From uncountable abelian groups to uncountable nonabelian groups
aa r X i v : . [ m a t h . A T ] M a y FROM UNCOUNTABLE ABELIAN GROUPS TOUNCOUNTABLE NONABELIAN GROUPS
KATSUYA EDA
Abstract.
The present note surveys my research related to gen-eralizing notions of abelian group theory to non-commutative groupcase and applying them particularly to the investigation of funda-mental groups. Introduction
This paper is an account on my studies of topics in mathematicsand, although they are rooted in abelian group theory, they mostlyonly indirectly are related to abelian groups themselves. The emphasisis to show connections between my study of abelian groups to that offundamental groups, which are non-abelian. To state theorems exactlywe need to use technical terms from algebraic topology, for which werefer the reader to [19]. But, I take care so that the reader can under-stand the outline without understanding precise definitions, which isthe main content of this paper.2.
The Specker theorem
My joining into abelian group people started from my attendanceof the Honolulu conference in 1982-1983. Before then, R¨udiger G¨obel,who passed away in 2014, contacted me as one of the organizers of theconference. Laszlo, who by then already was a central person in abeliangroup theory, was present so that I met him there. After then I workedon abelian groups for several years. The reason why I started my studyof abelian groups is my interest to the Specker theorem about Z ω [20],which is a unique theorem about infinitely generated discrete groupssupporting a duality theorem, i.e.Hom(Hom( ⊕ ω Z , Z ) , Z ) ∼ = ⊕ ω Z . Mathematics Subject Classification.
Key words and phrases. one-dimensional, Peano continuum, fundamental group,singular homology, Hawaiian earring, algebraic compactness, cotorsion, shapegroup.
I felt that there should exist good mathematics around this theorem.Since the ˇCech homology group ˇ H ( H ) of the Hawaiian earring H isisomorphic to Z ω , I aimed at analyzing the algebraic structure of theHawaiian earring (see the figure). . . . I tried to find applications of the Specker theorem and also the Chaselemma to algebraic topology. Since the singular homology group of theHawaiian earring is not isomorphic to Z ω , contrasting to the ˇCech ho-mology group, I introduced a canonical factor of the singular homology[13] and investigated such groups for spaces of certain types. This willbe explained in Section 3. But, I felt it is not sufficient as what Ifelt from the Specker theorem and tried to investigate the fundamentalgroup of the Hawaiian earring and to prove a non-commutative versionof the Specker theorem. After having obtained a proof around 1985 Ionly found out that G. Higman’s work [17] from 1952 already containeda demonstration of this fact: Namely, there he first proves that everyhomomorphism from the unrestricted free product, i.e. the canonicalinverse limit of free groups of finite rank, factors through a free group offinite rank with the projection, which can be seen a non-commutativeversion of the Specker theorem. Then, Higman mentions the validityof this result also for a certain subgroup P of the inverse limit. Threeyears later, H. B. Griffiths [15] proved P to be isomorphic to π ( H ). Iintroduced a new notion free σ -product ×× σi ∈ I G i of groups G i to investi-gate fundamental groups of spaces like the Hawaiian earring [6]. A free σ -product ×× σi ∈ I G i is a subgroup of the unrestricted free product in [17]consisting of elements expressed by words defined on countable linearlyordered sets, while an element of a usual free product is expressed bywords defined on finite linearly ordered sets. There, I proved a non-commutative version of the Chase lemma, which I’ve mentioned above,i.e. Theorem 2.1. [6, Theorem 2.1]
Let h : ×× σi ∈ I G i → ∗ j ∈ J H j ROM UNCOUNTABLE ABELIAN GROUPS TO UNCOUNTABLE NONABELIAN GROUPS3 be a homomorphism for groups G i and H j . Then, there exist finitesubsets F of I and G of J respectively such that h ( ×× σi ∈ I \ F G i ) ≤ ∗ j ∈ G H j . The original Chase lemma is about homomorphisms from directproducts to direct sums. In the category of abelian groups there existinjective objects and consequently the statement is more complicated,but in the non-abelian case it becomes simpler. Actually the conclu-sion of Theorem 2.1 can be considerably sharpened as we will see inTheorem 2.5.Applying Higman’s theorem I proved that every endomorphism on π ( H ) is conjugate to an endomorphism induced from a continuous self-map on H . Though it has become a seminal result, I felt that it wasvery far from results about the Sierpinski carpet and Menger spongeat that time, since the Hawaiian earring has only one wild point, whileall the points are wild in the others (see the sequel to Theorem 2.3 forthe word ”wild”).At that time, conjugators about the fundamental groups were trou-blesome for me. But, a few years later I found that conjugators becomethe key for finding points of the above spaces in their fundamentalgroups, fundamental groupoids more exactly. Actually we have: Theorem 2.2. [7, Theorem 1.1]
Let X be a one-dimensional metricspace and h : π ( H , o ) → π ( X, x ) a homomorphism. Then, thereexists a continuous map f : H → X and a point x ∈ X and a path p from x to x such that f ( o ) = x and h = ϕ p ◦ f ∗ , where ϕ p : π ( X, x ) → π ( X, x ) is the point change isomorphism and f ∗ is the homomorphisminduced by f . If the image of h is uncountable, the point x is uniqueand p is unique up to homotopy relative to end points. A conjugator in the fundamental group, which was troublesome, cor-responds to the homotopy type of a loop p in the statement. Pointscan be restored from fundamental groups as the maximal compactiblefamilies of subgroups which are the homomorphic images of π ( H ) [1].Based on this we have, Theorem 2.3. [7, Theorem 1.3]
Let X and Y be one-dimensional, lo-cally path-connected, path-connected metric spaces which are not semi-locally simply connected at any point. If the fundamental groups areisomorphic, then X and Y are homeomorphic. A one-dimensional space space is called semi-locally simply con-nected , if any point has a neighborhood without a circle. We call a space wild , if the space contains a point at which the space is not semi-locally
KATSUYA EDA simply-connected. Theorem 2.3 implies that the fundamental groupsof the Sierpinski carpet and the Menger sponge are not isomorphic toeach other, since the two spaces are not homeomorphic. This result wasquite unexpectable: The homotopy equivalence of spaces implies theisomorphic-ness of fundamental groups but in general the homotopyequivalence is much weaker than the homeomorphism type. Poincar´eintroduced the notion of fundamental groups, as a much rough equiva-lence in comparison with a homeomorphism type. Therefore, though itis a very restricted case, this was unexpected. Although I several timeshad already checked my proof of Theorem 2.3, I still distrusted it andconsequently even tried to manufacture a counter example at least afew times, and I’ve heard that several topologists did not believe The-orem 2.3.Then, I worked on this line, i.e. to investigate relationship betweenproperties of groups and those of spaces. This duality between spacesand groups through the fundamental groups is extended to at mostcountable direct products and other constructions [1] of spaces. Butit was difficult to publish such papers, since no one except me wasworking in this area. Since my retirement year was 2017, I neededto publish my papers as a researcher and worked on other subjects.Among them I proved:
Theorem 2.4. [8, Theorem 1.1]
For one dimensional Peano continua X and Y , X and Y are homotopy equivalent, if and only if π ( X ) and π ( Y ) are isomorphic. Although this result looks like a standard statement in algebraictopology, but the proof of this theorem depends on Theorem 2.3, whichis extraordinary. Consequently this theorem is actually an extraordi-nary theorem.Before then, as I mentioned, Theorem 2.1 was strengthened as fol-lows.
Theorem 2.5. [9, Theorem 1.3]
Under the same assumption of Theo-rem 2.1, there exist a finite subset F of I and an element j ∈ J suchthat h ( ×× σi ∈ I \ F G i ) is contained in a subgroup conjugate to H j . As a variant of this theorem, we have
Theorem 2.6. [9, Theorem 1.4]
Let X be a path-connected, locallypath-connected, first countable space which is not semi-locally simply-connected at any point. If h : π ( X ) → ∗ j ∈ J H j is an injective homo-morphism, then the image of h is contained in a subgroup conjugate tosome H j . ROM UNCOUNTABLE ABELIAN GROUPS TO UNCOUNTABLE NONABELIAN GROUPS5
These are results where I translated known facts from abelian grouptheory to not necessarily abelian goups, and, in particular, to funda-mental groups. 3.
Algebraically compact groups
As is well-known, the singular homology group of a path-connectedspace is the abelianizations of the fundamental group for a path-connectedspace. In algebraic topology it is well-known that all groups appear asfundamental groups and, consequently, all abelian groups appear ashomology groups. However, the corresponding spaces constructed byusing group theoretic data are artificial ones. On the other hand, thefundamental groups of spaces which have local complexities, e.g. frac-tals, are out of studies for a long time. Also, divisible, or algebraicallycompact groups did not occur as homology groups or their subgroupsof spaces, which are familiar as topological spaces, i.e. not artificial orformal objects.Back to around 1990, I found out that the singular homology groupsof certain spaces are complete modulo the Ulm subgroup [5] (a notionwhich had been introduced by Dugas-Goebel [2]). In particular, the di-visible group Q occurs as a subgroup. Since torsion-free groups whichare complete modulo the Ulm subgroup are algebraically compact, re-duced algebraically compact groups possibly occur as subgroups of thesingular homology group of the Hawaiian earring at that time [6], seeFigure.Actually the first singular homology group of the Hawaiian earring,i.e. H ( H ), turned out to be isomorphic to Z ω ⊕ ⊕ c Q ⊕ Π p :prime A p , where c is the cardinality of the continuum and A p is the p -completionof the free abelian group of rank c [11]. After proving Theorem 2.4,I investigated the singular homology groups of one dimensional Peanocontinua. My assessment was that he singular homology group of theHawaiian earring can hardly isomorphic to any of the Menger spongeor the Sierpinski carpet - because of the so much simpler topologicalstructure of the Hawaiian earring. In spite of my many trials to showthis, I had failed to do it. After changing my mind I proved, Theorem 3.1. [10]
The singular homology group H ( X ) of a one-dimension Peano continuum X is isomorphic to a free abelian groupof finite rank or the singular homology group H ( H ) of the Hawaiianearring. KATSUYA EDA
That is, if a Peano continuum X contains a single wild point, then H ( X ) is already isomorphic to H ( H ), no matter how many additionalwild points are there in X . Remark 3.2.
In [6] I used the notion ”complete modulo the Ulmsubgroup” due to Dugas-Goebel [2]. Very recently Herfort-Hojka [16]have characterized the notion ”cotorsion” using equation systems. Ac-cording to it results in [6] can be improved to being cotorsion. Thischaracterization is a commutative version of the equation system dueto G. Higman [17] for non-commutative groups.4.
The Reid class and ˇCech systems
The Reid class consists of the integer group Z and the groups ob-tained by iterating use of forming direct sums and direct products[18]. Back to 1983, answering a question in [3, 21], whether Hom( A, Z )belongs to the Reid class for every abelian group A , I proved thatHom( C ( Q , Z ) , Z ) does not belong to the Reid class, where C ( Q , Z ) is agroup of continuous functions. Before then I was interested in algebraictopology and so I tried to apply the Reid class to algebraic topology.There are two typical and distinct ways of attaching infinitely manycircles with a common point. The one is so-called a bouquet, wherea basic open neighborhood consists of open neighborhoods of all cir-cles. A basic open neighborhood of the common point in the other wayconsists of almost all copies of the circle and open neighborhoods inthe remaining finite circles. When the number of copies are countable,the latter space is homeomorphic to the Hawaiian earring, while theformer one is called a countable bouquet. Passing to the factor groupof singular homology as described in [13] one obtains respectively freeabelian groups ⊕ κ Z and direct products Z κ . Let us recall this factor-ization [13]: The singular chain group S n ( X ) is the free abelian groupgenerating by the set C (∆ n , X ), where ∆ n is the n -simplex. Since C (∆ n , X ) has the compact open topology, S n ( X ) can be regarded asthe free abelian topological group over C (∆ n , X ). Then, the bound-ary operator ∂ n : S n +1 ( X ) → S n ( X ) becomes continuous. Therefore,Ker( ∂ n − ) is closed, but Im( ∂ n ) may not be closed. We take its closureand consider Ker( ∂ n ) / Im( ∂ n +1 ) = H n ( X ), which is the factor of singu-lar homology. By taking the closure, information due to the wildnessof a space disappears. For instance, H T ( H ) is isomorphic to Z ω . Onthe other hand, if X is locally good, e.g. locally contractible, Im( ∂ n +1 )is closed and we have H Tn ( X ) = H n ( X ). Therefore, for spaces usu-ally appearing in algebraic topology it gives us the same as singularhomology. ROM UNCOUNTABLE ABELIAN GROUPS TO UNCOUNTABLE NONABELIAN GROUPS7
When attaching infinitely many component spaces with one commonpoint we have two typical types of one point unions. A neighborhoodof the common point is a union of neighborhoods of the point in com-ponent space in one type and is a union of neighborhoods of the pointin finitely many component spaces and the whole spaces for remainingcomponent spaces. These construction can be done alternately anditerated. In such constructions with the same common point the com-plexity of the topology around the common point should increase. Thecomplexity can be expressed by the hierarchy theorem of the Reid classvia the factor of singular homology [13] and [4, 14] (I talked about thisin the Oberwolfach conference in 1989).Many years later I, with J. Nakamura, tried to classify the inverselimits of sequences of finitely generated free groups. Such inverse lim-its in the abelian case become finitely generated free abelian groupsor Z ω , which is a consequence of the fact that any subgroup of afinitely generated free abelian group is also finitely generated. Re-mark that a subgroup of a finitely generated free group may not befinitely generated. The inverse limits of sequences of finitely gener-ated free groups are precisely the first ˇCech homotopy groups of one-dimensional connected compact metric spaces. When such spaces arelocally connected, they are Peano continua. Then, we have an inversesystem of surjective homomorphisms and the inverse limits are isomor-phic to finitely generated free groups F n or the inverse limit accordingto the canonical projections of finitely generated free groups. That is, G = lim ←− n →∞ ∗ ni =0 Z i , where the bonding map from ∗ ni =0 Z i to ∗ n − i =0 Z i isthe projection. In general, we have three other groups. Let F ω be thecountable free group. Let G = lim ←− n →∞ F ω ∗ ∗ ni =0 Z i , where the bondingmap from F ω ∗∗ ni =0 Z i to F ω ∗∗ n − i =0 Z i is the projection. Let F ω,n be copiesof F ω and G = lim ←− n →∞ ∗ ni =0 F ω,i where the bonding map from ∗ ni =0 F ω,i to ∗ n − i =0 F ω,i is the projection, is the inverse limit according to the projec-tions of finite free products of copies of F ω . Now, F ω , G and G arethe three groups. As you can see from these results, the classificationof the inverse limit of at most countable free groups is the same as thecorresponding one to finitely generated free groups [12]. Therefore, theclassification of shape groups of one dimensional connected, compactmetric spaces , i.e. their first ˇCech homotopy groups, is the same asthat of the corresponding groups of connected separable metric spaces. KATSUYA EDA
We explain how we proved that these uncountable groups G , G and G are not isomorphic to each others. Let [ G, G ] be the commu-tator subgroup of a group G and Ab( G ) = G/ [ G, G ]. Let R Z ( A ) = T { Ker( h ) | h ∈ Hom( A, Z ) } . As is well-known, A/R Z ( A ) is isomor-phic to a subgroup of Z κ for some κ . We consider a functor F( G ) =Ab( G ) /R Z (Ab( G )). Then, we have F( G ) = Z ω , F( G ) = ⊕ ω Z ⊕ Z ω and F( G ) = ( ⊕ ω Z ) ω . Then, by the hierarchy theorem of the Reidclass we conclude these groups are not isomorphic and consequently soare G , G and G .Together with the results in Sections 2 and 3, I feel the follow-ing. Though abelian groups are related to many areas of mathematicsthrough homology and cohomology, apart from the finitely generatedcase the relationships are formal ones. Therefore, what I’ve explainedin the preceding are new aspects in the relationships between infinitelygenerated abelian groups, non-abelian groups and topological spaces.In particular, divisible subgroups or algebraically compact (cotorsion)subgroups of singular homology groups imply the existence of wildpoints in the spaces.I am still working on the topic discussed here and so please do notconsider this note as my final report on these issues. I hope I can con-tinue my research until the age of Laszlo, i.e. twenty years more. WhenI wrote to him ”I understand that one can continue doing mathematicsover 90,” he shot back a proposal to argue that doing mathematicsactively would get me over 90. Acknowledgement
The author thanks the referee for improving wording to express con-tents exactly and reading this manuscript thoroughly. It seems thatthe referee also read some old references.
References [1] G. R. Conner and K. Eda,
Fundamental groups having the whole informationof spaces , Topology Appl. (2005), 317–328.[2] M. Dugas and R. G¨obel,
Algebraisch kompakte Faktorgruppen , J. reine angew.Math. (1981), 341–352.[3] M. Dugas and B. Zimmermann-Huisgen,
Iterated direct sums and products ,Abelian group theory (Oberwolfach, 1981), Springer, 1981, pp. 179–193.[4] K. Eda, On Z -kernel groups , Archiv Math. (1983), 289–293.[5] , The first integral singular homology groups of one point unions , Quart.J. Math. Oxford (1991), 443–456.[6] , Free σ -products and noncommutatively slender groups , J. Algebra (1992), 243–263. ROM UNCOUNTABLE ABELIAN GROUPS TO UNCOUNTABLE NONABELIAN GROUPS9 [7] ,
The fundamental groups of one-dimensional spaces and spatial homo-morphisms , Topology Appl. (2002), 479–505.[8] ,
Homotopy types of one-dimensional peano continua , Fund. Math. (2010), 27–45.[9] ,
Atomic property of the fundamental group of the hawaiian earring andwild peano continua , J. Math. Soc. Japan (2011), 769–787.[10] , Singular homology of one-dimensional peano continua , Fund. Math. (2016), 99–115.[11] K. Eda and K. Kawamura,
The singular homology of the hawaiian earring , J.London Math. Soc. (2000), 305–310.[12] K. Eda and J. Nakamura, The classification of the inverse limits of free groupsof finite rank , Bull. London Math. Soc. (2013), 671–676.[13] K. Eda and K. Sakai, A factor of singular homology , Tsukuba J. Math. (1991), 351–387.[14] P. Eklof and A. Mekler, Almost free modules: Set-theoretic methods , North-Holland, 1990.[15] H. B. Griffiths,
Infinite products of semigroups and local connectivity , Proc.London Math. Soc. (1956), 455–485.[16] W. Herfort and W. Hojka, Cotorsion and wild topology , Israel J. Math. (2017), 275–290.[17] G. Higman,
Unrestricted free products, and variety of topological groups , J.London Math. Soc. (1952), 73–81.[18] G. A. Reid, Almost free groups , Tulane University, 1966/67.[19] E. H. Spanier,
Algebraic topology , McGraw-Hill, 1966.[20] E. Specker,
Additive Gruppen von folgen ganzer Zahlen , Portugal. Math. (1950), 131–140.[21] B. Zimmermann-Huisgen, On Fuchs’ problem 76 direct sums of free cycles , J.Reine Angew. Math. (1979), 86–91.
Department of Mathematics, Waseda University, Tokyo 169-8555,JAPAN
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