On manifolds with nonhomogeneous factors

arXiv:1203.4151v2 [math.GT] 20 Mar 2012 ON MANIFOLDS WITH NONHOMOGENEOUS FACTORS M. C´ ARDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVˇ S Abstract. 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. 1. Introduction A topological space Xis said to be homogeneous if for every pair of points x, y ∈Xthere exists a homeomorphism h: (X, {x})→(X, {y}). This very classical topological notion became very important when in the 1960’s Bing and Borsuk proved that in dimensions below 3, homogeneity can actually detect topological manifolds among all finite-dimensional absolute neighborhood retracts (ANR’s). Bing and Borsuk also conjectured that this is true in all dimensions, and this conjecture remains a formidable open problem (in dimension 3 it implies the Poincar´e Conjecture). Recently, homogeneity has gained renewed attention among geometric topologists, since it turned out that the so-called Busemann G-spaces (which have also been conjectured to be topological manifolds) possess homogeneity among other key properties [Bry02, HR08]. It was in our recent investigations of Bing-Borsuk and Busemann Conjectures that we came upon some observations on the homogeneity of products of nonhomogeneous spaces which we have collected in the present paper. In particular, we give (uncountably many) connected nonrigid finite-dimensional positive answers to the following question from §1.7 listed on page 125 of [Ar03]: ”Is there a nonhomogeneous (compact) space whose square is homogeneous?”. Clearly, there is a dimensional restriction to such examples, namely n≥3. Several positive answers to this question are already known. In 2003 a nonconnected example was given by Rosicki [Ros03] in the realm of topological groups. Earlier, an infinite-dimensional connected rigid example was constructed by van Mill [Mil81] in 1981, whereas in 1983 Ancel and Singh [AS83] constructed finitedimensional rigid examples Xwith dim X ≥4 and Ancel, Duvall and Singh [ADS83] produced such an example also for the case dim X = 3. Recall that a space is said to be rigid if it does not have any self-homeomorphism other than the identity. The results in this paper provide alternative finite-dimensional answers to the question above which are easier to construct than the rigid ones, as a straightforward application of the theory of decompositions of manifolds. In a more general setting, we say that a space Xis k-homogeneous,k≥2, if for any given k-element sets {x1,...,xk}and {y1,...,yk}, there exists a homeomorphism h:X→Xsuch that h(xi) = yifor i= 1,...,n. The case k= 2 is simply referred to as bihomogeneity. Our results can also be used to get examples Date: 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. 1 2 M. C´ ARDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVˇ S of manifold factors that fail to be bihomogeneous and (k≥3)-homogeneous since any of them implies homogeneity. 2. Wild cells of arbitrary dimensions and codimensions In geometric topology, a wild k-cell in Rn(0 < k < n) is a topological embedding of the unit k-ball of Rkwhich cannot be mapped onto the canonical embedding Bk⊂Rk⊂Rnby a homeomorphism of Rnonto itself. First examples of wild 1-cells in R3(called wild arcs) were constructed by Artin and Fox ([FA48]). In fact, there are uncountably many wild arcs (see [FH62, Lom68, Mye00]). In [Mye00] different arcs were distinguished by the fundamental groups of their complements: i.e., two arcs α,βare not equivalent if and only if π1(S3\α)6∼ =π1(S3\β). Recall that two arcs are called equivalent if there is a self-homeomorphism of R3taking one arc to the other. Notice that we can consider these arcs to be wild also in S3. Well-known methods based on elementary properties of the suspension of a space lead to the construction of wild cells in arbitrary dimensions. To illustrate this we shall give some details. Let F3,1be any uncountable family of wild arcs in S3such that their complements in S3are not simply connected (for instance, the one given in [Mye00]), and let α∈ F3,1. Then for each k≥1 one can construct from αa sequence of wild arcs (αk)⊂S3+k. Indeed, if we already have a wild arc αk−1in S3+(k−1), then by Corollary 2.6.4 of [DV09], the sphere S3+kis homeomorphic to the suspension of the quotient space S3+(k−1)/αk−1. Now, using Lemma 2.7.2 of [DV09], let αkbe the arc in S3+kthat corresponds to the suspension of the class of the points of αk−1in the quotient S3+(k−1)/αk−1. Notice that S3+k\αkis homotopically equivalent to S3+(k−1) \αk−1, hence αkis wild. So for each n≥3, there is a family Fn,1of uncountably many distinct wild arcs in Sn. Now, given n≥3 and 0 < k < n, let αbe a wild arc in Sn−k+1 from the collection Fn−k+1,1. By Lemma 1.4.1 of [DV09], the (k−1)-th suspension Σk−1α of αis a wild k-cell in Sn. Hence, for each n≥3 and 0 < k < n, there is a family Fn,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 and 0< k < n (the failure of cellularity follows from Exercise 2.7.4 of [DV09] and Exercise 2.6.2.(a) of [Rus73]). As above, these k-cells can be taken to be embedded either in Rnor in Sn. 3. Products of Generalized Manifolds A generalized n-manifold Xis defined as a finite-dimensional Euclidean neighborhood retract (ENR) whose local Z-homology groups agree with those of the Euclidean 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×Ylis a homology n-manifold then Xkand Ylare homology k- and l-manifolds, respectively, where k+l=n(see also Theorem 2 of [Bra58]). Also, by Problem K.3 on p. 30 of [Hu59], X×Yis a metrizable ANR if and only if Xand Yare metrizable ANR’s. Combining both results we get the following: Proposition 3.1. If Xk×Ylis a generalized (k+l)-manifold then Xkand Ylare generalized k- and l-manifolds, respectively. ON MANIFOLDS WITH NONHOMOGENEOUS FACTORS 3 4. Examples Unless otherwise stated, all manifolds in this section will be assumed to be connected. Theorem 4.1. There exist uncountably many distinct topological n-manifolds Mn such that Mn=X×Y, where Xis a nonhomogeneous nonmanifold factor, if and only if n≥4. Proof. (⇐) If n≥4, then by Theorem 1 of [AC62] it suffices to pick any wild arc α⊂Rn−1such that Rn−1\αis not simply connected, from the family Fn,1 defined in Section 2, and consider the quotient space Xn−1=Rn−1/α. Then Xn−1 is a generalized (n−1)-manifold with one singular point (hence a nonhomogeneous space), since the space Xn−1fails to be locally Euclidean at π(α)∈X, where π:Rn−1→Xn−1=Rn−1/α is the quotient map. On the other hand, X×Ris homeomorphic to Rn, so Xn−1is an n-manifold factor. By letting αrange over the uncountable 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)×R is an n-manifold, and so (Sn−1/α)×S1is a closed n-manifold. In a similar way, but in a more general setting, one can apply Theorem 1.1 of [Bry68]: Let Dbe any k-cell in the family Fn,k defined in Section 2, with n≥4 and 0 < k < n. Then (Rn−1/D)×Ris homeomorphic to Rn. Again, the space X=Rn−1/D is not a manifold since it has a unique singular point which makes the space nonhomogeneous. By varying Din Fn,k we get uncountably many distinct examples. (⇒) Let n≤3 and Mn=Xk×Ylso that k+l=n. Then by Proposition 3.1, Xkand Ylare generalized k- and l-manifolds, respectively. Since Mis connected, we may assume that 0 < k and l < 3. Hence these low-dimensional generalized manifolds Xand Yare 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)-manifold Mn−2choose any cell-like usc-decomposition Gsuch that Xn−2= Mn−2/Gis finite-dimensional. (This dimensionality condition is necessary due to examples of Dranishnikov [Dra88] and Dydak-Walsh [DW93]). By Theorem 26.8 of [Dav07], Xn−2×R2has the Disjoint Disks Property and hence, by Edwards’ Theorem (see e.g. Theorem 2.2 of [HR08]): Nn=Xn−2×R2 is a topological n-manifold, whereas Xis nonhomogeneous if one assumes that the singular set is not dense in X, i.e. S(X)6=X. Here S(X) denotes the singular set of X, i.e., the set of all points in Xhaving no Euclidean neighborhood. Theorem 4.3. There exist uncountably many topological 2n-dimensional manifolds M2nsuch that M2n=X×X, where Xis a nonhomogeneous manifold factor, if and only if n≥3. Proof. (⇐) For n≥3, we apply Corollary 3 from [Bas81] (see also [Smi73]) to a cell-like decomposition Gof a manifold Mof dimension dim M ≥3 in order to obtain that (M/G)×(M/G) is homeomorphic to M×M. Hence, it is enough to take M=Snand Gto be the cell-like decomposition, whose only nondegenerate element 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 3.1 that Xis a generalized n-manifold. Therefore, for n= 2, Xis a generalized 2- manifold and hence a surface; while for n= 1, Xis a generalized 1-manifold and 4 M. C´ ARDENAS, F.F. LASHERAS, A. QUINTERO AND D. REPOVˇ S hence a circle. Recall that (n < 3)-dimensional homology manifolds are genuine manifolds (see Ch. IX of [Wil79]).  Remark 4.4. In fact, Bass’ result used in the proof of Theorem 4.3 shows that given two manifolds of dimensions ≥3 and cell-like decompositions Gand G′of M and N, respectively, satisfying certain mild conditions, it follows that M×N∼ = (M/G)×(N/G′). Hence, these constructions provide affirmative answers in all dimensions ≥6 to the second question from §1.7 posed on p. 125 of [Ar03]: “Can the product of two nonhomogeneous spaces be homogeneous?”. 5. Epilogue Question 5.1. What can one say about homogeneous continua with nonhomogeneous factors in arbitrary dimensions? More explicitly, we state the following questions: (1) Can an (n≤3)-dimensional homogeneous continuum Kbe written as a product K=X×Y, where at least one of the factors Xand Yis not homogeneous? (2) Can an (n≤5)-dimensional homogeneous continuum Kbe written as a product of two nonhomogeneous factors? (3) Can an (n≤5)-dimensional homogeneous continuum Kbe written as K= X×X, where Xis 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 Tpare homogeneous. Recall that these celebrated compacta, which have the property that dim Tp= 2 for all prime p, but dim(Tp×Tq) = 3, whenever p6=q, show that the Logarithmic Law fails if Xand Yare not ANR’s. Recent work [Bry02] was believed to lead to a positive answer to Question 5.2 (see [F05]). However, last year Bryant discovered a serious gap in the proof of Theorem 2 from [Bry02]. Remark 5.3. The most famous problem still open in decomposition theory is the classical R. L. Moore Problem from the 1930’s, concerning the characterization of topological n-manifolds. It asks if every (finite-dimensional) cell-like decomposition Rn/Gof Rnis 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], which asks if the product of a homology manifold and Ris always homogeneous? 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MR 598636 (82a:57001) Departamento de Geometr´ ıa y Topolog´ ıa, Universidad de Sevilla, Apdo 1160, 41080- Sevilla, Spain E-mail address:[email protected]s, [email protected], quint[email protected] Faculty of Mathematics and Physics, and Faculty of Education, University of Ljubljana, P.O. Box 2964, Ljubljana 1001, Slovenia E-mail address:dusan.repo[email protected]rnes.si