn x_ ... x. . and x. x0 ... x. lie in F (X ) , and x. e X 1 2 i-I 12 l n n in £1 £2 i-1 Noting that if 6^=1 then c^ = (j) we x! x2 **• xi-l therefore see that c. e (b(F (X ) x X ) = B . Similarly, if e. = -1 , iY n n n n y i -1 £1 £2 £: then c e 4>(F (X ) x x ) = B . Hence X1 x2 * * * Xi n n n n h = c. cn ... c. e gp (B ) , since k < n : that is, 12 k n n H H f (X)Cgp (B) for all n . This proves the theorem, n n — n n The following example shows that the condition in Theorem 3.2.4 for the freeness of a subgroup of FG(X) is not a necessary condition. 3.2.5. Example. Let (X,e) be a pointed compact Hausdorff space, and Y a closed subset of X containing e . By an easy application of Theorem 3.2.1, the subgroup H = gp(Y) of FG(X) is FG(Y) (cf. Corollary 1.2.10) . Suppose that there exists a continuous section s of the projection p as in Theorem 3.2.4. Let x e X\Y and suppose that £1 £2 £n ^1 e2 £n —1 s(Hx) = x, x« ... x in reduced form. Then x. x0 ... x x e H 1 2 n 1 2 n £n £1 e2 en —1 If x * x , then x. x_ ... x x is reduced (as written) with n 12 n respect to X , and therefore with respect to Y , and so x^, x^, x^ , x e Y , contradicting x e X\Y . We must therefore £1 e2 'n-1 have x = x . But this implies that x. x_ ... x . € H and n 1 2 n-1 39. ^1 £2 en_i since we know already that x. x_ ... x - is in the Schreier trans- J 12 n-1 ei C2 £n versal for H (because it is an initial segment of x^ x^ ... x^ ) , £1 G2 £n £n we have x. x~ ... x = e ; that is, s(Hx) = x = x , for x e X\Y . 1 2 n n Of course, for x e Y , x £ H also, and so s(Hx) = e . Since X is compact, and s and p are continuous, we find that s(p(X)) = X\Y U {e} is compact. Taking X = [0,1] , Y = [0,^] and e = 0 for example, we have X\Y U {e} = {0} U (^,1] , which is certainly not compact. Thus, although H is FG(Y) , our assumption that a Schreier transversal for H can be chosen continuously is false, and we see that Theorem 3.2.4 cannot be applied here. Under some circumstances, however, it is possible to derive a somewhat stronger result from the proof of Theorem 3.2.4. Corollary 3.2.6. Let (X,e) be a pointed k -Apace with decompoAitlon U Xn and let H be a Mibgsioup ofi G = FG(X) . Suppose, that theAz H Ia FG(Y) and Ia cloAed In G . Proof. By Theorem 3.2.1 we have only to show that for each n there is an m such that H H Fn(Xn) C gp^(Y) . This statement is proved exactly as was the corresponding statement in Theorem 3.2.4, except that the functions s and (p need no longer be continuous. Returning now to Example 3.2.5 above, it is easy to see that, while Theorem 3.2.4 could not be applied, Corollary 3.2.6 may nevertheless be used to show that H = FG(Y) . A Schreier transversal for H in FG(X) 40. certainly exists. (It may be constructed by the usual inductive argument (Hall [1]).) Then if r(g) denotes the representative of an element g e FG(X) , the free basis defined by the Schreier transversal contains the elements r(e)yr(y) ^ for each y e Y . But since y e H , r(e) = r(y) = e , and so the free basis contains Y : because Y is already a free basis for H , however, the new basis must be precisely Y . The hypotheses of the corollary are therefore satisfied, and H = FG(Y) . §3. Subgroups of the free topological group on [0,1]. Our main aim in this section is to classify some of the subgroups of FM([0,1]) . The principal result is that for a compact space X , FM(X) occurs as a subgroup of FM([0,1]) if and only if X is finite dimensional and metrizable. Two steps are involved in the proof. The first is to show that for any k^-space X , FM(X) contains copies of the groups FM(Xn) for each n (where Xn denotes the topological product of n copies of X ) . The second step involves finding an expression for the dimension of certain subspaces of a free topological group, which shows in particular that the dimension of F ([0,1]) is precisely n . The results of §3 appear in Nickolas [2]. To show that FM(Xn) is a subgroup of FM(X) , we must first find a copy of Xn in FM(X) which is algebraically a free basis for the subgroup it generates. Hardy, Morris and Thompson [1] and Thomas [1] have shown that for any completely regular Hausdorff space X , the map 41. r\ n11 1 0 : (x. , x0, .. . , x ) ^ x. x0 . .. x of Xn into FM(X) is a 1 2 n 1 2 n homeomorphism onto a closed subspace, and it might be conjectured that (J)(Xn) is a free basis. This is not the case. Indeed, under no mapping kl k2 kn of the form 6 : (x. , x_, ..., x ) ^x. x~ ... x (with fixed 1 z n I z n integers {k_^}) is (Xn) a free basis for gp((J)(Xn)) . For, choosing distinct x, y, z, t e X , y v y \ is a reduced word in letters from A slightly more complicated kind of map, however, achieves what we want. Proposition 3.3.1. 1^ x ti> a compleXeZy azguZan. HauAdoafifi 4pace, the, map 4> : x x X -► FM(X) de,fitne,d by -lb a kome,omon.pklAm onto ttA tmage,. Proof. It is clear that (p is continuous and injective. We shall show that (j) (as a map onto its image) is open. Let U x V be a basic open neighbourhood of (x,y) in X * X , where U and V are open in X . By Theorem 1.1.7, the set UVU is a neighbourhood of (})(x,y) in the subspace F^(X) . (Here UVU is the product with respect to the group operation of FM(X) .) But clearly UVU H 4>(x,y) in the subspace (j)(X x x) . Hence (p : X x X ^(X x x) is a homeomorphism as required. 42. Lemma 3.3.2. Veitne (J) : X x X ^ fm(x) ah above,, and het e £? e B = <|>(x x x) . Let w = b^1 b2 ... bnn be. an element oi gp(B) Whitten i.n n,edueed ionm wtth n,ehpeet to B . (That th, b., bOJ ... b € B , £,, e0, ..., e =±1, and b. = b... tmplteh 12 n 12 n i i+l that e± = ei+1 ion, i = 1, 2, ..., n-1 .) Fon, i = 1, 2, ..., n huppohe that b^ = (Kx^,y^) = x_^ y_^ , wheAe x_^ , y^ e X . Then the deduced ionm oi w muXh n,ehpeet to x e. (1) n,etatnh eaeh oi the iaetonh y 1 , i = l, 2, ..., n , s-y1 » 'i-i — ^ » (2) hah ah tth itnht two iaetonh -s ^x^ yl , H £]_ = -!» h{\ £ = I y n xn u n and (3) hah ah tth laht two iaetonh -1 -1 w { y x , hi £ = -1 . n n u n Vonthenmon.e, (a) gp(B) n f (x) c gpn_2 (B) ion, eaeh n , and (b) gp(B) th algebnxxleallg the in.ee gn.oup on the het B . Proof. We first prove (1), (2) and (3), by induction on the length n of w with respect to B . Clearly (1), (2) and (3) hold if n = 1 . Suppose that they also 1 e2 hold for n = k-1 , for some integer k . Now let w = b^ b2 ... b^ be an element of gp(B) of reduced length k with respect to B . By £ £ £ 1 2 k-1 hypothesis, the reduced form with respect to X of b^ b2 ... satisfies (1), (2) and (3). Suppose that = 1 , so that the reduced form with respect to £ £ £ 1 2 k-1 X of b^ b2 ... ends with the symbols y^ ^ x^_^ . Multiply 43. k r . k this word on the right by = (x x^J and consider the possible cancellation (with respect to X) in the product. If £. = 1 , it is clear that no cancellation will occur, and that k the reduced form of w satisfies (1), (2) and (3). If = -1 , similar comments apply, unless x^ ^ = x^ • Even in this case, however, cancellation can proceed no further than the deletion of the pair x^_i x^ . For if further cancellation occurs, we must have ^k 1 = Xk * and so b^_^ = b^ , which contradicts our assumption that w is reduced with respect to B , since £k-l = ^ anc* £k = f°H°ws that in this case also, the reduced form of w with respect to X satisfies (1), (2) and (3). A similar proof applies if £k-l = * anc* t*ie truth of (1), (2) and (3) follows by induction. If w is now an element of gp(B) whose reduced length with respect to X is less than or equal to n , the lemma so far clearly shows that the reduced length of w with respect to B cannot be greater than n-2 . That is, gp(B) H F^(X) — ^pn-2 ^ * T^is proves (a), and (b) follows in a similar way. Theorem 3.3.3. Voti any k^-Apace. X , FM(X) fi Proof. If X = U is the decomposition of X , then X x X is also a k^-space, with decomposition U (X^ x X^) (Proposition 3.1.2 (4)). If we define B = (j)(X x X) as earlier, it follows that B is a k^-space with decomposition U B^ , where B^ = (j)(Xn x X^) , since (f) is a homeomorphism. Furthermore, Lemma 3.3.2 (b) shows that gp(B) is algebraically the free group on the set B . By Theorem 3.2.1 it will 44. follow that gp(B) is FM(B) if for each n there exists an m such that gp(B) H f (x ) C gp (b ) , We claim in fact that this inclusion n n — m m holds for m = n . Fix an integer n . By Lemma 3.3.2 (a), gP(B) n Fn(xn) £ §P(B) n Fn(X) £ 8Pn_2 * so if w 6 §P(B) n Fn(xn> » the reduced form of w with respect to B may be written e, e2 e w = b. b ... b r (for b. e B and £. = ± 1 ) with r ^ n-2 . If 1 2 r i i b^ = x_^ y^ x^ for each i (with x^, y^ e X ) , an inductive argument similar to that of the previous lemma shows that x^, y^ e for i= 1, 2, ...» r , and so b. £ B^ , i = 1, 2, ..., r . Hence w e &Pn_2 (Bn) — §Pn(Bn) * anc* SP(B) is topologically isomorphic to FM(X x X) as required. 2 As remarked earlier, X = X x X is a k^-space, so our argument 2 4 thus far shows that FM(X ) contains a copy of FM(X ) . So FM(X) 4 also contains a copy of FM(X ) , and, continuing the argument, we find 2n that FM(X) contains a copy of FM(X ) for each n . Finally, we note £ that for an integer k , and any £ < k , X may be identified with a 1c closed subspace of X , so by another application of Theorem 3.2.1, k £ FM(X ) contains a copy of FM(X ) . The result follows. To prove our main theorem we now need some information on the dimension of Fn([0,l]) , the subspace of FM([0,1]) consisting of words of lengths less than or equal to n . If X is a k^-space with decomposition U X , where each X is a metric space, then (for each n n n ) the disjoint union X^ U X ^ U {e} and its topological product (X U X ^ U {e})n are compact metric spaces. But F (X ) is the n n r n n image of this product under the obvious continuous map, and so F (Xn) 45. is compact and metrizable (by XI, 5.2 of Dugundji [1] for example). In particular it is separable and metrizable, and since the standard definitions of dimension coincide for such spaces (see Nagata [1]) we may speak unambiguously of the dimension of F (X ) , and denote it n n unambiguously by dim F^X^) . Since a subspace of a separable metric space is again separable, similar comments apply to subspaces of F (X ) . n n We shall also need to refer to the Sum Theorem of dimension theory (Nagata [1], Theorem II.1; see also Theorem II.7). The special case of this result that we use states that if a separable metric space M can be covered by countably many closed sets each of dimension less than or equal to n , then the dimension of M is also less than or equal to n . Theorem 3.3.4. 1^ X a kp-ipace uliAk decomposition U Xn , wk&ie each 4 pace Xn ti mct/ilzablc and o^Ivutc cUmcmlon, then &on each n dim Fn(Xn) = dim ((x^)11) , ickene (xn)n ^ topologtcat product o£ n COpt Proof. Our proof proceeds by induction on n . If n = 1 the result is certainly true, since F^(xp = X^ U X^ U {e} , the dimension of which is equal to dim X^ . Under the assumption that dim F ,(X .) = dim ((X for p-1 p-1 p-1 some p , we construct a countable covering of the space F^(X^) by closed sets, each of dimension less than or equal to dim ((X )^) . By an application of the Sum Theorem it then follows that dim F (X ) < dim ((X )^) . The induction is completed once we note that P P P the set Xp*Xp...... ^p (the product in the group FM(X) of p copies 46. of Xp ) lies in F (X^) and is homeomorphic to the product space (X )p , whence dim F (X ) > dim (X )P . P P P P Assume now that for some integer p , £1 £2 dim F ,(X .) = dim ((X . Let w be a p-1 p-1 p-T X1 X2 reduced word of length precisely p in F^(X^) . Using Theorem 1.1.7 we see that a set of the form ... U^P is a neighbourhood of w in the subspace F^(X) » where (for i = 1, 2, ..., p ) U_^ is an open neighbourhood of x_^ in X . Further, using the complete regularity of X , let us choose these open sets in such a way that if x. * x.L1 then U. H U.(1 0 , for i = 1, 2, ..., p - 1 . As l l+l l l+l a result of this choice, and because w is in reduced form, it follows e2 e that every element of U. U„ ... U P and indeed of 1 2 p _ o1 o„ _ c. (up (up z ... (U ) p is of reduced length precisely p . It also £. e2 e follows that ... UpP is now a neighbourhood in F^(X) of (Lack of its points - that is, it is open in F (X) . Now set P U. n X and C. u. n x and observe that CL is compact l p l i p £ for each i . Noting that U. Un ... U p Gi F (X ) = V. V_ V ? 12 p P P 12 P £, e2 e we see from our comments above that V. ... V p is an open 1 2 p neighbourhood of w in the subspace F (X ) \ F ,(X .) . Set P P P-1 P-1 £ £ £ G = V 1 V 2 V p w 1 2 P If we carry out this process for each w e F^(X^) \ F^_^(X^ , the sets {G } form an open covering of F (X ) \ F .(X .) But w r a p p p-1 P-1 this space is separable and metrizable since F^(X^) is, and consequently has the Lindelof property (Kelley [1]). Therefore a countable subclass of the sets {G } covers F (X ) \ F ,(X .) , and so do a countable w P P p-1 P“1 47. e2 e number of the sets C. C« ... C ^ 1 2 p Now the obvious continuous map from C. x C„ x ... x c onto 12 p ei £2 S C.. C0 ... C P is closed since C. x r x ... x c is compact, and 1 2 p 12 P is injective by our choice of the sets (lK) and {C_^} , and is therefore a homeomorphism. Hence, since C for i = 1, 2, ...,p, £1 £2 £p each set C, C„ ... C ^ has dimension less than or equal to 1 2 p dim ((Xp)^) . Consequently, the countable collection of these sets, together with the closed set F .(X ,) , is a covering of F (X ) of p-1 p-1 p p the type required, since we have assumed that dim F n (X .. ) = dim ((X < dim ((X )^) . Then by the Sum Theorem p-1 p-1 p-1 p J (quoted above), dim F (X ) < dim ((X )^) . Our earlier remarks show P P P that the proof is now complete. Corollary 3.3.5. 1^ X tb a compact met/vic Apace oft fitntte cUmenAhon, then each n dim F^(X) = dim Xn . Corollary 3.3.6. In FM([0,1]) , dim F^([0,1]) = n , ^OH cack tntCQQA. n . We are now able to prove the main result. Theorem 3.3.7. Let x be a compact: HauAdosififi Apace. Then FM([0,1]) kaA a AubgAoup topotogteatty Tj>omofiphtc to FM(X) i.fa and onty X tA fitntte-cUmenAtonal and met/Uzable. Proof. We know that FM([0,1]) is a k^-space with decomposition U F ([0,1]) (Theorem 3.1.3). If FM(X) is a subgroup of FM([0,1]) , 48. the compact space X must then lie in ([0,1]) for some n , by Proposition 3.1.2 (2). So X is metrizable (by the remarks preceding Theorem 3.3.4) and finite-dimensional (by Corollary 3.3.6). Conversely, if X is a compact finite-dimensional metric space it may be identified with a (closed) subspace of [0,l]n for some integer n , by Theorem IV.8 of Nagata [1]. Then by Theorem 3.2.1 the subgroup it generates in FM([0,l]n) is topologically isomorphic to FM(X) . But it follows from Theorem 3.3.3 that FM([0,l]n) occurs as a subgroup of FM([0,1]) , and so FM(X) does too. It is now easy to prove the following result. Theorem 3.3.8. Let Y be a completely n.egulan. HauAdon.fa 4pace. The fallowing conditio via cute equivalent: (1) Y Ia not totally patkwlA e- dlA connected, (2) fm(y) kaA a Au.bgn.oup topologically lAomon.phlc to FM([0,1]) , and (3) FM(Y) kaA Aubgn.oupA topologically lAomon.pklc to FM(x) fan. all compact finite-dimenAlonal metnlc ApaceA X . Proof. Theorem 3.3.7 shows that (2) and (3) are equivalent; we show the equivalence of (1) and (2). If (1) holds, Y contains a non-trivial continuous image of [0,1] , and Theorems 3-15 and 3-24 of Hocking and Young [1] then show that Y contains a homeomorphic image of [0,1] . Since this image is compact, it follows from Theorem 1.10 of Morris [5] that it generates a subgroup of FM(Y) topologically isomorphic to FM([0,1]) . 49. Conversely, suppose that (2) holds, and denote by I the copy of [0,1] which generates the subgroup in question. By an argument in the spirit of those of Hardy, Morris and Thompson [1] we see that for some n (which we choose to be minimal) I C F (Y) . (In fact, let (j) : Y $Y be the embedding of Y in its Stone-£ech compactification 3Y , and extend FM(3Y) is a k^-space with decomposition U F^(3Y) and 0(1) is compact, 0(1) C Fn(3Y) for some n (Proposition 3.1.2 (2)). Then clearly I C Fn(Y) . ) e^I e2 Now let w = x^ x^ ... x^ be a reduced word of length precisely n in I , for x^eY,e_=±l,i=l, 2, ...,n. As in the proof of Theorem 3.3.4, choose for each i an open neighbourhood Ik of x^ in Y such that x. * x.., implies U. H u.., = 0 , for l l+l l l+l i = 1, 2, ...» n-1 . This choice of {lL} ensures that every element of £1 G2 £n U = Uf U2 . . . is of reduced length n , and in fact has reduced £1 C2 £n form y^ y^ .. . y^ > f°r Y^ e IL , i = 1, 2, ..., n . Furthermore, by Theorem 1.1.7, U is a neighbourhood of w in the subspace F (Y) , and so U H I contains another copy of [0,1] , which we denote, say, ei e2 C by J . If for each i we define p^ : U Y by p^(y^ Y 2 ••• Y^) = » Theorem 1.1.7 may be used again to show that each p^ is continuous. But it is clear that for at least one value of i , p^(J) must contain more than one point, and so Y contains a non-trivial continuous image of [0,1] , thus proving (1). The kind of arguments we used in Proposition 3.3.1 and Lemma 3.3.2 do not immediately lend themselves to use in the case of the Graev free topological group. The Graev analogue of Theorem 3.3.4, however is valid, as can be seen by making appropriate minor changes in the proof. With its 50. aid we can easily prove the analogue of Theorem 3.3.7 for FG([0,1]) . Theorem 3.3.9. Let X be a [pointed] compact HauAdoa^ Apace. Then FG([0,1]) ha6 a Aubgaoup topologically u>omoh.phlc to FG(X) Ifa and only l{> X It finite- dimensional and metnlzable. The proof is a consequence of the above remarks, Theorem 3.3.7, and the following facts (cf. section 1.3.2) : (1) For any completely regular Hausdorff space X , FM(X) contains a copy of FG(X) , and (2) FG([0,1]) contains a copy of FM([0,1]) . (Any interval [a,b] in [0,1] not containing the basepoint generates a copy of FM([a,b]) . ) 3.3.10. Remark It would be interesting to have an analogue of Theorem 3.3.7 in the case of AM([0,1]) . The remarks preceding Proposition 3.3.1 show that the methods used in that proposition and in Lemma 3.3.2 do not extend to the Abelian case. Theorem 3.3.7 leaves unanswered some interesting questions. We might, for example, wish to weaken the compactness assumption on X in the theorem and ask which k^-spaces X have the property that FM(X) is a subgroup of FM([0,1]) . In particular, does FM([0,1]) contain the free topological group on an open interval? We will be able to provide answers to questions such as these in the next chapter, where a wider range of techniques is available to us. 51. CHAPTER 4 Subgroups of Free Products of Topological Groups PART 1 : Preliminary Material Our main aim in this chapter is to prove a topological version of the Kurosh subgroup theorem. The arguments we give make use of the theory of topological groupoids and the chapter therefore begins with a review of certain parts of this work. Because of this, and the length of the work on the Kurosh theorem itself, it has seemed advisable to divide the chapter in two, with §§1-3 in Part 1 and §§4-7 in Part 2. The summary of the theory of topological groupoids occupies §1, and §2 as far as Proposition 4.2.3, and is adapted from the contents of Brown and Hardy [1] and Hardy [1]; it is thus in no way original and is included simply in order that the chapter be fairly self-contained. Related material appears in Brown and Hardy [2] and Brown, Danesh-Naruie and Hardy [1]. Underlying all of this work is the theory of (abstract) categories and groupoids expounded in the monograph of Higgins [1]; the reader should consult this and Brown [1] for any terms left underfined in Part 1. After Proposition 4.2.3 the contents of §2 are original, although their principal interest here lies in their applications to our work on the Kurosh theorem. In the short §3 we digress for a moment and note one or two miscellaneous consequences of the results of §2. 52. §1. Topological groupoids and universal morphisms. 4.1.1. Definitions and notation. If C is a category we use Ob(C) to denote its class of objects and C its class of arrows, while 3', 9 : C ^ObCC) are the initial and final maps, sending an arrow to (respectively) its initial and final points. We denote by u : Ob(C) C the unit function mapping an object x e Ob(C) to the identity arrow at x , which is usually written as 1^ . Composition in C will be denoted by the symbol 0 which, in practice, will usually be omitted. If in addition there is a function c c ^ defined on C such that cc * = 1^, and c ^c = 1~ for all c e C , then C is said 3 c 3c to be a groupoid; the map c •"* c ^ is the inverse function. A small category C (MacLane [1]) is a topological category if Ob(C) and C have topologies with respect to which 3',3,u and o are continuous on their respective domains. If G is a topological category whose underlying category is a groupoid and whose inverse function is continuous, then G is a topological groupoid. If the space of objects of a topological category (groupoid) C is X we say that C is a topological category (groupoid) over X. Topological subcategories (subgroupoids) of C are defined in the obvious way; such a subcategory (subgroupoid) is wide in C if its object space is the whole of X . If C is a groupoid and x e X then the set of arrows c e C such that 3'(c) = 3(c) is a topological group, the vertex group of C at x , and is denoted by C{x} . 53. A morphism of topological categories or groupoids is a functor in the usual sense which is continuous on both objects and arrows. It is clear that there are categories TC and TG of (respectively) topological categories and topological groupoids, with morphisms as above. If X is any topological space we can construct from it the pointlike topological groupoid over X , having X both as its object space and its arrow space, and with 9’,3 and u being the identity map on X . This construction clearly defines a functor P from the category of topological spaces to TG (or to TC ). If G is a topological groupoid we frequently identify POb(G) , written simply as Ob(G) , with the subgroupoid of G consisting of just the identities of G . We often abbreviate PX to X for any space X . Definition 4.1.2. A moSiplvLAm 0 : G H Ml TC (TG) universal Tfi tlie cornmmtcutLve. kquoAd Ob (6) Ob (G) ------» Ob (H) G ------► H 0 cl puAkouX mi TC (TG) , ookeAe tkd v&uUcal map* a/ie. the. TncluATonA. Hardy [1] (Corollary 3.2.1 of Chapter 1) has shown that TC and TG are cocomplete. It follows that if we are given a topological groupoid G and a continuous map O : Ob(G) -+ X for some space X , then a topological groupoid C^(G) and a morphism a* : G U^(G) 54. exist making the following square a pushout in TG : a Ob (G) ------* X G ------> u (G) a* a That is, G* is a universal morphism. We call G* the universal morphism induced by G . We also refer loosely to U^(G) as a universal topological groupoid, although this terminology is ambiguous unless G is specified. The underlying groupoid structure of UQ(G) is well-known (Higgins [1], Chapter 8), but, as with free topological groups (Chapter 3), the topology of ^(G) is not in general easily described. Brown and Hardy [1], however, have given an explicit construction of Uo(G) when G is a k -space, and we now review their work. U) Let G be a k^-groupoid (a topological groupoid which is a k^-space) and suppose that G : Ob(G) “► X is continuous, where X is some k^—space. Defining G to be X , let W^(G) be the subspace of 00 n 0 the disjoint union LI G consisting of G and all n-tuples or "words" n=0 (g^> g2» •••» §n) (for n > 1) , where g_^ and g^+^ are G-composable — that is, where G3(g^) = g3’ (g^^) — for each i . If we set §2’ 8n^ = S2» •••» gn) = C73(gn) and 3T(x) = 3(x) = x for x e X , and define composition to be juxtaposition, with the elements of G° acting as identities, we see that is a 00 topological category over X and (because it is closed in U Gn ) is a n=0 k^-space. 55. Now with any word in W (G) there is associated a uniquely defined "reduced word" (see Chapter 10 of Higgins [1]); this is a word (§1> §2’ **•» 8n) such that for each i g^ and g^+^ are CF-composable, but not composable — that is, aSCg^ = (gi_hl) but 8(gi) * 3* (g±+i) • Then, if U (G) denotes the set of all such reduced words, there is a natural map p : W^CG) -► Ua(G) , and we give U (G) the quotient topology relative to p . Then we have Theorem 4.1.3 (Brown and Hardy [1]). Ij{ G and X OA.e k^-4pace4 AO oZao Za Uq(G) ; fiufitheAmofie, u (G) ^ & topoZogZcaJL gfioapoZd ovqa X and the map o* : G -► u (G) 6ending g to p(g) Z& the unZveAAal motLpbum Zndueed by a . The following definition and theorem of Brown and Hardy [1] (see also Brown, Danesh-Naruie and Hardy [1]) are essential for our application of topological groupoids to subgroups of free products. Definition 4.1.4. Let q : G -► G be a moA.phZt>m otopotogZeal gfioupoZdA, and ^ohm the palZbaek G x Ob (G) ------> Ob (G) Ob (q) ' ' W G ------» 0b(G) tn the category ofi topoZogZeaZ ApaceA. Then q Za a covering morphism *-i the map (q,3T) : G -► G x ob(G) Za a homeomo/iphlAm. 56. Theorem 4.1.5. I^ iA a piMback cUagAam mi TG , tn which H,G and G a/ic k^-gAoupotdt, q -6s a cov&Ung moAphUm and 0 -ga unlv&u>a£, then the moAphum 0 : H -* G t6 oJU>o univeJi&aJt. §2. Some specific universal morphisms. If G is a topological groupoid and O is the (unique) map from Ob(G) to a singleton space X , we can as usual form U (G) ; now, however, U (G) is a topological group and reference to G is unnecessary. The group is denoted U(G) and is called the universal topological group on G . Clearly o* : G U(G) is universal for morphisms of G into topological groups. Both free topological groups and free products of topological groups can be expressed as universal groups on appropriate groupoids. Specifically, if (X,e) is a pointed topological space we can form a topological groupoid Tx with objects X , arrows X x X , 3' and 9 the projections onto the first and second factors, and with composition (x,y) ° (y,z) = (x,z) : TX is called the tree topological groupoid over X. It is easy to check that if o* : TX U(TX) is the universal morphism and g : X U(TX) is defined by gv(x) = G*(e,x) , then the pair A A 57. (gx , U(TX)) is a Graev free topological group on X (Hardy [1], Proposition 3.6.1 of Chapter 1). If G is a topological groupoid over a space X then we say G is globally trivial if there is a wide, tree topological subgroupoid of G . In such a groupoid we can define a morphism p : G G{x} known as a retraction, for any x e X : if x is the unique arrow from x to y in the tree subgroupoid, and g is an arrow in G with 3'(g) = y ) 9(g) = z , then, by definition, p(g) = T gx ^ . y z To express free products as universal topological groups, suppose that {G^} is a collection of topological groups. Then (regarding each G-, as a groupoid with one object) U G, is also a topological A A A groupoid, and it is easy to see that U( U G^) is the free product These constructions can be generalised as follows. A topological graph is a graph whose object set and edge set have topologies with respect to which the initial and final maps 3’ and 3 are continuous. A morphism of topological graphs is a continuous incidence-preserving map. A topological graph T is distinguished if there is a continuous function u : Ob(F) Y such that 3’u and 3u are both equal to the identity map on Ob(T) , while a morphism f : T T* is distinguished if f(u(x)) = u(f(x)) for each x e Ob(T) There is clearly a category TV of distinguished topological graphs and distinguished morphisms, and a forgetful functor from TG to TV ; we may thus regard a topological groupoid as an object of TV , and speak of a subgraph of a groupoid. 58. Definition 4.2.1. Let ip : V F(T) foe. a TV-mon.phism in.om cl distinguished topological gnaph r into a topological gn.ou.poid F(T) . li ip i6 univensal with aespect to TV-mon.phisms into topological gSiOupoicU, then F(T) is the (Graev) free topological groupoid on T . (Markov free topological groupoids can be similarly defined, but will not concern us here.) Note that if T is a tree topological groupoid over X then T is F(A) , where A is the distinguished subgraph of T consisting of the identities of T and all edges t e T with 3’(t) = x , for a fixed but arbitrary x e X . If T is a distinguished subgraph of a globally trivial topological groupoid G with wide tree subgroupoid T , we say that T conjoins T if T is (algebraically) freely generated (Higgins [1]) by the non-identity arrows of T which are also in T . Definition 4.2.2. Let {G,} be a iamily oi topological gaoupoids AeA and let : G^ G be (ion. each A) a moaphism into a topological gnoupoid. Then G t6 the free product oi the iamily [with aespect to the maps {j,}) , denoted IT G, , ii ion. any topological gaoupoid AeA A H and any collection oi mon.phisms : G^ H satisiying Ob(k^) = m o Ob(j^) ion, some continuous map m : Ob(G) -* Ob(H) , thene is a unique moaphism k : G -► H such that k^ = k ° ion. each A , and Ob (k) = m . Usually we are interested in the case where the groupoids {G-^} are subgroupoids of G and the morphisms {j-^} are the inclusions. 59. Both free topological groupoids and free products of topological groupoids can easily be constructed as universal topological groupoids on appropriate topological groupoids (Hardy [1]). As a consequence, and using Theorem 4.1.5, Brown and Hardy obtain the following result, which is crucial for applications of the theory to subgroup questions. Proposition 4.2.3. Let q : G G be. CL COVeAtng moAphiAm ofi k^-gAoupolds. (i) 1^ G = F(T) oa cl k^-gAOLph r , then G = F(f) , wkeAe T aa the. pullback ofi the. IncluAlon ip : r -* G and q : G G . (ii) 1£ G aa the. &Aee pAoduct n G^ o{ a countable, family ofi k^-gAoupoldA {Gx} , then G aa oIao a ^Aee pAoduct II G^ , wheAe G^ aa the pullback ofi the Inclusion jx : G^ -> G and q : G -► G . Recall from §1 that we denote the pointlike topological groupoid on a space Y by FY . Proposition 4.2.4. Let a : Ob(G) x be continuous, uiheAe G aa a k^-gAoupold and x aa a k^-Apace, and ^oAm the unlveAAal moAphlAm a* : G ua(G) . I £ Y aa Aome k^-Apace then the fallowing AquaAe is a pushout In TG : ox i Ob (G) x Y------X x Y ' ' G x PY ------U (G) x PY a*. x i, O In otheA rnAdA, theAe aa a TG-AAomoApklAm »5* $ : U (G) x PY ^ , (G x PY) Auch that $ ° (a* x l) = (a x l) , a u axi 60. and U^(G) x Py aj> the, imiveAAaZ topologtcat gfioupotd on G x Py tnduczd by a x l . Proof. The following square is easily seen to be a pullback in TG : a* x l G x PY ------> U (G) x PY a > »■ ^ G ------* U (G) a* and a* is universal. The result will therefore follow from Theorem 4.1.5 if we can show that the projection q : U^G) x Py U^CG) is a covering morphism. Looking at the required pullback (of topological spaces) U (G) x (x x Y) ------* X x Y U (G)------X o 3. we see immediately that (q,3’) : U^(G) x PY U^(G) x (X x Y) is a homeomorphism as required. This proposition also follows from a result to be proved in Chapter 5, as we shall see at the appropriate time. The result also translates naturally into the contexts both of free topological groupoids and of free products of topological groupoids. Corollary 4.2.5. ip : T -► F(T) aj> the, natuAal tnctaiton o£ the, k^-gsiaph T tn ttA ^ee topologtcat gscoupotd, and tfi X a k^-4pace, 61. thzn theJiz aJ> a TG-sUomosiphsUm T : F(T) x px -> f(T x Px) Audi that v o (ir x i) = irxpx : T x pX -> F(r x pX). (2) Proof. Brown and Hardy [1] define a topological groupoid T with objects T , arrows T x r = {(a,b) e T x r : 9*(a) = 9’(b)} , initial and final maps the two projections, and composition (a,b) ° (b,c) = (a,c) . If G : Ob(r^)) = p Ob(T) is the final map 9 they then show that (2) ip : r Ua(Tv ) , defined as the identity on objects and given by a^o*(u9'(a),a) on arrows, is the inclusion of V in its free topological groupoid. By Proposition 4.2.4 there is a TG-isomorphism $ : Ua(r(2)) X Px = F(D X px -► U^xl (r(2^ X PX) such that k $ o (q* x i) = (a x i) Now there is obviously a TG-isomorphism to : r(2) X Px (T x PX)(2) defined by ((a,b),x) ((a,x) , (b,x)) , and the two pushouts G x 1 r x x ------* 0b(O x x 2 (2) ( ) x PX U . PX) * Gxl (r (G x 1) and G x 1 r x x------^Ob(T) X x 2 2 (r x px) ( ) > u , ((r x px)(v O) = F(r x px) •kk 9 : uaxl x Px> F(r x Px) such that 9 ° (a x l) = (a x 1) ° oo . (We distinguish the second of the two * morphisms which should be called ( We thus have an isomorphism T = ^ ° $ : F(F) x Px F(F x Px) . To check that T ° (ip x 1) = ipxpx , let (a,x) e T x Px . Then T ° (ip x l)(a,x) = £2 ° $(a*(uB'(a),a),x) = 9 ° $((a* x 1)((u3’(a),a),x)) = fi((a x l)*((u3'(a),a),x)) = (a x i) o co((u3’ (a) ,a) ,x) = (a x i) ((u3?(a),x),(a,x)) = (a x l) (u3’ (a,x) , (a,x)) = ipxpx (a,x) , and the proof is complete. Corollary 4.2.6. I fa G tke fah.ee product ofa a countable. faamiZy ofa k^-gh.oupoldA {ga} , and x a k -4pace, #iew G * Px ti the fah.ee product ofa the gh.oupolcU> {G-^ x px} mXk aetpeet to the obvlouA moh.phtj>m. The proof is trivial. In the rest of this chapter we shall frequently have occasion to deal with groups of form U(G x Px) , where G is a k^-group (regarded as a groupoid on one object) and X is a k^-space. We will use the convenient expression U(G,X) to denote U(G x Px) . If : G H is a morphism of topological groups and f : X “► Y a continuous map then ((J),f) : (G,X) (H,Y) induces an obvious morphism U((J),f) : U(G,X) U(H,Y) , and with this definition of U we see that 63. U is a functor (of the two variables G and X) . We now record a few of its properties. Note that U(G,X) is algebraically the free product of copies of G , one copy for each x e X ; it is the topological free product if X is discrete (when X is also, of course, countable). Proposition 4.2.7. Fo'i any k^-gtioup G and. k^-4pac.&4 X and Y tkeAe, U> a TG-dAomohphAAm $ : U(U(G,X),Y) -► u(G,x x y) commuting mAh the. imiv&uaJt mo^.pkl6m. Proof. Denoting the singleton space by 1 we have a pushout in TG O X ------> 1 G x px ------►U(G,X) a* and then Proposition 4.2.4 shows that the following square is also a pushout: a x l X x Y ------> Y G x P(x x Y) ------» U(G,X) x PY a* x 1 (We replace PX x py by P(X x Y).) If we now form the composite of the preceding square with the pushout 64. T Y------* 1 T ▼ U(G,X) x PY ------►U(U(G,X),Y) T* (with T the unique map from Y to 1) we obtain another pushout T o (a X 1) X X Y------► 1 G x P(x x Y)------►U(U(G,X),Y) t* ° (a* x i) The pushout defining U(G,X x Y) then gives rise to a TG-isomorphism $ : U(U(G,X),Y) -* U(G,X x y) such that * $> ° T* ° (Q* x 1) = (x ° (a x 1)) as required. We shall denote by Z the infinite cyclic group with the discrete topology; for concreteness we regard it as the group {..., -1, 0, 1, ...} of integers under addition. Of course, Z is a k^-group. Proposition 4.2.8. let: x be a topological Apace, and faonm t\le unlveAAal mosipfuAm a* : z x Px u(z,x) , wtteste a map-5 X to a Alngleton Apace,. Then the palJi (f ,u(z,x)) , wheAe f : X -► U(z,x) t5 the map xh a*(l,x) , t5 a Markov fifiee topological gfioup on X . Proof. Let (j> : X H be a continuous map into some topological group H . Since Z is discrete it is clear that the map (p1 : Z x Px -*• H defined by (n,x) (})(x)n is a (continuous) morphism. The pushout defining U(Z,X) therefore gives us a morphism $ : U(Z,X) H such that 65. $ ° O* = (J)’ . Then, for x e X , $> ° fv(x) = $ ° a*(l,x) = (j)'(l,x) = (f)(x), X so that $ ° f = (j) . The uniqueness of the extension $ is easily X verified. This result can be generalized in the following fashion (noting that Z is the free Markov topological group on a singleton space). Proposition 4.2.9. LeX (f„, FM(X)) be. the. MaAkov (\Ae.e. topoloQlcal X gAoup on the. k^-Apace. X , let: Y be. anotheA k^-Apace., and. fioAm the. unlveAAal moAphlbm x* : FM(X) x Py -^u(FM(x),y) , whetie. x map-6 Y to a singleton Apace.. Then the. pcuA (f ,u(fm(x) ,Y)) , wheAe f : X X Y -► U(FM(X) ,Y) lA the. wap (x,y) ^ x*(f (x),y) , lA a MoAkov fafUML topological gAoup on X x y . Proof. By Proposition 4.2.8 we replace the pair (f„, FM(X)) by X (f , U(Z,X)) , where f (x) = G*(l,x) and o* : Z x Px ^ U(Z,X) is X X the universal morphism. But Proposition 4.2.7 shows that there is an isomorphism $ : U(U(Z,X),Y) U(Z,X x Y) such that k $ ° x* ° (a* x i) = (a x x) , while Proposition 4.2.8 shows that (f ,U(Z,X x Y)) is a Markov free topological group on X x Y , where X* i k f v(x,y) = (a x x) (l,(x,y)) . Therefore so is the pair Xx i (f,U(U(Z,X),Y)) , where f = $ ^ ° f , and we have Xx i f(x,y) = $ 1 ° (G x x) (1, (x,y)) = x* ° (G* x 1)((l,x),y) = x*(g*(1,x),y) = x*(fx(x),y) , completing the proof. 66. It is worthwhile noting here that the main result of Gildenhuys and Ribes [1] is an open subgroup theorem for free products in the category of pro-C-groups, and that one of the tools used in its proof is a construction in that category analogous to (although somewhat more general than) our construction of U(G,X) . §3. Equivalence under the functor FM. The preceding work has a couple of unexpected consequences which we note here, although they are not otherwise related to the main concern of this chapter. We will say that two k^-spaces X and Y are FM-equivalent, written X ~ Y , if the groups FM(X) and FM(Y) are topologically m isomorphic. (Of course, ~ is an equivalence relation.) See §3 of Chapter 1 for examples of such spaces. Theorem 4.3.1. Let X^ and Y_^ (^OA. i = 1, 2, ..., n) and X and Y be pajJtt o£ FM-equivalent k -4paceA. Then (1) X, x x„ x ... x X and Y. x y0 x ... x y a/ie FM-equivalent, 12 n 12 n n and (2) xm and Ym an.e FM-e.qiUvaZe.nt on each. positive Znte.geA m . Proof. By Proposition 4.2.9 there are topological isomorphisms FM(XX x x2) = U(FM(X1),X2) = u(fm(y1),x2) = FM(Y1 x x2) , 67. so that x ^2 FM Y1 x X2 * an<^ we t^ien have equivalences X1 X X2 EM Y1 X X2 FM X2 X Y1 FM Y2 X Y1 ~ Y x y FM 1 2 The extension to the product of n factors follows by a simple induction, proving (1), and (2) is then immediate. We have already remarked (Chapter 1) that it is unknown whether FM-equivalence implies FG-equivalence (defined in the obvious way) although the reverse implication is true. In particular, we do not know whether there is an FG-analogue of the above theorem. 68. PART 2 : The Topological Kurosh Theorem Having disposed of the preliminaries, we proceed in Part 2 with our work on the Kurosh theorem itself. The result is stated in §4 and proved in §5, while in § §6 and 7 we list a number of its corollaries. §4. The statement of the Kurosh theorem. Throughout §4 we shall consider a free product G II G, of a 'A AeA collection {G^}^ ^ topological groups, and a subgroup H of G . Restrictions will be placed on G and H shortly but for the moment there are none. We begin by setting down the notation and definitions needed for the formulation of the subgroup theorem. For each A e A a double coset HgG-^ (for any g e G ) will be called a A-double coset ; the collection of all distinct A-double cosets will be denoted by . The natural map from G to G/H sending g e G to the (right) coset Hg will be denoted by p , and the map from G/H to D-^ sending a coset C to the A-double coset CG^ will be denoted by p^ . Further, G/H will always carry the quotient topology relative to p , while D-^ will carry that relative t° pA . We find it convenient to introduce a function 0) : G\{l} A defined as follows: if g e G\{1} has reduced form g = g^ ... g^ , then oo(g) is the (unique) A e A for which g^ e G^ . In fact, we formally extend the definition of U) to the whole of G by setting U)(l) equal to any object not in A (such as A , for example). 69. We wish now to define certain transversals for H in G . As in §2 of Chapter 3, however, it suits our purposes to regard these as sections of the projection p , rather than just as sets of coset representatives. Definition 4.4.1 (cf. Weir [1], MacLane [2] and Kurosh [2]). A A-section (for H in G ) iA a Auction : g/h -* G ofa the projection p (ao that p ° tA the identity) AatiA fiying (1) ra(h) = 1 , and (2) ifi c e G/H and g e G^ then R^CCg) e ra(c)Ga . A collection o{± X-AectionA {r^}^ ^ ^ called a Schreier system (of A-sections) ifc it AatiA^ieA the condition: (3) 1^ o)Ra(C) = A and RA(C) = sg , with g e G^ , s * 1 and 0)(s) * A , then (a) RA(Hs) = s and (b) R^s) (Hs) = s . The reader should note the similarity between condition (3) (and particularly (3)(b), which inter-relates the different A-sections) and the definition of a Schreier transversal for a subgroup of a free group (§2 of Chapter 3). Roughly speaking, (3) says that initial segments of represent atives are again representatives, for appropriate indices. Given a Schreier system (R, } as above we have Proposition 4.4.2. (1) In each X-double coAet D e , there iA precisely one coAet sa(d) Auch that o)Rasa(d) * A ; thuA SA : da G/H detfineA a Aection ofi the projection pA : G/H . (2) Define or each A a function r^ : G/H G by the formula 70. r^(C) = R^S-^p^(C) 1 R-^(C) , C e G/H . Then ion. any C , r^(C) e , and io>l D e , r^S^(D) = 1 . Proof. Suppose that C is a coset in D with wR-^(C) = A , so that R-^(C) = sg , where g £ G^ and to(s) * A . Since C = Hsg and Hs C Dg ^ = D , it follows from 4.4.1 (1) or (3)(a) (according to whether or not s is equal to 1 ) that R^(Hs) = s , and so we may set S^(D) = Hs . If there are two cosets C^, D satisfying U)R^(C^) * A * 03R^(C2) , there is clearly a g e G^ such that = C2g • Then by 4.4.1 (2) R^(C^) = = R^(C2) gT for some g’ € G^ . But since neither R^(C^) nor R^(C2) ends with a letter from G-^ we must have g' = 1 , and so R^(C^) = R^(C2) and = C2 • Thus the function S^ : -+ G/H is well-defined, and since S-^(D) ^ D we have p^ 0 S^ = 1^ . This proves (1) . A The proof so far shows that whatever coset C we begin with, either ooR^(C) * A , in which case C = S-^p-^(C) and r-^(C) = 1 e G-^ , or else ooR-^(C) = A and R-^(C) = SS with g e G^\{l} and U)(s) * A . In this case, whether or not s = 1 , we have R-^(Hs) = s and Hs = S-^p-^(C) , so that r-^(C) = s ^ sg = g e G^ as required. If D e we can write D = p^(C) for some C e G/H , and then rASApA(C) = RASApASApA(C) RASApA(C) = RASApA(C)_1 RASApA(C) = 1 . Definition 4.4.3. The. system {R^} is a continuous Schreier system of A-sections li each function R^ is continuous on G/H AND li each oi the associated functions is continuous on . 71. From now on we shall include the functions {S^} in the definition: our Schreier system will thus be the pair ({R^} » {S^})x- . We are now in a position to state our version of the Kurosh theorem. Theorem 4.4.4. Let G be the fan.ee product II G, ofa a countable XeA collection {g^}^ ^ ofa k^-gn.oupt and let H be a doted tubgaoup ofa G . Suppote that (a) then.e It a contlnuout Schnelen tyttm ({r^} , {s^}) X-tectlont faof1 H In G , and (b) faon. each. XeA the double cotet tpace can be mitten at a disjoint union D, = I—I D, with the faollowlng paopenty: Von. each A yeMx Ay y e Mx then.e it a [doted) tubgfioup g^ ofa G^ tuck that faon. any -1 X-double cotet D e D, toe have s Hs n G faon. s = R-^S-^(D) Ay X TXy Then H it the fan.ee topological product H n* n* h. * fg(x) XeA yeMx Ay whe/ie {h, } and FG(x) one at faoUtom : Ay (1) it the tubgaoup ofa H genenated by the conjugatet {RXSX(D) G^R^S^D)-1 : D e } ; and the TG-mon.phitm -1 °Xy = % X P% -HXy d^lmd b* V8’D) = Va(D)SRASa' unlventd, to that = u(Gx^ , ) . (2) fg(x) it the Gnaev fan.ee topological gn.oup on the tet ofa dementt -1 X {R,(C)R, (C) : C e G/H , X e A} , A A0 whesie Xq e A it falxed but an.bltnan.y. Vunthenmoae, Ifa u)e n.egan.d A at a ditcAete topological tpace, the map fan.om G/H x A onto x tending (C,X) to R,(C)R, (C)"1 it an Identlfalcatlon map. X XQ 72. §5. The proof of the Kurosh theorem. (1) Following Brown and Hardy [1] we define a topological groupoid G with object space G/H , arrow space G x G/H , 9’(g,C) = C , 3(g,C) = Cg , and with composition (g1,C)(g2,Cg^ = (gjg^C) . It is easy to see that the vertex group of G at any vertex is topologically isomorphic to H . Our aim is to find a decomposition of G as a free product of appropriate subgroupoids and then to obtain from this a decomposition of its vertex group, and hence of H . The first step in reaching the desired decomposition of G is taken by noting that the projection q : G G mapping (g,C) to g is a covering morphism in TG . We ~ k ~ then invoke Proposition 4.2.3 to show that G = IT G^ , where G^ = q ^ (G-^) is the wide subgroupoid of G with arrows G-^ x G/H . (2) We wish now to obtain a suitable free decomposition of each G^ To this end we fix for the time being a A e A . It is not difficult to see that if D e then the full subgroupoid of G^ over the objects p^ (D) is an algebraic component of G^ , and it is clear that all the algebraic components of G-^ are given in this way. Since D = I—I we also have Ob(G^) = G/H = I—I , where yema y y y C = p^ (D^y) . If we set G-^y equal to the (full) subgroupoid of G^ over C, we therefore see that G^ = I—I G^ (the disjoint union y y y topologically as well as algebraically). (By the remarks of the first sentence, G, is the union of precisely those algebraic components of Ay -1 G^ with object spaces p^ (D) , for D e .) In particular, 73. = II G^ , the free product with respect to the inclusion maps. (3) For a fixed A and y arbitrarily select some A-double coset DG = HSq G-^ e (with Sq = R^S^(D^)) and consider the full subgroupoid G^ of G-^ on the objects = p^ (D^) . Then G^ is globally trivial: the map y : G^ defined by Ch (r^(C) , S^p-^(C)) , where r-^(C) = RX^ARA^G^ ^ R^(C) (as RroPos:*-t::‘-on 4.4.2), is continuous and its image generates a wide tree subgroupoid T^ in G^ . If we let be the subgraph of G^ consisting of y(C^ ) together with the identity at each object in , then T^ = F(T^ ) . Hence, if is the vertex group of G^ at Hs^ = S^(D^) , we have G? = VG * T? = V° * F(r^ ) . Ay Ay Ay Ay Ay Note that V, = {(g,HsQ) : g £ G^ , HsQg = HsQ} -1 {(g,Hs0) : g e sQ HsQ H G^} {(g,HsQ) : g £ G-^ } , which is isomorphic to G^ . (4) Our next task is to establish a TG-isomorphism $ between G, x PD, and G-, ; once this is done, Corollaries 4.2.5 and 4.2.6 Ay Ay Ay * ’ enable us to transfer the free decomposition of G? given above to a Ay decomposition of G Ay By the corollaries just mentioned we know that G^y x Pd^ = (V°y x P%) * F(TG^ x ^D^y) » and it is therefore sufficient to specify our isomorphism $ : x PD-^ G-^ on the subgroupoid VAy X ^Ay and the subSraPh x PDX)J . To define $ on x PD-^ , recall that VAy = ^§»Hso> : 8 e GX\? ; then we set $((S»Hsq)»d) = (g»S^(D)) , and $ is clearly a (continuous) morphism. On the other hand, = y(C^ ) 74. (together with the appropriate identities), with y(C) = (r^(C) , S^p^(C)) and here we define $((g,HsQ),D) = (r-^p(R^S^(D)g) , S-^(D)) . Next we write down the inverse $ ^ : G, G? x PD, . If Ay Ay Ay (g,C) 6 G^ x CA (the arrow space of G^ ) we set $_1(g,C) = ((g1,Hs0)'1(g2>Hs0)(g3,Hs0) , pA(C)) , where = r^(HsQr^(C)) , g2 = r^(C)gr^(Cg) 1 , and g3 = r^(Hs0r^(Cg)) . It is clear that $ ^ is continuous. We note the following facts. Lemma 4.5.1. The. oJUiom (g1 ,HsQ) and (g3,HsQ) axe. In , white. (g2,Hs0) e V°u . Proof. If we set = HsQr-^(C) we see that ?A(C].) = HsOrA^C^GA = Hs0GA = D° 9 since r^(c) € GA by ProPosit:Lon 4.4.2. Hence Y(CX) — r^(C^) , SxpA(C1)) = (gl,Hs0) <■ r°y . Similarly, (g^jMs^) = y(C2) » with = HsQr^Cg) , and is therefore r0 in I Ay -1 It remains to be shown that g^ £ G-^ = Sq HSq H But g2 = rx(C)grx(Cg) 1 = RASApA(C)_1 RA(C)gRA(Cg)_1 RASApA(Cg) RASApA(C) 1 RA(C)gRA(Cg) 1 RASApA(C) -1 by definition of rA . Since clearly RA(C)gRA(Cg) £ H condition 4.4.4 (b) shows that g2 £ GA as required. The following lemma is essentially a restatement of the condition 4.4.4 (b). 75. Lemma 4.5.2. 1^ D , D? e Dx and g , g' e , £h p[RxSx(D)g] = p[RASA(D)g’] and only £{> P[RASx(Df)g] = p[RASA(Df)g’] . Proof. The first equality clearly implies that g'g 1 e Rasa(d) 1 H Rxsx(D) n ga -1 But the right hand side is equal to RXSX(DT) H R^^D') H Gx by condition 4.4.4 (b), and this yields the second equality. The other implication is proved similarly. ,-l Lemma 4.5.3. $ ° $ ^C6 the. Identity on G Ay Proof. Let (g,C) e G-^ , so that g e Gx , C e C-^ . Then with g^ = r^HsQr^C)) g2 = rA(C)grA(Cg)"1 and g3 = rx(Hs0rx(Cg)) we have = ((g1#Hs0) , pA(C))~1((g2,Hs0) , pA(C))((g3,HsQ) PX(C)) By Lemma 4.5.1, (g^Hs^) e , and so Now p[RASApA(Hs0rA(C)) rx(Hs0rx(C))] = pRA(HsQrA(C)) (by definition of rA ) ■ Hsorx(c) > and RXSAPA(HsOrA(C)) = RXSXPX(Hs0) = S0 ’ so that Hsorx(Hsorx(c)) = Hs0rX(C) ' 76. Then Lemma 4.5.2 implies that p[R^S^p-^(C)r^(Hs0r^(C))] = p[R^S^p^(C)r^(C) ] = pRx(C) = C . Therefore $( (g^ ,HSq) , p^CC)) = (r-^ (C) , S^p^(C)) . An exactly analogous argument shows that $((g3,HsQ) , pA(C)) = (r^(Cg) , S^p^CO) , and so we see that (since S^p^(C)r^(C) = C ) $ ° $_1(g,C) = (rA(C)_1 r-^(C)gr^(Cg) 1 r^(Cg) , C) = (g,C) , which proves the lemma. ,-l Lemma 4.5.4. u* thu yidnivtity on Proof. Since x PD^ is the free product (V^ X ^DXy) * F^Xy X ^DXy^ We neec* only check the action of $ 1 o $ on V-^ x pD^ and on F-^ x PD^ . It is easy to see that $ o $ is the identity on x Pd^ and so we confine our attention to its action on x PD^ . Supposing that (g,HSg) e and D e D-^ , we find that $ 1 ° $((g,Hs0),D) = ((gxjHSq) 1(g2,HsQ)(g3,HsQ) , P^(D)) , (note that p-^S^(D) = D) , where 81 = rX(HsOrASA(D)) ’ g2 = rASA(D)rxp(RASx(D)g)rA[Sx(D)rAp(RASA(D)g)]'1 and g3 = r^(HsQr^[S^(D)r^p(R^S^(D)g)]) . It is clear from Proposition 4.4.2 (2) that g^ = 1 and also that the first term, r^S^(D) , in the expression for g^ is 1 . 77. Now p[RAS^pAp(R^S^(D)g)r^p(R^S^(D)g)] = pR-,p(R-^S-^(D)g) (by definition of ) = p(RxSx(D)g) = pCR^S^p-^p (R^S-^(D) g) g] , and so Lemma 4.5.2 shows that SA(D)rAp(RxSx(D)g) = p[RASA(D)rAp(RASA(D)g)] = p[RASA(D)g] . Therefore = 1 . By another argument of the same kind we find that g^ = r^Hs^g) . But since (g,!^) e , g is of the form rA(C) (for some C e and we have g^ = g . Thus $ 1 ° $((g,HsQ),D) = ((g,HsQ),D) , as required. (5) We now set 4>(V°y x PDXp) = VXy and 4»(r°y x PDXy) = rXy : the above work then shows that the subgroupoid generated by rA is F(Ta^) , and that G-^ = V-^ * F(TA^) . Since we have seen already ~ ■k ~ that G-vA = IIy G,Ay we therefore see that 5A ' n*(VXy * F(V> U " n*VAy * n*F(rAy> y y “ "\i * F(FA> ’ y where T,A = U T,Ay , and hence that y ~ k ~ g = n g. = n*(n*v * F(rx)) A y = n*n*v, * n*F(r,) Ay, AyK A, X' = nVv, * F(D , A y Ay 78. where T = U r, . Furthermore, it is easy to verify the following result. X A Lemma 4.5.5. Tfie gSiapk T C.OVU>AJ>&> 0^ tkz 2*dg2A {(rA(C) , Sxpx(C)) : C £ G/H , A e A} , togeXk&i uuXk the. IdzwUXy aX zack object; tko. gsioupodd vx fau objucXA {s^(D) : D e Dx } and aJiAom {(g,S^(D» : S e , D e D^} . (6) For A e A , let A^ be the graph {(RX(C),H) : C e G/H> , and set A = U A, . Note that, expressed in this fashion, A contains A A repeated elements: if wR^CC) = A and R^CC) = SS with g £ G^ , 0)(s) * A and s * 1 , then R^CHs) = R^g^ (Hs) = s (by Definition A.4.1 (3)). It is easily checked that the expression A = {(RX(C),H) : C £ G/H , A £ A , rx(C) * 1} gives the elements of A without repetition. We also note that A C F(T) . For if R^CC) is written as a reduced word g^ g^ ... gn in G then (RX(C),H) = (g1,H)(g2,Hg1) .... (gn,Hg1g2...gn_1) , and the Schreier condition 4.4.1 (3) (b) shows that if g^ £ G-^ , then i (g1,Hg1g2...gi_1) = (rx (Hg1g2...gi) , sxPx(Hglg2...g±)) , i which is in T for each i , and so A C F(T) . Furthermore, A generates F(T) , since if (rx(C) , S^p^CC)) e T then (r-^ (C) , S^p^CC)) = (R^S^p^ (C) ,H) *(RX(C),H) , and both of the bracketed terms on the right are in A . Since the map from G/H to A^ mapping C to (RX(C),H) is 79. continuous we also see that A^ generates a wide tree subgroupoid F(A-^) which is free on A^ . We now have Lemma 4.5.6. Given a topo logical. gsioupoid A and. TV-mosipPuAmA ip^ : A^ -> A with the psiopeAty that \p = U\p^:UA^ = A->A u> a well- defined gsiaph mosiphiim (aZge.bsuitcalt.ij at ZeaAt), then theAc iA a unique TG-mosiphiAm T : F(T) -► A sioAtsvictlng to ip^ on A^ . In pasiticuZaA, F(H iA equal, to F(A) . We are really saying that, in an appropriately defined sense, F(T) (or F(A) ) is the free product of the groupoids (F(A^)} with certain subgroupoids amalgamated (cf. Ordman [6]). Proof. Our proof above that A generates F(T) shows in fact that rA C F(Aa) for each X , and hence that F(rA) S. F(A^) . Consequently, the extension of \p^ to a TG-morphism from F(A^) to A induces a TG-morphism (j)^ : FCT^) -► A for each X . Since {(f^} clearly agree on objects and F(T) = II F(T-^) , we therefore obtain a TG-morphism ¥ : F(T) “► A extending each (j)^ . Using the fact that \p = U\p^ is well-defined, it is also easy to verify that T extends each ip^ . The uniqueness of ¥ is clear. It follows immediately that F(T) is F(A) . (7) We now have the decomposition G = II II V-v * F(A) of G which X y Ay we desire , and have only to obtain from this a decomposition of its vertex group, and hence of H . To do this we require first the following 80. proposition. Its contents are not essentially new, but are not stated elsewhere in exactly the form we need. Related work is given in Brown and Hardy [1, Proposition 7], Brown [1, 8.2.3] and Higgins [1, Theorem 6]. Proposition 4.5.7. Suppose, that the k^-gsioupotd A t6 the {^nee product [oi tt6 AubgfiOLipotdA) A = F(T) * B whe/ie Ob(B) aj> eio&ed tn A , and that th2Ae t6 a u)tde tsiee i,abgfioapotd T = F(A) ofj A eonjotned by T and defitntng a net/iactton p : A -► V1 , u)hene V’ t6 the, vertex gtioup ofi A at -6ome vertex v . Then (i) the Aubgsioupotd ofi A generated by T and B t6 T * B and aj> atoned tn A ; (ii) t£ v t6 the vertex gnoap ofa T * B at v then T * B = T * V and A = F(H * V ; (iii) p : B -► V t6 a imtvetuaZ mon.pht6m tn TG , -60 that V = U(B) (iv) V’ = FG(p(r)) * V ; and (v) pr : r -► p(r> , the fie6t/itctton ofi p to r , t6 an tdenttfitcatton map. Proof. The tree T = F(A) is the kernel of p : A Vf and is therefore closed in A . The remarks on p.62 of Hardy [1] then show that the subgroupoid of A generated by T and B is T * B , and is closed in A , proving (i). The fact that T * B = T * V follows immediately from the fact that T is a tree groupoid. Further, if F^ denotes the vertex group of F(T) at v , then F(T) = F^ * T , and so a = f0*t*b = fq*t*v = F(P) * V , proving (ii). 81. By 6.7.3 of Brown [1] the square T------» {v} T * B------> V P is abstractly a pushout, and it follows easily that it is a pushout in TG because p is a retraction. Furthermore, the square Ob (B)------* T B------* T * B j is also a pushout, with j the inclusion, since Ob(B) = Ob(B) H Ob(T) . Hence the composite square Ob (B)------>{v] B------* V P ° j is also a pushout in TG , so that p ° j : B V is universal, and V = U(B) , proving (iii). Proposition 7 of Brown and Hardy [1] shows that F^ (the vertex group of F(T) at v ) is FG(p(T)) . Then, since A = (Fq * V) * T = V* * T (note that Fq and V are subgroups of V' ) , we have V’ = F^ * V = FG(p(T)) * V , proving (iv). To prove (v) note firstly that p : F(F) -► FG(p(T)) is an identification map, since it is the restriction of the retraction p : A V' , and is therefore a retraction itself (considered now as a 82. map of topological spaces). Now let Q be the set p(T) with the quotient topology under py : T p(T) , and form FG(Q) . (Note that Q is a (Hausdorff) k^-space.) Then in the diagram i P r------► F(r)----- * FG(p(r>) i i i i i i FG(Q) the TTA-morphism py induces a TG-morphism (f) : F(T) FG(Q) such that (j) ° i = pp ; and clearly 1 ° p = (j) , so since p is an identification 1 : FG(p(T)) FG(Q) is continuous. That is, p(T) = Q , and Pp is an identification. Returning now to the decomposition G = II II V^y * F(A) , it is trivial that A conjoins the tree F(A^) for any A , and we have a retraction p^ : G -► V' , where V1 is the vertex group of G at the object H . Furthermore the proposition shows that for each A and y VXy * F(A) = UXy * F(A) , where UXy is the vertex group of VAy * F(AX) , and that p^ : V^y UXy is universal, so that UXy = U(VXy) . In fact we have * * Lemma 4.5.8. G = II II U, * F(A) . A y Ay Proof. Suppose that we are given morphisms (f)^ : UXy A and (j) : F(A) A for some topological groupoid A , and that {(J), } and Ay (j) agree on objects. Define maps ^Xy : V^y A by setting -1 ^Xy(v) = ^((R^O'v) ,H)) (|)Xyp-)v(v)(l)((RxO,v) ,H)) for v £ V^y . For convenience we write 6 = (Rx(3’v),H) , so that 83. ij^y(v) = (j)(6) ^ (j)(6) . It is clear that i|^y is well-defined and continuous, and (since 9’v = 9v for each v e is a TG-morphism. Furthermore {iK } and (() agree on objects and therefore Ay give rise to a TG-morphism $ : G = II II V, * F(A) A , restricting Ay A y to on VXy and to (J) on F(A) . And for v e V^y as above, -L $ ° p^(v) = $(<5 v 6 ) (by definition of p^ ) -1 <1>(6)$(v)$(6)' -1 4>(5)^y(v)<(>(5) (j)(5)(j)(5) 1 c^xyPx (v) cj)(6) 4? (6) 1 % 0 pX(v) ’ so that 0 ° p Ay PA Xy Ay Now the inner square in the diagram is a pushout, so since the outer square commutes, there is a unique TG-morphism ¥ : U^y “► A such that ¥ ° p^ ; but Ay PA Ay Ay both d>, and $ have this property, so $ (j), for each A Xy Ay Xy ’Xy and y . It is easy to check that $ is unique, and this proves the lemma. * * Note that II II U, is the free product of the topological groups Xy X y 84. {lLAy } and is a subgroup of V' , the vertex group of G at H . (8) We now select arbitrarily an index e A and write p ■k k Then (iv) of Proposition 4.5.7 shows that V' = II II U, * FG(p(A)) A y Ay Recalling that q : G G is the projection (g,C) g s we set <(V HA and q (p(T)) = X , and we claim that * * q(V’) = H = II II H, * FG(X) is the required decomposition of H . A y Ay We have already noted that V-^ is the groupoid with objects (S^(D) : D e DA } anc* arrows {(g,S^(D)) : g e GA , D e DA } an<^ t^iat the universal morphism pA : UA is defined by -1 Px(g,Sx(D)) = (RxSx(D),H)(g,Sx(D))(RxSx(D),H) = (RxSA(D)gRxSA(D)_1 , H) . We further defined a TG-isomorphism $ : Vx x PD-^ V-^ by $((g,Hs0),D) = (g,Sx(D)) so that PA ° $ : V°y X PDXy - UAy : mapping -1 ((g,Hs„),D) to (RASA(D)gRASA(D) , H) , is also universal. Then the ■k composite q ° pA ° $ is clearly the universal morphism CT-^ required in Theorem 4.4.4. And again if (RA(C),H) is an edge of A then -1 p(Rx(C),H) = (Rx (H),H)(RX(C),H)(RX (C),H) -1 (RX(C),H)(RX (C),H) -1 (ra(c)ra (C) , H) , -1 so that X = {R,(C)R, (C) : C e G/H , A e A} as required. A A0 It now remains only to prove Lemma 4.5.9. The map f : G/H x A -> x (icke/ic A ficgaAdcd cu> a diAcneXc topologloat Apace), defined by f(c,A) = R,(C)R, (c)-1 aj> an A Ao Identification map. 85. Proof. Since by Proposition 4.5.7 (v) p : A p(A) is an identification, it suffices to prove that the map from G/H x A to A mapping (C,A) to (R-^(C) ,H) is an identification. Moreover, the fact that each R^ is a homeomorphism into G gives rise to an obvious homeomorphism between G/H x A and I—I A-, , so we need only consider the map A A i : I—I A-^ A , equal to the inclusion i-^ : A-^ A on each A-^ . A But if A^ denotes A with the quotient topology under i , then A^ is a (Hausdorff) k^-graph, and the TTA-morphisms i-^ : A-^ F(A^) extend, by Lemma 4.5.6, to a TG-morphism I : F(A) F(A^) . In fact I is a TG-isomorphism, and A = A^ , which proves the lemma. The proof of Theorem 4.4.4 is now complete. §6. Some consequences of the Kurosh theorem. We now deduce various corollaries of our version of the Kurosh theorem, some previously known and others unknown. The first such result is the open subgroup theorem of Brown and Hardy [1], Theorem 4.6.1. Let G be the iK.ee product II G^ oi a countable. A collection {G,} oi k -gKJOupA, and let H be an open AubgK.oup oi A tO Then H Ia the iK.ee pK.odu.ct H n*n*H, * fg(x) A y Ay -1 wheAe (1) H^ = H n G^s^ and ioK. each A e A , {s^ } Ia a suitable collection oi X-double coAet K.epK.eAentatlveA, and (2) X Ia a countable dlAcAete Apace. 86. Proof. Since H is open, G/H and are discrete and we write = I—I , where {D^ } are simply the points of . A y y y Schreier system ({R^},{S^}) of A-sections for H in G always exists (see Weir [1] or MacLane [2]), and in the present case is trivially continuous. Condition 4.4.4 (b) is also clear. Setting s^ = R^S^(D-^) we have the desired decomposition of H . The space G/H x A is obviously countable and discrete, and so is its quotient X . When the subgroup H in Theorem 4.4.4 is normal the result takes on the following particularly simple form. Theorem 4.6.2. Let G be t\le {\Aee product II G^ o{\ a countable, X A collection {G-^} ofi k^-gAoups, and let H be a doted nomal subgroup ofi G . Suppose that thetie It a continuous Sch/ieieA system ({Rx},{sx}) ofa X-sections ^oa H in G . Then H is the faAee product H = II*H, * FG(X) , X A wheAe (1) H^ is the subgroup ofi H generated by the conjugates {raSa(D)(H n ga)RaSa(D)_1 : D € Da> , and the TG-moAphitm o* : (H n Gx) X Pda ^ Ha defined by o*(g,D) = RASA(D)gRASA(D)_1 is universal, so that = U(H n , D^) ; and (2) x is exactly as in Theorem 4.4.4. Proof. Here the groups s ^Hs H G^ (s = R^S^CD)) are all equal to H H , and for each X the set {y} = is singleton, so the result follows immediately from Theorem 4.4.4. It is worth noting that when H is normal, G/H (the space of right cosets of H in G ) is a topological group, and that may 87. be looked upon as the space of td^t cosets of the subgroup p(G^) in G/H . Suppose that for i = 1, 2, n is a k^-group and that : G^ G^/A^ is the natural projection, for some closed normal subgroup A_^ of G^ . Let K be the kernel of the homomorphism q : G = G1 * G2 * . . . * Gn (G^A^ x (G2/A2) x ... x (Gn/A ) induced by q1# q2, ...» qR • Then the preceding paragraph shows that we may identify the product (G1/A1) x ... x (G^/A^) x (G1+1/A±+1) x ... x (Gn/An) with the space of i-double cosets. If for each i there is a continuous section s. : G./A. G. of l ii l q^ , then a continuous Schreier system of i-sections for K in G may be defined as follows: R. : G/K = (G./A.) x ... x (G /A ) ^ G l 11 n n is the map (C1,...,^) H- s1(C1)...s1_1(C1'1)s. + 1(C1+1)...sn(Cn)s1(C:L) > where denotes generically an element of G. /A_. , while G/K is the obvious inclusion. The composite s^ = ° is thus given by (cl-.-.C1 1,C1+1,...,Cn) H-s1(C1)...s1_1(Cx 1)si+1(C1+1)...sn(Cn) Now Theorem 4.6.2 implies Corollary 4.6.3. In the. AdXuatton juAt deACAsibdd, K thd ^Kdd product K = K * K * . . . * K * FG(X) , 12 n 88. u)kene (1) {on each. i , Ki is genenated by the conjugates {s1(D1)Ais1(D1)"1} , whzne. ... ,cn) fuxrUs through the eJt&m&ntA of •Jj Di ; and the monphism o : A± x Pb± -> K± de{tned by (ajD1) ^ si(D1)asi(Di)_1 ^6 untvensal, so that K± = u(A jD^ ; and (2) x the set { s1(C1)...s1_1(C1'1)si+1(C1+1)...sn(Cn)s.(C1)sn(Cn)'1.. .s^C1)”1 : (C1,...,Cn) € G/K , i = 1,2.... n } and is a quotient Apace o{ G/K x {i,2,...,n} unden the obvtous map. 4.6.4. Remark We could instead have defined our sections {R^} by the formula Ri(C1,...,Cn) = si+1(C1+1)...sn(Cn)s1(C1)...si(C1) . In this case the corollary takes on a slightly different form, but we refrain from writing down the details. In §3 of Chapter 1 we defined the Cartesian subgroup of G * H to be the kernel of the natural homomorphism from G * H to G x H . Similarly, the kernel of the natural map from G^ * G£ * ... * G^ to G, x go x ... x G is the Cartesian subgroup of G. * G_ * .. . * G 12 n &r12 n If in Corollary 4.6.3 we put = {1} for each i we obtain the following result, which is the main theorem of Hardy and Morris [2]. Corollary 4.6.5. Let G^ G0, ..., g be k -gnoups. Then the Contes tan subgnoup K o{ g=g*g0*...*g is the Gnaev {nee l 2 n u topologicat gnoup FG(X) , whene x is the set -1 : (gl,...,gn) « Gx x ... x Gn . { ••8i-18i+1-**8n8i8n ‘ ■h i = 1,...,n } , 89. and thd obviouA map fi/iom x ... x Gn x {l,...,n} onto X iA an iddntihication. In the case of a free product of two factors we have the following result of Morris, Ordman and Thompson [1]. Corollary 4.6.6. lh G and H aAd k^-gAou.pA, thdn thd CaAteAian MibgAoup oh G * H iA FG(x) , u)hdAd X = {hgh"1g"1 : g e G , h c H} and X iA homdomoAphic to G A H . If in Corollary 4.6.3 we put n = 2, G^ = G, = H, = G and = {l} we obtain Corollary 4.6.7. lh G and H aAd k^-gAou.pA, thdn thd noAmal AubgAoup N(G) oh G * H gundAatdd by G iA u(G,H) , wfieAe we AdgaAd H oa a topological hpaac. The univzAAal TG-moAphiAm a : G x Ph -> N(G) iA thd map a (g,h) = hgh Recall from §3 of Chapter 1 that if X is completely regular and Hausdorff, and if Xq e X , then the subgroup of FM(X) generated by Xq1 X is FG(X) and there are isomorphisms FM(X) = gp(x~X X) * gp({xQ}) S FG(X) * Z . We apply the preceding corollary to this decomposition to show that the noAmal subgroup generated by x^ X is the Graev free topological group on a countable wedge of copies of X . Corollary 4.6.8. Ih X iA a k -4pacd and xQ e X thdn thd noAmal AubgAoup N oh FM(X) gdncAatdd by x”1 X iA FG(Y) , whdAd Y iA 90. the AubApace {xq 1 x Xq1 :-oo and Xi dj> a copy ofi X ^ok each i , then f : V Xi -► Y i_ i -± ~co< 1 <0° defitned by (x,i) x^ x Xq1 Ia a homeomoKpfuAm. Proof. By Corollary 4.6.7 N is U(gp(xQ^ X),gp({xQ})) , which is isomorphic to U(FG(X),Z) , and there are clearly isomorphisms U(FG(X),Z) = U( U FG(X.)) i£Z 1 = II* FG(X.) ieZ X = FG( V X.) . ieZ 1 The remaining statement is easily checked. We could analyse the normal subgroup generated by x^ in the same way, but we first prove a more general result. It is easy to check (cf. §3 of Chapter 1) that for completely regular Hausdorff spaces X and Y there are isomorphisms FM(X V Y) = gp(Y) * SPCx”1 X) = FM(Y) * FG(X) where X and Y are wedged at Xq . The main theorem of Gildenhuys and Lim [1] is the analogue in the category of pro-C-groups of the following result. Corollary 4.6.9. 1^ X and Y axe k^-ApaczA ooedged at Aome potnt xQ then the nohmal Aubgxoup N generated by Y tn FM(X V Y) tb topotoglcatZy tbomoxphtc to fm(y x fg(x)) (Kegadding FG(x) oa a topologtcat Apace). (f) : FG(X) -► gpCx”1 X) tA the tsomoKphtAm tnduced by x Xq1 x , then the tbomoxphtAm $ : FM(Y x FG(X)) -► N 91. is Induced by the map on Y x fg(x) defined by (y,w) f* Proof. Corollary 4.6.7 shows that N is U(FM(Y),FG(X)) , where the universal morphism is (v,w) wvw ^ . Then proposition 4.2.9 applies, showing that U(FM(Y),FG(X)) is FM(Y x FG(X)) , and that the injection of Y x FG(X) into N is (y,w) (J)(w)y (w) ^ . In the case where Y is a single point we have Corollary 4.6.10. Ifj X is a k^-^pace then the normal subgroup N ojj FM(x) generated by a point xQ e x Ia FM(FG(X)) ; the, embedding otf FG(X) Into N -C6 the map w (j)(w)xQ ^(w)-1 , where Hardy and Morris [2] (see also Morris, Ordman and Thompson [1]) have shown that a free product G = G^ * G^ * ... * G^ is homeomorphic to G^ x x ... x G^ x K , where K is the Cartesian subgroup of G . The following proposition shows this to be an instance of a more general result. Proposition 4.6.11. Let H be a Aubgroup ofi the topological group G , and AuppoAe that there Is a continuous section s : G/H G 0j( the projection p : G G/H . Then G Is homeomorphic to the product Apace g/h x h : the homeomorphlsm 4>(g) = (p(g),gs(p(g)) 1) . Proof. It is clear that (p is continuous, and it is straightforward to check that its inverse is the continuous map (C,h) ^ hs(C), C £ G/H, h £ H 92. The reader should note the similarity between (J> and the isomorphism $ defined between x PD^ and G-^ in §5. 4.6.12. Remarks. (1) Proposition 4.6.11 points out a fairly strong limitation on the applicability of any kind of subgroup theorem expressed in terms of continuous sections (such as Theorem 3.2.4 or 4.4.4). If, for example, the group G in the proposition were connected, then H would be connected as well, so that Theorem 3.2.4 can give no information on disconnected subgroups of FG(X) if X is connected, and Theorem 4.4.4 can tell us nothing about disconnected subgroups of a free product of connected groups, although such subgroups may still be (respectively) free or free products. (2) The proposition also shows that if the group G is Hausdorff then the subgroup H must be closed. This can be seen more directly by noting that H = (s ° p) ^({l}) . In particular, the hypothesis that H be closed in Theorem 4.4.4 (and in Theorem 4.6.2) is not essential. §7. More on FM([0,1]) . Corollary 4.6.10 shows that, for a k^-space X , FM(X) contains the subgroup FM(FG(X)). Since we also know that FG([0,1]) contains a copy of FM([0,1]) as a closed subgroup, we see that FM([0,1]) contains a copy of FM(FM([0,1])) , and we therefore have 93. Theorem 4.7.1. X tA CL k^-ApCLOH, FM([0,1]) COntatnA CL [cJtoAzd] Aubgsioup topologtccMy dAomosipktc to FM(X) tfi and onZy FM([0,1]) contatviA a cJtoAnd AubApcLce, kom&omofipktc to X . In §3 of Chapter 3 we mentioned the result of Hardy, Morris and Thompson [1] and Thomas [1] that FM(X) has closed subspaces homeomorphic to Xn for each n . Putting X = [0,1] we therefore see that FM([0,1]) contains FM([0,l]n) as a subgroup for each n (cf. Theorem 3.3.3) . 4.7.2. Example. We can also use the theorem to answer the question posed at the end of Chapter 3 : Does FM([0,1]) contain the free topological group on an open interval? To show that it does, it suffices to find a copy of (0,1) as a cZoA&d subspace of FM([0,1]) . (Of course, copies of (0,1) which are not closed are easy to come by. If a copy is closed, however, it cannot be contained in Fn([0,l]) for any n , because it would then be compact.) We regard the interval [0,1] as a subspace of FM([0,1]) and choose in it a doubly-infinite sequence ••• c2 < C1 < C0 < a0 < al < a2 ••• ' We then set B = [c^a^] , and (for n > 1 ) A = [a ,,a ] and 00 n n-1 n C = [c ,c .] . We also set x. = (cn)(a1) 1 and y. = (aA)(c1) 1 , n n n-1 I u 1 l u i and inductively define x = x , (a _)(a ) ^ and n n-1 n-2 n Y = y , (c _)(c ) 1 for n > 2 . (Here multiplication and inversion n •'n-l n-2 n 94. outside the brackets ( ) are relative to the group structure of FM([0,1]) . ) Then we claim that the set 00 oo J = ( u x A ) U B U ( U y c v , n n' v . n n n=1 n=1 is closed in FM([0,1]) and is homeomorphic to (0,1) . In fact, for convenience, we establish a homeomorphism between J and the space R of real numbers. Clearly each term in the expression for J is homeomorphic to a closed interval, and we fix homeomorphisms (d) } „ as follows: n neZ (})q : [-1,1] B such that V !) c0 and <(>0(1) 0 * and for n > 1 (J> : C-(n + l),-n] x A such that -n n n + 1)) = xn an_x and d) (-n) = x a , -n n n and d) : [n, n + 1] y C such that n n n +n(n) y n cn and d)(n+l)=y c . . n ;n n-1 Then we wish to show that the bijection (j) : R J built up from the functions i-s a homeomorphism. Since each d) is continuous it is clear that d) is continuous, n To show that (j) ^ is continuous we need to know first that J is closed in FM([0,1]) . But FM([0,1]) is a k^-space with decomposition U F ([0,1]) , and for any k ^ 1 95. J n F2k_x ([0,1]) = j n F2k ([0,1]) k-i k-i =(U x a)ubu(u y c v . n n' v . yn n n=l n=l = Jk , say, 00 is compact, implying that J is closed. Furthermore, U J is a k=l k k^-decomposition of J , and it clearly suffices to prove the continuity of (j) on each Jk . Once again though, this reduces to checking its continuity on each x A and y C , and on B , so there is nothing J n n yn n ° left to prove. Thus we see that FM([0,1]) contains a copy of (0,1) as a closed subspace, and so, by Theorem 4.7.1, it has a (closed) subgroup topologically isomorphic to FM((0,1)) . 96. CHAPTER 5 Universal Constructions and Function Spaces §1. The universal topological group and spaces of morphisms. Suppose that G is a topological groupoid, and let H be a topological group. Then it is easy to see that the pushout in TG defining the universal morphism i : G U(G) induces a bisection i between the set of TG-morphisms from U(G) to H and the set of TG-morphisms from G to H . If we now equip these sets of morphisms with the compact-open topology, then the main result of §1 states that ■k i is in fact a homeomorphism, provided that G is a k^-groupoid. Of course, both free products and free topological groups can be expressed as universal topological groups on appropriate groupoids, so that the result specialises to give information on each of these constructions. We are also able to deduce facts about free Abelian topological groups. The contents of this section will appear in Brown and Nickolas [1]. Before proceeding, we establish some notation. If G and H are arbitrary topological groupoids, then the set of TG-morphisms from G to H will be denoted by M(G,H) . Its elements will usually be regarded as maps from the arrow space of G to that of H , so that M(G,H) can be identified with a subset of C(G,H) , the set of continuous maps between (the arrow spaces of) G and H . In this way 97. we give M(G,H) its topology as a subspace of C(G,H) with the compact-open topology: M(G,H) thus has a subbase of open sets of the form N(K,U) = {f € M(G,H) : f(K) C u} , where K runs through the compact subsets of G and U through the open subsets of H . We now prove a couple of preliminary results, the first of which specifies the location of compact sets in U(G) . The reader is referred to §1 of Chapter 4 for the construction of the universal topological groupoid U^(G) and of the quotient map p : W^(G) U (G) . Proposition 5.1.1. 1^ tlic k^-gtioupold G haA k^-decompoAItlon U Gr and Ifi the fieAt/ilctlon o^ p : W(G) u(G) to (Gn)m denoted by p then any compact AubAet K ofi u(G) HeA tn a finite union ofi the compact ActA p (G )ra , and Ia a filnlXe union K = u Ki compact i=l ActA K. , whoAe each. K. LLet) In a Aultable p (G )m . i i rnm n Proof. It is easy to see that W(G) has k^-decomposition U W^(G) , n ^ where W (G) = I—I (G ) , and therefore that U(G) has k -decomposition n . A n w i=0 U U (G) , where n U (G) = p(W (G)) = u p (G )1 . n n . A ni n i=0 Proposition 3.1.2 (2) then shows that KC U^(G) for some n , from which the result follows. We require also the following lemma, whose proof is straightforward. 98. Lemma 5.1.2. Let A^, A^, ..., A be compact AubAetA and u an open AubAet oa topological. gkoup H , and AuppoAe that ... An C u . Then thene a/ie open AQjt6 IL D k± ^on each i Auch that U. Un ... U C 12 n — u . Theorem 5.1.3. Ton. any k^-gn.oupoid G and any [not neccbAasiiJLy HauAdok^) topological gkoup H the natu/ial map 1* : M(U(G),H) M(G,H) induced by the univenAol mosiptuAm i : G ^ U(G) iA a homeomokphiAm. •k Proof. We have already remarked that i is a bisection, and it is continuous since i is continuous, so we have only the continuity of * -1 (i ) to prove. Let N(K,U) be a subbasic open neighbourhood of some f e M(U(G),H) thus K is a compact subset of U(G) , U is an open subset of H , and f £ N(K,U) = {d g M(U(G),H) : d(K) C u} . By Proposition 5.1.1 we may choose K with no loss of generality to be a subset of p (G )m for some n and m , where U G is the nm n n k -decomposition of G . For convenience write p = p . 0) r *nm r Then setting V = f ^(U) we have P-1(K) c p_1(V n p(G )m) C (G )m , — n — n and C = p ^(K) is compact and W = p ^(V H p(Gn)m) is open in the product (G )m since K is closed and V is open in U(G) . For any m-tuple w = (g^, g^, ...» gm) e W we can therefore find open sets ,w „w Bn, ..., B in the space G such that g. g B. for each i and 12m r n l l w £ B™ x B^ x ... x BWC bY x B! x ... x BW ^ W . 12 m — 12 m — 99. www w m In particular B = x x ... x is open in (G ) while WWW w C = B1 x B0 x ... x B is compact. 12 m r Clearly C C w = U BW = U CW , so since C is compact we weW weW For each j set = C ^ and for each i set C"? = . Then 1 l k p(C) = K c U p(CJ) C p (W) = ¥ n p(G )m C V , j=l and for any j p(Cj) = i(C^) UCp ... i«d) c V . * * Writing f = i (f) = f ° i we therefore have f*(ch f*(d) ... f*(ch = f(p(cj)) c U , since f(V) C u . * -j n Now the sets {f (C^)} are compact subsets of H , so by the lemma there are open sets {U^} in H such that f (c|) C and U~j U;j ... C u . Therefore, setting 12m — km . n = n n n(c~? , it?) , j=i i=i 1 1 we see that f e N and that N is open in M(G,H) . Furthermore if x € K then x e p(C^) for some j , and so if d e N , with d = i (d) for some d e M(U(G),H) , we have d(x) £ d(i(ch i(ch ... i(ch) 12 m = d d ... d (d)12 (d) (d)m c uj ... £ u , so that d e N(K,U) . Thus (i*)_1(N) C N(K,U) and (i*)_1 is continuous, completing the proof. 100. Corollary 5.1.4. IjJ {G^} It) a countable, collection oupA then the natural map 4> : M(II*G, , H) -* n M(G. , H) A A A A lt> a homeomotvphlAm ^ok any topological gfioup H . ■k Proof. As we mentioned in §2 of Chapter 4 II G-* is the universal group A A U(l—I G-,) , and the corollary follows from the theorem once it has been A A checked that M(l—I G, , H) and II M(G, , H) are homeomorphic. A A A A For pointed topological spaces X and Y , C^(X,Y) denotes the space of basepoint-preserving continuous maps from X to Y , with the compact-open topology. The space of maps between unpointed spaces X and Y is denoted by C(X,Y) . Corollary 5.1.5. 1^ g : X FG(x) It) the Inclusion o^ the pointed k^-Apace In Gnaev fatiee topological gtioup then the Induced map g* : M(FG(X),H) -► C*(X,H) lt> a komeomotiphlim fioti any topological gtioup H . Simlla/ily, the map f* : M(FM(X),H) C(X,H) Induced by the Inclusion f : X -► FM(X) lt> a homeomotiphlim. This is proved as was Corollary 5.1.4, by using any of the standard constructions of FG(X) and FM(X) as universal topological groups (see Brown and Hardy [1] and our Proposition 4.2.8, for example). If the group H in Corollary 5.1.5 is Abelian, any morphism from FG(X) to H can clearly be factored through the canonical quotient 101. morphism p : FG(X) AG(X) in such a way that the induced map p : M(AG(X) ,H) ^M(FG(X),H) is a bisection. Moreover, it is easy to see that a quotient map of k^-spaces is compact-covering (that is, any compact set in the codomain is the image of a compact set in the domain), and Proposition 3.5 of Brown [3] then shows that p is a homeomorphism. We therefore have Corollary 5.1.6. 1^ j : X AG(X) Is the Inclusion otf the, pointed k^-space x In Its Graev firee Abelian topological group then the Induced map j* : M(AG(X),H) “► C*(X,H) Is a homeomorphlsm faor any Abelian topological group H . The analogous result tfor am(x) also holds. In fact, both M(AG(X),H) and C^(X,H) are topological groups * under pointwise multiplication and j is a homomorphism and hence an isomorphism of topological groups. Using a similar observation, combined with an argument like that used to prove the previous corollary, we can deduce the following result from Corollary 5.1.4. Corollary 5.1.7. l{\ (G, } Is a countable collection o{\ Abelian k -groups ------A GO then the natural map Ab ip : M( H G, , H) n M(G, , H) X A x A Is a topological Isomorphism faor any Abelian topological group H , Ah where U denotes the coproduct In the category oft Abelian topological groups. 102. The case where H is the group T of complex numbers of modulus one under multiplication is of particular interest, for M(G,T) is then the character group of the group G , so that we have Corollary 5.1.8. l{\ {G^} a countable. collection o{\ Abelian k. -gnoup* A tO then the natuAal map ip : ( nAb g,)a n g; A A A A a topological lAomofiphlbm. This corollary is a special case of a result of Kaplan [1] (see also La Martin [1]). 5.1.9. Counterexample. Corollary 5.1.6 in the case of AM(X) provides a class of counter examples to Theorem 8 of Yang [1], which states that, under mild conditions on X, Y and H , an algebraic isomorphism between C(X,H) and C(Y,H) which preserves constant functions induces a homeomorphism between X and Y . Consider the group AM(I) , where I is the interval [0,1] . It is not difficult to see that AM(I) is also AM(J) , where J is the subspace [0,^] U (I4) ([l*;) ^ [%,1] of AM(I) . It is clear that I and J are compact and that (I,R) and (J,R) are SQ-pairs in the sense of Yang [1], where R is the additive group of real numbers, and Corollary 5.1.6 shows that there is an isomorphism $ between C(I,R) and C(J,R) . Indeed, if we parametrize J as the subspace ([-1,1] * {0}) U ({0} x [0,1]) of R , we can write $ down explicitly as follows: for f e C(I,R) , 103. f $(f)(x,0) = f(%(x +1)) , x € [-1,1] ^ $(f) (0,y) = f(h(y + D) + Hk) - f(h) , y € [0,1] . Clearly $ preserves constant functions but I and J are not homeomorphic, contradicting the theorem of Yang mentioned above. §2. An exponential law for k-categories. The work of this section is aimed towards setting the results of §1 in a more general context, by proving an exponential law for topological categories and groupoids, from which results such as those of §1 would follow. In practice, however, we have to resort to the categories of k-categories and k-groupoids to obtain the exponential law, and the corollaries then derived turn out not to be as strong as those given by the methods of §1, although their scope is much broader. The author wishes to acknowledge that the main ideas of this section are due to R. Brown. In particular, the idea of proving an exponential law and applying it to situations such as those considered in §1 is his. Most of the results of §2 will appear in the joint paper Brown and Nickolas [1]. 5.2.1. Definitions and notation. A k-category is a category equipped with a Hausdorff topology making the structure functions k-continuous (that is, continuous on compact sets), and a morphism of k-categories is a functor which is k-continuous on both objects and arrows. There is clearly a category KC of k-categories and, 104. analogously, a category KG of k-groupoids. There is also a category K of Hausdorff spaces and k-continuous maps; this category has been studied in some depth by Brown [3] and we refer the reader to this paper for a list of its fundamental properties. For k-categories (or k-groupoids) C and D we denote by M(C,D) the set of (k-continuous) morphisms from C to D . (The same notation is of course used in § 1 for sets of morphisms in TG , but there is little risk of confusion, since if C is a k^-space then the two possible meanings of the expression M(C,D) coincide.) Once again we give M(C,D) the compact-open topology as a subspace K(C,D) , the set of maps in K between (the arrow spaces of) C and D . Brown [3] has shown that an exponential law holds in K ; that is, that the exponential map e : K(X x Y,Z) K(X,K(Y,Z)) given by e(f)(x)(y) = f(x,y) is well-defined and is a homeomorphism for each X, Y, Z in K . We now set about formulating and proving such a law in KC and KG , analogous to the exponential law for abstract categories (MacLane [1]). To do this we must first construct a k-category (C,D) having M(C,D) as its space of objects: such a construction is well-known in the case of abstract categories and we find a rather similar although formally different procedure convenient in the topological case. o 4 For a k-category D , D is the subspace of D consisting of quadruples (p,q,r,s) of arrows of D such that pq , rs are defined and equal. Such a quadruple is called a commuting square in D and it 105. is frequently helpful to write it as r q We regard D° as a k-category with object space D , with 9’(p,q,r,s) = r and 9(p,q,r,s) = q , with unit function u(p) = (u9’p,p,p,u9p) , and with "horizontal" composition (p,q,r,s)(u,v,q,w) = (pu,v,r,sw) . Following Ehresmann we write this composition Q] . Thus q CD There is an analogously defined "vertical" category structure on with vertical composition 0 . The following result is proved in Brown and Nickolas [1]. Proposition 5.2.2. ¥o>i k-categories c and D t(zere Is a k-category (c,D) with object space M(C,D) and arrow space m(c,dD) [where kat> the horizontal composition), and with structure maps Induced by the vertical (category structure on dd ; and l{ d Is a k-groupold so Is (c,d) with the Induced Inverse map. Ift, further, d Is a topological category or groupold, so also Is (c,d) . 106. With the definition of (C,D) now behind us we can prove Theorem 5.2.3. (The exponential law for k-categories.) I^ C, D and E ajic k-categories there is a natural KC-isomorphism 0 : (C x d,E) (C,(D,E)) which, it* also continuous. Ijj E is also a topological category then 0 1 as continuous. Proof. For k-categories A and B we may regard the space (A,B) 4 as a subspace of K(A,B) by Proposition 3.6 of Brown [3]. In 4 4 particular we may regard (C,(D,E)) as a subspace of K(C,/C(D,E) ) 16 and of K(C,K(D,E)) , and by the exponential law in K the latter 16 space may be replaced by K(C x D,E) To define 0 note that an arrow f e (C x D,E) is a morphism from C x D to E111 (with the horizontal composition) so that for each (c,d) e C x d we may write f(c,d) r(c,d) q(c,d) s(c,d) a commuting square in E . Note also that since 3’f = p and 3f = s , p and s are morphisms from C x D to E . We require that 0f be 107. L_J a morphism from C to (D,E) , so for any given c e C define k^c) 0f(c) = k^(c) Y Y k2(c) k4(c) a commuting square in (D,E) , where k_(c) e M(D,E ) (i = 1,2,3,4) are defined as follows: for d e D p(u3* c,d) k^(c) (d) = p(c,u3'd) v p(c,u3d) p(u3c,d) p(u3c,d) k2(c)(d) r(u3c,d) w q(u3c,d) s(u3c,d) f(u3c,d) p(u3fc,d) k3(c)(d) r(u3'c,d) w u q(u3'c,d) s(u3'c,d) f(u3'c,d) 108. s(u9'c,d) and k^(c)(d) s(c,u9’d) u v s(c,u3d) s(uBc,d) Certainly each k^(c)(d) is in , since p and s are morphisms and f maps into ED . To show that (say) k^(c) e M(D,ED) we note that the four components of k^(c) (d ^ p(uB'c,d),d ^ p(c,u9d) and so on) are k-continuous since p is , so that k^(c) is itself k-continuous. If d = d^d^ in D then we have u9?d^ = uB’d , uBd^ = uBd and uBd^ = uB'd^ , and so clearly k^(c)(d^) CD k^(c)(d2) = k^(c)(d) as required. The proof that k^(c) e M(D,E°) is similar, and the proofs for k2(c) and k^Cc) > though a little different, are equally straightforward. To see that 0f(c) is a commuting square in (D,E) , let d e D and note that Ck1(c)k2(c)](d) = k1(c)(d) B k2(c)(d) and Ck3(c)k^(c)](d) =k3(c)(d) B k4(c)(d) . Then the first of these expressions is p(u9’c,d) ---- =>---- p(c,uB'd)r(uBc,d) \f V p(c,u9d)q(uBc,d) ■> s(u3c,d) 109. while the second is p(u3* c,d) r (u3T c, d) s (c,u3' d) v V' q (u3' c,d) s (c, u3d) s(u3c,d) and these squares are equal since r(u3c,d) = q(c,u3’d) , r(u3'c,d) = r(c,u3’d) , q(u3c,d) = q(c,u3d) and q(u3’c,d) = r(c,u3d) , and since p(cf,d’)q(c’,d') = r(c',d')s(c',d’) for every c’ e C , df e D . The proof that 0f : C (D,E)D is a (k-continuous) morphism for each f e (C x D,E) is straightforward but tedious, as is the proof that ©(f^f^) = 0(f^)©(f2) for ^^>^2 composable arrows in (C x D,E) , and these proofs are omitted. To check the continuity of 0 , recall that we may regard 1 6 (C,(D,E)) as a subspace of K(C x D,E) and that it therefore suffices to prove the continuity of the sixteen components of © . The first of these, for example, is the map f *"> p(u3’-,-) e K(C x D,E) , where p(u3'-,-)(c,d) = p(u3Tc,d) . But this is simply the composite of the continuous projection f p and the continuous map (u3’ x 1) : K(C x D,E) K(C x D,E) induced by u3! x 1 rCxD^CxD (see Brown [3]), and is thus continuous. Proceeding in this way one sees that all the components of 0 are continuous and hence that 0 is itself continuous. no. To construct 0 ^ , first define functions : ED E by (respectively) (p,q,r,s) ^ q, r, pq . Thus and y^ are the final and initial maps of the category ED with the horizontal composition and y is, in effect, the composition in E . In particular, y^ and y3 are continuous and y is k-continuous. Given g e (C,(D,E)) write g(c) a commuting square in (D,E) for each c e C , and define ygx(c)(d) 0_1g(c,d) = y3g3(c)(d) w t Y2§2 yg4(c)(d) for (c,d) e C x D . It is not difficult to check that 0 ^ is well- defined, and its k-continuity follows easily from the continuity of y3 and y3 and the k-continuity of y . If E is a topological category then y is continuous, and so is 0 ^ . The proof that 0 really is the inverse of 0 is again routine, as is the proof of the naturality of 0 (in the three variables C, D, E ) . 111. The theorem and Proposition 5.2.2 immediately give Corollary 5.2.4. (The exponential law for k-groupoids.) I fa C, D and E aste k-gsioupolds thesie Is a natusual KG-is>omosipkism 0 : (C x D,E) (C,(D,E)) which. is also continuous. 1^ E is also a topological gnoupold then @ 1 is continuous. From here it is possible to use the Yoneda lemma to deduce the following result (Brown and Nickolas [1]). Corollary 5.2.5. Suppose that the k-categosiy c ts a cotimtt lim C^ o^ the k-categosu.es {C-^} . Then ^osi any k-categosiy D the natuAal map (J) : M(C,D) -► lim M(CX , D) is a continuous bljection with k-contlnuous Invests e. The analogous siebult holds In KG . This corollary can now of course be applied to the case of any specific colimit in KC or KG . If G is a k-groupoid and O : Ob(G) X is a K-map to a space X in K , the universal k-groupoid on G induced by O , UK^(G) , is defined by the following pushout (if it exists) in KG : O Ob (G)------» X > r > ' G------;------UKa(G) a 112. In the case where X is a singleton space, UK^(G) is a k-group, the universal k-group on G , and is denoted by UK(G) . The universal morphism is here denoted by i : G UK(G) . Ignoring for a moment the question of the existence of UK(G) we have the following analogue in KG of Theorem 5.1.3. Corollary 5.2.6. The natuxat map i* : M(UK(G) ,H) M(G,H) tndaeed by i a conttnuouA btjectton wtth k-conttnuouA tnveue, ^ofi any k-gxoup H . This corollary can be made to give information about universal constructions in TG , and at the same time we can gain limited information on the question of the existence of universal k-groupoids, by means of the next result. Note that where necessary any object or arrow of TG may be regarded in a natural way as an object or arrow (respectively) of KG Proposition 5.2.7. Let the fioLtoootng puAhout tn TG defitne the untvexiat topologtcat gxoupotd u^CG) tnduced by a (i and j axe the tnctiutonA). - VG) 1£ G and X axe k^-ApaeeA then the Aquaxe ti> alAo a pushout tn KG 113. TkuA U^CG) 'Ci the ulviLvqaaclL k-gfioupok.d UK^(G) on the, k-gnou.pod.d G , k.nduczd by o . Proof. Suppose that H is a k-groupoid and that (f>^ and (j>2 are KG-morphisms as shown below such that (J)^ ° i = (j^ 0 CJ . a Ob (G) U (G) Then there is a unique morphism of abstract groupoids $ : U^(G) H •k such that 0 ° a = (j>^ and $ ° j = $2 > since the square is abstractly a pushout. We wish to show that $ is k-continuous (and therefore continuous in fact since U a (G) is a ka) -space)r . We claim that cj)^ and 2 induce a (continuous) morphism (j) : W^(G) H such that cj) = $ 0 p , where p : W^(G) -► U^(G) is the canonical quotient map; from this of course it follows that $ is continuous. For (g1# g2, ..., gn) € Wa(G) define 4>(gx» g2> •••» gn) = ^1(g1)^1(g2) • • •4)1(gn) > and note that defined since u3(j)1(gi) = u9(g±) = = = 4>2 a9' (g±+1) 114. for each i . It is clear that (J) is a morphism. If U G is the k -decomposition of G and U X E U is the n to n n decomposition of X = , we see that W^(G) has k^-decomposition U W (G) , with n W (G) = W (G) n (G° L_J G1 I—I ... U Gn) n o n n n = I—I (W (G) n g^) , i=0 0 and it suffices to check the continuity of $ on each set W^(G) H G^ . But the restriction (j) : W^(G) H G^ H can be written as the composite W (G) n G1 ------► H X ... X H -- * H O n where H x . . . x H is the set of i-tuples of composable arrows in H and m is the i-fold composition. Thus we see that (f> is k-continuous, and hence continuous, on the (compact) set W (G) H G^ , and is therefore continuous on W (G) . a Finally, noting that $ 0 P(g1# g2» •••» 8n) = ^(P(g1)p(g2)••-P(gn)> = = $a*(gl)$a*(g2)...$a*(gn) = — ^ (g ^ > §2* * * * * g^) » we conclude that $ is continuous, finishing the proof. We note that the square defining U^(G) is also a pushout in KC ; this follows from essentially the above argument and the fact (Hardy [1]) that U (G) is the universal topological category on G induced by O . 115. Now of course Corollary 5.2.6 applies to U(G) as well as UK(G) ; the version of Theorem 5.1.3 thus obtained however is weaker than the ■k theorem itself - the map i is a continuous bisection with k-continuous inverse rather than a homeomorphism. Lastly we use the ideas of this section to re-prove a result of Chapter 4. Proposition 5.2.8 ( = Proposition 4.2.4). Let O : Ob(G) X be continuous, whe/ie G ts a k^-gfioupotd and x ts a k^-Apace, and onm the unto a x 1 Ob (G) x Y ------* X x Y i X 1 j x 1 G x PY * U (G) x PY a * x i a That ts, u (G) x Py ts the untoensaJl topologtcat g/toupotd on G x Py tnduced by a x l . Proof. By the previous proposition the square Ob (G) defining U^(G) is a pushout not only in TG but also in KG , and 116. so Corollary 5.2.5 applies, showing that the commutative square /\ j M(Ua(G),H) ------* M(X,H) M(G,H) M(Ob(G),H) is a pullback in K for any k-groupoid H , where the morphisms are induced by those in the first diagram. Now if H is a topological groupoid and we are given TG-morphisms : G x PY H and (j)^ : X x Y H such that (j) o (i x l) = (f>^ o (a x l) 9 the exponential map e in K (described earlier in §2; see Brown [3]) gives us /(-maps e((f>^) : Y M(G,H) and e((p^) : Y M(X,H) such that i ° e( A and j o e((j)) = ^(^2^ » and as easily checked that the K-map e * e (cj)) = (j) : U^(G) x PY H is in fact a KG-morphism (and hence a TG-morphism) such that (j) ° (o x 1) = §3. Reflexivity and free Abelian topological groups. Several authors have studied the class of Abelian topological groups which are reflexive (that is, which satisfy the Pontrjagin duality theorem) and have shown that it contains many non-locally compact groups (Kaplan [1] and [2], Varopoulos [1], Noble [1], Venkataraman [1] and 117. Brown, Higgins and Morris [1]). Kaplan, for example, has shown that an arbitrary product of reflexive groups is reflexive and that the inverse limit of a sequence of locally compact Abelian groups is reflexive, while Noble has extended the latter result by establishing the reflexivity of any closed subgroup of a countable product of locally compact Abelian groups. Thus, in spite of the results of Chapter 2 showing that free Abelian topological groups are not locally compact unless discrete, it is still reasonable to ask if these groups are ever reflexive. In this section we derive a fairly general necessary condition for reflexivity which yields as a special case the fact that a large class of free Abelian topological groups fail to be reflexive. In Noble [1] it was asked: Is every complete Abelian k-group with sufficiently many characters reflexive? (Noble defines a k-group as a topological group on which every k-continuous homomorphism is continuous - a definition differing from that used elsewhere in this chapter.) Another consequence of our work here is a negative answer to Noble's question. All the results of §3 will appear in Nickolas [3]. We shall use additive notation for all groups except the circle group T , which we regard as the (compact) group of complex numbers of modulus one under multiplication. Recall that for any Abelian topological group G , the dual group of G is the space M(G,T) of continuous characters of G . Theorem 5.3.1. the. kboJjjLYl topologicaZ g/LOLLp G it> cl HCLUbdoH.^ k-4 pace the.n the. pcuth- component oft the. ide.ntiXy tn itA duuZ gtioup G^ iA tke. union oft aJUL one.-pcummeJieJi AubgstoupA ofi G^ . 118. Proof. The union of the one-parameter subgroups of is clearly contained in the path component P of the identity, and so only the reverse inclusion requires proof. Fix a character y e P : we wish to construct a one-parameter subgroup of in which y lies, and we begin by using an argument similar to that of (4.73)of Hofmann [1] to lift y to a real character 3 : G ”*• R . Let f : [0,1] -► G^ , with f(0) = 0 (the trivial character on G) and f(1) = y , be a path joining 0 and y . If we regard G~ as a subspace of the function space C(G,T) , f may be regarded as a map from [0,1] into C(G,T) , and since G is a Hausdorff k-space, XV, 3.1 of Dugundji [1] shows that f induces a homotopy H : G x [0,1] T between the trivial character 0 and y . Then the homotopy lifting property (2.2 of Spanier [1]) applied to the covering projection p : R T defined by p(x) = exp(27Tix) shows that in the following commutative square a homotopy F can be found making the resulting triangles commute. G x G x [0 H For any g^, g^ e G consider the map FT : [0,1] R defined by F'(x) = F(g1,x) + F(g2,x) - F(g1 + g2»x) for x € [0,1] . It is clear that F’ is continuous, that F’(0) = 0 , and that F' is a lifting of the map H* : [0,1] T given by H’(x) = H(g^,x)H(g2,x)H(g^ 4- g2»x) * • But since H is induced by f : [0,1] G^ we see that H'(x) = 1 for all x , and F’ is thus 119. a lifting of the trivial path in T . The trivial path in R is also such a lifting, so since F1(0) = 0 , the unique path lifting property of p (Spanier [1], 2.2) implies that F'(x) = 0 for all x . Hence for x e [0,1] , g1# g2 e G , we have F(gx + g2,x) = F(g1#x) + F(g2,x) . In particular, 3 = F : G “► R is a continuous homomorphism Gx{l} making the following triangle commute: If we now define (J) : R M(G,R) by cf)(x) = x3 , where (x3) (g) = x(3g) for g c G , it is easy to check that (f) is a continuous homomorphism. Composition with the continuous homomorphism p^ : M(G,R) “► M(G,T) = G~ induced by p then gives us ip - p* 0 (f> : R “*■ G~ , and i|j(R) is a one-parameter subgroup of G~ . Since ^(1) = P* ( Since R is a divisible group the following corollary is immediate. Corollary 5.3.2. Wtth. G 06 mi the theonm, tho.: path-component ofi the tdcntity tn G^ ti> cLLvtbtble. The corollaries we now wish to obtain are derived by applying the theorem to the dual G~ of some group G ; to do this however we require first that G^ be a k-space. This is the case, for example, if 120. G is locally compact, for then so is , or if G is hemicompact (Noble [1]), for then G^ is first countable. A k^-space is both hemicompact and a k-space. The theorem and Corollary 5.3.2 now give Corollary 5.3.3. Suppose, that the. AbeXXan topologtcal g/ioup G Ia n.c^le.xXoe,. Ijj g~ a k-Apacc then the. path-component ofi the, Identity tvi G iA the, union the, one-pa/iameXeA. AubgsioupA ofi G [and hence Ia dlvtilblej. The. concluAton kotdA tn paAXlculaA hfi G Ia a k^-Apace. The next corollary shows that the study of reflexivity for many locally path-connected groups may be restricted to those groups which are also path-connected. Corollary 5.3.4. Suppose that G Ia tiefilexAve. Ifj G Ia a localZy path-connected kApace then IX, Ia topologlcalXy lAomofiphXc to p © D , wheAe P Ia ne^lexXoc and tA a path-connected, localZy path-connected kApace, and D Ia countable, and cUACAeXe. Proof. The path-component P of the identity in G is a (locally path-connected) open subgroup, and the work of Venkataraman [1] (see also Corollary 3.4 of Noble [1]) therefore shows that P is reflexive. It is also a k^-space because it is closed in G and thus Corollary 5.3.3 applies, showing that P is divisible. By (6.22)(b) of Hewitt and Ross [1] any open divisible subgroup of G is a topological direct summand of G , whence G = P © D , where D is the discrete group G/P . Since the quotient D is also a k^-space, it is countable. 121. Corollary 5.3.5. LeX x be a k --6pace containing a non-tAtviaZ path. Tkcn AM(X) Zi> not AcfiZcxtvc. Proof. The underlying abstract group of AM(X) is a free Abelian group, which clearly has no non-trivial divisible subgroups. Of course, AM(X) is a k^-space, so if it were reflexive Corollary 5.3.3 would show it to be totally path-disconnected, contradicting our assumption on X . 5.3.6. Remark. Although the information was not needed to prove the last corollary, the work of §1 enables to write down the dual group of AM(X) : by Corollary 5.1.6 there is a topological isomorphism AM(X)" = M(AM(X),T) = C(X,T) . The analogue of Corollary 5.3.5 of course holds for AG(X) , and the dual group of AG(X) is C^(X,T) . 5.3.7. Example. Corollary 5.3.5 also provides us with a negative answer to the question of Noble posed earlier, once we note that AM(X) is complete and has sufficiently many characters to separate points. Completeness is immediate from Proposition 3.1.2 (5). On the other hand, if w = a, x, + a0 xn + ... + a x is a non-trivial element of AM(X) 112 2 n n (with n > 1 , a^, . .., non-zero integers and x^, x^, . .., x^ al distinct elements of X ) , choose z e T for which z * 1 , and 122. extend the map rxiKz , i = 2, 3, . . ., n to a continuous function (j) : X T . By definition of AM(X) , (J) al then extends to a character 0 : AM(X) T , and $(w) = z * 1 (cf. similar arguments in Gelbaum [1] and Morris [6]). 5.3.8. Remark. It is well-known that a character defined on a closed subgroup of a locally compact (Hausdorff) Abelian group can be extended to a character on the whole group. Without the assumption of local compactness, however, the situation is less clear and it therefore seems worth noting that at least some of the groups discussed in Corollary 5.3.5 fail to share this property. 2 Specifically, consider the group AM(I ) , where I is the interval [0,1] . The boundary B = ({0,1} x I) U (I x {0,1}) of 1^ is clearly homeomorphic to T — let i : B T be a homeomorphism — 2 and since it is closed in I it generates a copy of AM(B) as a 2 closed subgroup of AM(I ) (cf. Theorem 3.2.1). Furthermore, the 'k 2 character i : AM(B) T induced by i cannot be extended to AM(I ) , since such an extension would then induce, in effect, a retraction of 2 I onto its boundary, in contradiction of Brouwer’s theorem (Dugundji [1], XVI, 2.1). The proof of Theorem 3.3.8 shows that whenever a k^-space X contains a non-trivial path, AM(X) has a closed subgroup AM(I) , so 123. it would be of some interest to know whether an analogue of Theorem 3.3.3 holds for free Abelian topological groups and, in particular, 2 whether AM(I) contains a copy of AM(I ) . An affirmative answer would enable us to apply the above example to all the groups of Corollary 5.3.5. 124. REFERENCES ABELS, H. [1] Normen auf freien topologischen Gruppen. Math. Z. 129 (1972), 25-42. BING, R.H. 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