We present simple examples of finite-dimensional connected homogeneous spaces
(they are actually topological manifolds) with nonhomogeneous and nonrigid
factors. In particular, we give an elementary solution of an old problem in
general topology concerning homogeneous spaces

A.N. Dranishnikov

Antonio Quintero

B.J. Smith

C.D. Bass

D. Repovš

D.M. Halverson

Dušan Repovš

F. Quinn

F. Raymond

F.D. Ancel

Francisco F. Lasheras

G.E. Bredon

J. Dydak

J. Mill van

J.J. Andrews

J.L. Bryant

Manuel Cárdenas

R. Myers

R.H. Fox

R.J. Daverman

R.L. Wilder

S. Hu

S.J. Lomonaco

T.B. Rushing

T.R. Brahana

V.V. Fedorchuk

W. Rosicki

English

arXiv

Crossref

Central European Journal of Mathematics

On manifolds with nonhomogeneous factors

We present simple examples of finite-dimensional connected homogeneous
spaces (they are actually topological manifolds) with nonhomogeneous
and nonrigid factors. In particular, we give an elementary solution of
an old problem in general topology concerning homogeneous spaces.Ministerio de Ciencia e InnovaciónSlovenian Research Agenc

Cárdenas Escudero, Manuel Enrique

Fernández Lasheras, Francisco Jesús

Quintero Toscano, Antonio Rafael

Repovš, Dušan

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:1203.4151v2  [math.GT]  20 Mar 2012ON MANIFOLDS WITH NONHOMOGENEOUS FACTORSM. CA´RDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVSˇAbstract. We present simple examples of finite-dimensional connected ho-mogeneous spaces (they are actually topological manifolds) with nonhomoge-neous and nonrigid factors. In particular, we give an elementary solution ofan old problem in general topology concerning homogeneous spaces.1. IntroductionA topological space X is said to be homogeneous if for every pair of pointsx, y ∈ X there exists a homeomorphism h : (X, {x})→ (X, {y}). This very classicaltopological notion became very important when in the 1960’s Bing and Borsukproved that in dimensions below 3, homogeneity can actually detect topologicalmanifolds among all finite-dimensional absolute neighborhood retracts (ANR’s).Bing and Borsuk also conjectured that this is true in all dimensions, and this con-jecture remains a formidable open problem (in dimension 3 it implies the Poincare´Conjecture). Recently, homogeneity has gained renewed attention among geomet-ric topologists, since it turned out that the so-called Busemann G-spaces (whichhave also been conjectured to be topological manifolds) possess homogeneity amongother key properties [Bry02, HR08].It was in our recent investigations of Bing-Borsuk and Busemann Conjecturesthat we came upon some observations on the homogeneity of products of nonho-mogeneous spaces which we have collected in the present paper. In particular, wegive (uncountably many) connected nonrigid finite-dimensional positive answers tothe following question from §1.7 listed on page 125 of [Ar03]: ”Is there a nonho-mogeneous (compact) space whose square is homogeneous?”. Clearly, there is adimensional restriction to such examples, namely n ≥ 3.Several positive answers to this question are already known. In 2003 a noncon-nected example was given by Rosicki [Ros03] in the realm of topological groups.Earlier, an infinite-dimensional connected rigid example was constructed by vanMill [Mil81] in 1981, whereas in 1983 Ancel and Singh [AS83] constructed finite-dimensional rigid examples X with dim X ≥ 4 and Ancel, Duvall and Singh[ADS83] produced such an example also for the case dim X = 3. Recall thata space is said to be rigid if it does not have any self-homeomorphism other thanthe identity.The results in this paper provide alternative finite-dimensional answers to thequestion above which are easier to construct than the rigid ones, as a straightforwardapplication of the theory of decompositions of manifolds.In a more general setting, we say that a space X is k-homogeneous, k ≥ 2, iffor any given k-element sets {x1, . . . , xk} and {y1, . . . , yk}, there exists a homeo-morphism h : X → X such that h(xi) = yi for i = 1, . . . , n. The case k = 2 issimply referred to as bihomogeneity. Our results can also be used to get examplesDate: March 21, 2012.2010 Mathematics Subject Classification. Primary 57N15, 57N75; Secondary 57P99, 53C70.Key words and phrases. k-homogeneity, rigidity, ANR, upper semicontinuous decomposition,generalized manifold, cell-like resolution, general position property, manifold recognition theorem.12 M. CA´RDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVSˇof manifold factors that fail to be bihomogeneous and (k ≥ 3)-homogeneous sinceany of them implies homogeneity.2. Wild cells of arbitrary dimensions and codimensionsIn geometric topology, a wild k-cell in Rn (0 < k < n) is a topological embed-ding of the unit k-ball of Rk which cannot be mapped onto the canonical embeddingBk ⊂ Rk ⊂ Rn by a homeomorphism of Rn onto itself. First examples of wild 1-cellsin R3 (called wild arcs) were constructed by Artin and Fox ([FA48]). In fact, thereare uncountably many wild arcs (see [FH62, Lom68, Mye00]). In [Mye00] differentarcs were distinguished by the fundamental groups of their complements: i.e., twoarcs α, β are not equivalent if and only ifpi1(S3 \ α) 6∼= pi1(S3 \ β). Recall that two arcs are called equivalent if there is aself-homeomorphism of R3 taking one arc to the other. Notice that we can considerthese arcs to be wild also in S3.Well-known methods based on elementary properties of the suspension of a spacelead to the construction of wild cells in arbitrary dimensions. To illustrate this weshall give some details. Let F3,1 be any uncountable family of wild arcs in S3 suchthat their complements in S3 are not simply connected (for instance, the one givenin [Mye00]), and let α ∈ F3,1. Then for each k ≥ 1 one can construct from α asequence of wild arcs (αk) ⊂ S3+k. Indeed, if we already have a wild arc αk−1 inS3+(k−1), then by Corollary 2.6.4 of [DV09], the sphere S3+k is homeomorphic tothe suspension of the quotient space S3+(k−1)/αk−1.Now, using Lemma 2.7.2 of [DV09], let αk be the arc in S3+k that correspondsto the suspension of the class of the points of αk−1 in the quotient S3+(k−1)/αk−1.Notice that S3+k \ αk is homotopically equivalent to S3+(k−1) \ αk−1, hence αk iswild. So for each n ≥ 3, there is a family Fn,1 of uncountably many distinct wildarcs in Sn.Now, given n ≥ 3 and 0 < k < n, let α be a wild arc in Sn−k+1 from thecollection Fn−k+1,1. By Lemma 1.4.1 of [DV09], the (k − 1)-th suspension Σk−1αof α is a wild k-cell in Sn. Hence, for each n ≥ 3 and 0 < k < n, there is a familyFn,k of uncountably many wild k-cells embedded in Sn.Notice that the k-cells in Fn,k are cell-like non-cellular sets, for n ≥ 3 and0 < k < n (the failure of cellularity follows from Exercise 2.7.4 of [DV09] andExercise 2.6.2.(a) of [Rus73]). As above, these k-cells can be taken to be embeddedeither in Rn or in Sn.3. Products of Generalized ManifoldsA generalized n-manifold X is defined as a finite-dimensional Euclidean neigh-borhood retract (ENR) whose local Z-homology groups agree with those of theEuclidean n-space, i.e.H∗(X,X \ {x};Z) ∼= H∗(Rn;Rn \ {0};Z) for all x ∈ X.By Theorem 6 of [Ray60] and [Bre69], if Xk×Y l is a homology n-manifold thenXk and Y l are homology k- and l-manifolds, respectively, where k+ l = n (see alsoTheorem 2 of [Bra58]).Also, by Problem K.3 on p. 30 of [Hu59], X×Y is a metrizable ANR if and onlyif X and Y are metrizable ANR’s. Combining both results we get the following:Proposition 3.1. If Xk×Y l is a generalized (k+ l)-manifold then Xk and Y l aregeneralized k- and l-manifolds, respectively.ON MANIFOLDS WITH NONHOMOGENEOUS FACTORS 34. ExamplesUnless otherwise stated, all manifolds in this section will be assumed to beconnected.Theorem 4.1. There exist uncountably many distinct topological n-manifolds Mnsuch that Mn = X × Y , where X is a nonhomogeneous nonmanifold factor, if andonly if n ≥ 4.Proof. (⇐) If n ≥ 4, then by Theorem 1 of [AC62] it suffices to pick any wildarc α ⊂ Rn−1 such that Rn−1 \ α is not simply connected, from the family Fn,1defined in Section 2, and consider the quotient space Xn−1 = Rn−1/α. Then Xn−1is a generalized (n− 1)-manifold with one singular point (hence a nonhomogeneousspace), since the space Xn−1 fails to be locally Euclidean at pi(α) ∈ X , wherepi : Rn−1 → Xn−1 = Rn−1/α is the quotient map. On the other hand, X × R ishomeomorphic to Rn, so Xn−1 is an n-manifold factor. By letting α range over theuncountable family Fn,1, we obtain uncountably many distinct examples. Moreover,if one wants to obtain a closed manifold M , one just observes that (Sn−1/α) × Ris an n-manifold, and so (Sn−1/α)× S1 is a closed n-manifold.In a similar way, but in a more general setting, one can apply Theorem 1.1 of[Bry68]: Let D be any k-cell in the family Fn,k defined in Section 2, with n ≥ 4and 0 < k < n. Then (Rn−1/D) × R is homeomorphic to Rn. Again, the spaceX = Rn−1/D is not a manifold since it has a unique singular point which makes thespace nonhomogeneous. By varying D in Fn,k we get uncountably many distinctexamples.(⇒) Let n ≤ 3 and Mn = Xk × Y l so that k + l = n. Then by Proposition 3.1,Xk and Y l are generalized k- and l-manifolds, respectively. Since M is connected,we may assume that 0 < k and l < 3. Hence these low-dimensional generalizedmanifolds X and Y are actually genuine manifolds (see [Rep92] or Ch. IX of[Wil79]) and are therefore also homogeneous. Remark 4.2. For n ≥ 5, more examples can be constructed. Given a topological(n− 2)-manifoldMn−2 choose any cell-like usc-decomposition G such that Xn−2 =Mn−2/G is finite-dimensional. (This dimensionality condition is necessary due toexamples of Dranishnikov [Dra88] and Dydak-Walsh [DW93]). By Theorem 26.8of [Dav07], Xn−2 × R2 has the Disjoint Disks Property and hence, by Edwards’Theorem (see e.g. Theorem 2.2 of [HR08]):Nn = Xn−2 × R2is a topological n-manifold, whereas X is nonhomogeneous if one assumes that thesingular set is not dense in X, i.e. S(X) 6= X . Here S(X) denotes the singular setof X , i.e., the set of all points in X having no Euclidean neighborhood.Theorem 4.3. There exist uncountably many topological 2n-dimensional manifoldsM2n such that M2n = X ×X, where X is a nonhomogeneous manifold factor, ifand only if n ≥ 3.Proof. (⇐) For n ≥ 3, we apply Corollary 3 from [Bas81] (see also [Smi73]) to acell-like decomposition G of a manifold M of dimension dim M ≥ 3 in order toobtain that (M/G) × (M/G) is homeomorphic to M ×M . Hence, it is enough totake M = Sn and G to be the cell-like decomposition, whose only nondegenerateelement is one of the k-cells from the family Fn,k which was defined in Section 2.(⇒) Let n ≤ 2. Given N2n = X ×X , then (as above) it follows by Proposition 3.1that X is a generalized n-manifold. Therefore, for n = 2, X is a generalized 2-manifold and hence a surface; while for n = 1, X is a generalized 1-manifold and4 M. CA´RDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVSˇhence a circle. Recall that (n < 3)-dimensional homology manifolds are genuinemanifolds (see Ch. IX of [Wil79]). Remark 4.4. In fact, Bass’ result used in the proof of Theorem 4.3 shows thatgiven two manifolds of dimensions ≥ 3 and cell-like decompositions G and G′ of Mand N , respectively, satisfying certain mild conditions, it follows that M × N ∼=(M/G) × (N/G′). Hence, these constructions provide affirmative answers in alldimensions ≥ 6 to the second question from §1.7 posed on p. 125 of [Ar03]: “Canthe product of two nonhomogeneous spaces be homogeneous?”.5. EpilogueQuestion 5.1. What can one say about homogeneous continua with nonhomo-geneous factors in arbitrary dimensions? More explicitly, we state the followingquestions:(1) Can an (n ≤ 3)-dimensional homogeneous continuum K be written as aproduct K = X × Y , where at least one of the factors X and Y is nothomogeneous?(2) Can an (n ≤ 5)-dimensional homogeneous continuum K be written as aproduct of two nonhomogeneous factors?(3) Can an (n ≤ 5)-dimensional homogeneous continuum K be written as K =X ×X , where X is not homogeneous?Question 5.2. Does the Logarithmic Law hold for homogeneous compact ANR’s,i.e. does the following equality hold:dim(X×Y) = dimX+ dimY?According to the proof sketched in [F05], the so-called Pontryagin surfaces Tp arehomogeneous. Recall that these celebrated compacta, which have the property thatdimTp= 2 for all prime p, but dim(Tp × Tq) = 3, whenever p 6= q, show that theLogarithmic Law fails if X and Y are not ANR’s. Recent work [Bry02] was believedto lead to a positive answer to Question 5.2 (see [F05]). However, last year Bryantdiscovered a serious gap in the proof of Theorem 2 from [Bry02].Remark 5.3. The most famous problem still open in decomposition theory is theclassical R. L. Moore Problem from the 1930’s, concerning the characterization oftopological n-manifolds. It asks if every (finite-dimensional) cell-like decompositionRn/G of Rn is a topological factor of Rn+1, i.e.(Rn/G)× R ∼= Rn+1(see [HR11] for a recent survey on this difficult problem).In connection with the Moore Problem we mention Problem 9.5 of [Qui06], whichasks if the product of a homology manifold and R is always homogeneous? Manyexamples (in particular, those in Theorems 4.1 and 4.3) give partial affirmativeanswers to both of these questions, but there are still far more examples to beconsidered.AcknowledgmentsThe first three authors were supported by the MTM project 2010-20445. Thefourth author was supported by the SRA grants P1-0292-0101 and J1-2057-0101.We gratefully acknowledge remarks and suggestions from John L. Bryant, JanuszPrajs and both referees.ON MANIFOLDS WITH NONHOMOGENEOUS FACTORS 5References[ADS83] F. D. Ancel, P. F. Duvall, and S. Singh, Rigid 3-dimensional compacta whose squares aremanifolds, Proc. Amer. Math. Soc. 88 (1983), no. 2, 330–332. MR 695269 (84m:54039)[AS83] F. D. Ancel and S. Singh, Rigid finite-dimensional compacta whose squares are mani-folds, Proc. Amer. Math. Soc. 87 (1983), no. 2, 342–346. MR 681845 (83m:54074)[AC62] J. J. Andrews and M. L. Curtis, n-space modulo an arc, Ann. of Math. (2) 75 (1962),1–7. MR 0139153 (25 #2590)[Ar03] A. V. Arhangel′ski˘ı, Problems from A.V. 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MR 1196725(93j:57011)6 M. CA´RDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVSˇ[Ros03] W. Rosicki, On the uniqueness of the decomposition of continua into Cartesian products,Bull. Polish Acad. Sci. Math. 51 (2003), no. 3, 247–250. MR 2017000 (2004j:54012)[Rus73] T. B. Rushing, Topological embeddings, Academic Press, New York, 1973, Pure andApplied Mathematics, Vol. 52. MR 0348752 (50 #1247)[Smi73] B. J. Smith, Products of decompositions of En, Trans. Amer. Math. Soc. 184 (1973),31–41 (1974). MR 0328946 (48 #7288)[Wil79] R. L. Wilder, Topology of manifolds, American Mathematical Society Colloquium Publi-cations, vol. 32, American Mathematical Society, Providence, R.I., 1979, Reprint of 1963edition. MR 598636 (82a:57001)Departamento de Geometr´ıa y Topolog´ıa, Universidad de Sevilla, Apdo 1160, 41080-Sevilla, SpainE-mail address: mcard@us.es, lasheras@us.es, quintero@us.esFaculty of Mathematics and Physics, and Faculty of Education, University of Ljubl-jana, P.O. Box 2964, Ljubljana 1001, SloveniaE-mail address: dusan.repovs@guest.arnes.si

On manifolds with nonhomogeneous factors

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