Cohomogeneity one topological manifolds revisited
aa r X i v : . [ m a t h . G T ] J un COHOMOGENEITY ONE TOPOLOGICAL MANIFOLDSREVISITED
FERNANDO GALAZ-GARC´IA AND MASOUMEH ZAREI
Abstract.
We prove a structure theorem for closed topological mani-folds of cohomogeneity one; this result corrects an oversight in the lit-erature. We complete the equivariant classification of closed, simplyconnected cohomogeneity one topological manifolds in dimensions 5, 6,and 7 and obtain topological characterizations of these spaces. In thesedimensions, these manifolds are homeomorphic to smooth manifolds. Main Results
A topological manifold with an (effective) topological action of a compactLie group is of cohomogeneity one if its orbit space is one-dimensional. Thesemanifolds were introduced by Mostert [36] in 1957 and their topology andgeometry have been extensively studied in the smooth category (see, for ex-ample, [12, 13, 16, 24, 25, 26, 39, 41] and [3, 9, 10, 17, 18, 19, 20, 45, 44, 54]).Much less attention has been given to these spaces in the topological cate-gory, probably because of the assertion in [36] that every topological mani-fold of cohomogeneity one is equivariantly homeomorphic to a smooth mani-fold. This statement originates in the claim that an integral homology spherethat is also a homogeneous space for a compact Lie group must be a stan-dard sphere (see [36, Section 2, Corollary]). This, however, is not the case.Indeed, the Poincar´e homology sphere P is a homogeneous space for the Liegroups SU(2) and SO(3) and it can be written as P ≈ SU(2) / I ∗ ≈ SO(3) / I,where I ∗ is the binary icosahedral group and I is the icosahedral group (see[30] or [49, p. 89]). One can combine this fact with the Double SuspensionTheorem of Edwards and Cannon [8, 11] to construct topological manifoldswith cohomogeneity one actions that are not equivariantly homeomorphicto smooth actions (see Example 2.3). We point out that, by work of Bredon[4], the Poincar´e homology sphere is the only integral homology sphere thatis also a homogeneous space, besides the usual spheres. In the present articlewe fix the gap in [36] and explore some of its consequences.Our first result is a complete structure theorem for closed cohomogeneityone topological manifolds (cf. [36, Theorem 4]). As is customary, we say Date : June 9, 2015.2010
Mathematics Subject Classification.
Key words and phrases. topological manifold, cohomogeneity one, group action,smoothing. that a manifold is closed if it is compact and has no boundary. We brieflydiscuss the non-compact case in Section 2.
Theorem A.
Let M be a closed topological manifold with an (almost) ef-fective topological G action of cohomogeneity one with principal isotropy H .Then the orbit space is homeomorphic to either a closed interval or to acircle, and the following hold.(1) If the orbit space of the action is an interval, then M is the union oftwo fiber bundles over the two singular orbits whose fibers are conesover spheres or the Poincar´e homology sphere, that is, M = G × K − C (K − / H) ∪ G / H G × K + C (K + / H) . The group diagram of the action is given by (G , H , K − , K + ) , where K ± / H are spheres or the Poincar´e homology sphere. Conversely, agroup diagram (G , H , K − , K + ) , where K ± / H are homeomorphic to asphere, or to the Poincar´e homology sphere with dim G / H ≥ , de-termines a cohomogeneity one topological manifold.(2) If the orbit space of the action is a circle, then M is equivariantlyhomeomorphic to a G / H -bundle over a circle with structure group N (H) / H . This theorem stands in contrast with the corresponding statement in thesmooth category, where the fibers of the bundle decomposition are conesover spheres, i.e. disks, and the manifold decomposes as a union of two diskbundles. We prove Theorem A in Section 2.It is well known that a closed smooth cohomogeneity one G-manifoldadmits a G-invariant Riemannian metric with a lower sectional curvaturebound. Alexandrov spaces are synthetic generalizations of Riemannian man-ifolds with curvature bounded below (see [6, 7]). Theorem A, in combinationwith work of Galaz-Garc´ıa and Searle [15], implies:
Corollary B.
A closed topological manifold of cohomogeneity one admitsan invariant Alexandrov metric.
We prove this corollary in Section 3. It is an open question whether everytopological manifold admits an Alexandrov metric. Corollary B shows thatthis is true if M admits a cohomogeneity one action.A topological manifold M is smoothable if it is homeomorphic to a smoothmanifold. A topological G-action on a smoothable topological manifold Mis smoothable if it is equivalent to a smooth G-action on M. The followingresults are simple consequences of Theorem A.
Corollary C.
A closed cohomogeneity one topological manifold is equiv-ariantly homeomorphic to a smooth manifold if and only if every slice ishomeomorphic to a disk.
OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 3
Corollary D.
Let M be a closed topological manifold with a cohomogeneityone G -action.(1) If the codimension of the singular orbits is not , then the action issmoothable.(2) If the codimension of some singular orbit is , then the action is(a) smoothable if and only if the -dimensional slices are homeo-morphic to -dimensional disks.(b) non-smoothable if and only if some -dimensional slice is home-omorphic to the cone over the Poincar´e homology sphere. Corollary E.
Every cohomogeneity one action on a topological n -manifold, n ≤ , is smoothable. Closed, smooth manifolds of cohomogeneity one have been classified equiv-ariantly by Mostert [36] and Neumann [39] in dimensions 2 and 3, by Parker[41] in dimension 4, and, assuming simply connectedness, by Hoelscher [25]in dimensions 5, 6 and 7. Mostert, Neumann and Parker gave canonicalrepresentatives for the classes in the equivariant classification in dimension n ≤
4. This was also done by Hoelscher [24] in dimensions 5 and 6 inthe simply connected case. In dimension 7, Hoelscher [26], Escher and Ult-man [12] computed the homology groups of the manifolds appearing in theequivariant classification. By Corollary E, the classification of closed, co-homogeneity one topological manifolds is complete in dimension n ≤
4. Indimensions 5, 6 and 7, however, it follows from Corollary D that Hoelscher’sresults do not yield a classification in the topological category. Our secondtheorem completes the equivariant classification of closed, cohomogeneityone topological manifolds in these dimensions.
Theorem F.
Let M be a closed, simply connected topological n -manifold, n ≤ , with an (almost) effective cohomogeneity one action of a compactconnected Lie group G . If the action is non-smoothable, then it is givenby one of the diagrams in Table 1 and M can be exhibited as one of themanifolds in this table. The proof of Theorem F follows the outline of the proofs of the classifi-cation in the smooth case (see, for example, [25]). After determining theadmissible group diagrams, we can write the manifolds as joins, productsor bundles in terms of familiar spaces. The problem still remains whetherthe topological manifolds in Table 1 are smoothable. One can quickly settlethis question for the joins in Table 1 since, by the Double Suspension The-orem, these manifolds are homeomorphic to spheres and therefore they aresmoothable. The situation for the products Susp P × S and Susp P × S in Table 1, where Susp P denotes the suspension of P , is more delicate.For example, the 6-dimensional product Susp S × S is homotopy equiva-lent to S × S . By the classification of closed, oriented simply connected6-dimensional topological manifolds with torsion free homology, carried out F. GALAZ-GARC´IA AND M. ZAREI
Dimension Diagram Manifold5 (S × S , I ∗ × Z k , I ∗ × S , S × Z k ) P ∗ S ≈ S × S , I ∗ × S , S × S , S × S ) Susp P × S ≈ S × S (S × S , S × I ∗ , S × I ∗ , S × S ) P ∗ S ≈ S × S , I ∗ × , I ∗ × S , S × P ∗ S ≈ S (S × S , ∆I ∗ , I ∗ × S , ∆S ) P ∗ S ≈ S (S × S , I ∗ × Z k , I ∗ × S , S × Z k ) S bundle over S (S × S , I ∗ × I ∗ , S × I ∗ , I ∗ × S ) P ∗ P ≈ S (S × S , I ∗ × , S × , S ×
1) Susp P × S ≈ S × S (S × S , ∆I ∗ , ∆S , ∆S ) Susp P × S ≈ S × S (S × S × S , I ∗ × S × Z k , I ∗ × T , S × S × Z k ) S bundle over S Table 1.
Non-smoothable cohomogeneity one actions in di-mensions 5, 6 and 7by Wall [47], Jupp [27] and Zhubr [53], there exist infinitely many home-omorphism types for a homotopy S × S , parametrized by a nonnegativeinteger k . For k even, the corresponding homeomorphism type is smooth-able; for k odd, the corresponding homeomorphism type is non-smoothable(see Section 6). Our third theorem settles the smoothability of Susp P × S . Theorem G.
The manifold
Susp P × S is homeomorphic to S × S . The proof of Theorem G is an application of the classification of closed,oriented simply connected topological 6-manifolds with torsion free homol-ogy, and essentially reduces to computing the first Pontryagin class ofSusp P × S (see Section 6). To do this, we use results of Zagier [52], Atiyahand Singer’s G-signature theorem [2] and a signature formula of Atiyah andBott [1].Observe that Susp P × S is the total space of a principal S -bundle overSusp P × S . By Theorem G, Susp P × S ≈ S × S . Hence, Susp P × S is smoothable and, since the Euler class of the bundle is a generator of H ( S × S ), we obtain the following result. Corollary H.
The manifold
Susp P × S is homeomorphic to S × S . Let M be the total space of a topological S -bundle over S . Since H (M , Z ) = 0, the Kirby-Siebenmann class of M vanishes, so M admitsa PL structure (see [31]). Since, in dimensions n ≤
7, every PL n -manifoldadmits at least one compatible smooth structure (see [21, 29, 32] or [48, OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 5 p. 66]), it follows that M is smoothable. Thus, the homeomorphisms in thethird column of Table 1, combined with Corollary E, yield the followingresult.
Corollary I.
A closed, simply connected topological n -manifold of cohomo-geneity one is homeomorphic to a smooth manifold, provided n ≤ . Our paper is organized as follows. In Section 2 we discuss Mostert’s article[36] and prove Theorem A. We prove Corollary B in Section 3. In Section 4we collect some results on cohomogeneity one topological manifolds that wewill use in the proof of Theorem F. Sections 5 and 6 contain, respectively,the proofs of Theorems F and G.
Acknowledgements.
M. Zarei thanks the Institut f¨ur Algebra und Ge-ometrie at the Karlsruher Institut f¨ur Technologie (KIT) for its hospitalitywhile the work presented herein was carried out. Both authors would liketo thank Anand Dessai, Marco Radeschi and Alexey V. Zhubr for helpfulconversations on the proof of Theorem G, and Martin Herrmann, WilderichTuschmann, and Burkhard Wilking for conversations on the smoothabilityof manifolds. The authors also thank Martin Kerin and Wolfgang Ziller forsuggesting improvements to the exposition. M. Zarei was partially supportedby the Ministry of Science, Research and Technology of Iran.2.
Setup and proof of Theorem A
Notation.
Let M be a topological manifold and let x be a point inM. Given a topological (left) action G × M → M of a Lie group G, welet G( x ) = { gx | g ∈ G } be the orbit of x under the action of G. The isotropy group of x is the subgroup G x = { g ∈ G | gx = x } . Observe thatG( x ) ≈ G / G x . We will denote the orbit space of the action by M / G andlet π : M → M / G be the orbit projection map. The (ineffective) kernel ofthe action is the subgroup K = T x ∈ M G x . The action is effective if K is thetrivial subgroup { e } of G; the action is almost effective if K is finite.We will say that two G-manifolds are equivalent if they are equivariantlyhomeomorphic. From now on, we will suppose that G is compact and assumethat the reader is familiar with the basic notions of compact transformationgroups (see, for example, Bredon [5]). We will assume all manifolds to beconnected.As for locally smooth actions (see [4, Ch. IV, Section 3]), for a topologicalaction of G on M there also exists a maximum orbit type G / H, i.e. H isconjugate to a subgroup of each isotropy group. One sees this as follows.Let M be the set of points with isotropy group of smallest dimension andleast number of components. By work of Montgomery and Yang [34], M is an open, dense and connected subset of M. On the other hand, by workof Montgomery and Zippin [35], for every x ∈ M there is a neighborhood V such that G y is conjugate to a subgroup of G x for y ∈ V . It then follows fromthe connectedness of M that the isotropy groups G y , y ∈ M are conjugate F. GALAZ-GARC´IA AND M. ZAREI to each other. By the density of M and the existence of the neighborhood V , each group G y , for y ∈ M , is conjugate to a subgroup of every isotropygroup. Therefore, the orbit type G / G y , for y ∈ M , is maximal. We call thisorbit type the principal orbit type and orbits of this type principal orbits .A homology sphere is a closed topological n -manifold M n that is an in-tegral homology sphere, i.e. H ∗ (M n , Z ) ∼ = H ∗ ( S n , Z ). We will denote thesuspension of a topological space X by Susp X and the join of X with atopological space Y by X ∗ Y. Recall that Susp X ≈ X ∗ S and, in general,Susp n X ≈ X ∗ S n − for n ≥ P ; it is homeomorphicto the homogeneous spaces SU(2) / I ∗ and SO(3) / I, where I ∗ is the binaryicosahedral group and I is the icosahedral group. We will use some basicconcepts of piecewise-linear topology in the proof of Theorem A. We referthe reader to [43] for the relevant definitions.2.2. Cohomogeneity one topological manifolds.
In this subsection wecollect basic facts on cohomogeneity one topological manifolds, discuss theomission in Mostert’s work [36] that gave rise to the present article, andprove some preliminary results that we will use in the proof of Theorems Aand F.
Definition 2.1.
Let M be a topological n -manifold with a topological actionof a compact connected Lie group G. The action is of cohomogeneity one ifthe orbit space is one-dimensional or, equivalently, if there exists an orbitof dimension n −
1. A topological manifold with a topological action ofcohomogeneity one is a cohomogeneity one manifold .By [36, Theorem 1], the orbit space of a cohomogeneity one manifold ishomeomorphic to a circle, an open interval, a half open interval or a closedinterval [ − , +1]. We refer to orbits which map to endpoints as singular . Wecall the isotropy groups of points in these orbits singular isotropy groups.When the orbit space is homeomorphic to [ − , +1], we denote a singularisotropy group corresponding to a point in the orbit ± ± . Orbits thatare not singular are called regular orbits ; they all have the same isotropygroup H and project to interior points of the orbit space. The subgroup His called the principal isotropy groupAs indicated in the introduction, the oversight in [36] stems from theclaim that a homology sphere that is a homogeneous space must be a stan-dard sphere. More precisely, in [36, Sections 2 and 4] Mostert shows thatK / H, where K is a singular isotropy group, must be a homology sphere anda homogeneous space (see [36, Lemma 2 and proof of Theorem 2]) and con-cludes, erroneously, that K / H must be a standard sphere (see [36, Section2, Corollary]). This, as explained in the introduction, is not the case. Thefollowing result of Bredon [4] implies that the Poincar´e homology sphere P and standard spheres are the only possibilities for K / H. OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 7
Theorem 2.2 (Bredon) . Let G be a compact Lie group and H a closedsubgroup of G .(1) If G / H is a homology k -sphere, then G / H is homeomorphic to either S k or to the Poincar´e homology sphere P .(2) If G acts almost effectively and transitively on P , then G is iso-morphic to SU(2) or SO(3) , with I ∗ or I as the isotropy group, re-spectively. The following example shows that there are cohomogeneity one topologi-cal manifolds with K / H ≈ P . Example 2.3.
Let S × SO( n + 1), n ≥
1, act on P ∗ S n as the join actionof the standard transitive actions of S ∼ = SU(2) on P and SO( n + 1) on S n . The orbit space is homeomorphic to [ − , +1] and K + = S × SO( n ),K − = I ∗ × SO( n + 1) and H = I ∗ × SO( n ). Thus K + / H = P . By theDouble Suspension Theorem, Susp P ≈ S and it follows that P ∗ S n ≈ Susp n +1 P is homeomorphic to S n +4 .Taking into account Theorem 2.2, the comments preceding it, and Exam-ple 2.3, we conclude that one must amend [36, Theorem 4] (considering thecorrections in the Errata to [36]) by adding P as a second possibility forK / H in items (iii) and (iv) in [36, Theorem 4]. There is a fifth item in [36,Theorem 4]:
Claim 2.4.
The action of the group G on a space with structure as in [36,Theorem 4] is equivalent to a cohomogeneity one G -action on the manifold M and, conversely, a space M constructed in such a way is a topologicalmanifold with a cohomogeneity one action of G . This claim is true when all the K ± / H are spheres and follows as in [36]. Inthe case where at least one of the K ± / H is homeomorphic to P , one mustprove Claim 2.4. We do this at the end of this section, in the case where Mis closed (i.e. compact and without boundary). This yields Theorem A. Theremaining cases, where the orbit space is not compact, can be dealt with inan analogous way, and we leave this task to the interested reader.By item (iv) in [36, Theorem 4], a cohomogeneity one G action on aclosed topological manifold with orbit space an interval determines a groupdiagram GK − j − > > ⑤⑤⑤⑤⑤⑤⑤ K + j + ` ` ❇❇❇❇❇❇❇ H i − ` ` ❇❇❇❇❇❇❇❇ i + > > ⑤⑤⑤⑤⑤⑤⑤⑤ where i ± and j ± are the inclusion maps, K ± are the isotropy groups ofthe singular orbits at the endpoints of the interval, and H is the principal F. GALAZ-GARC´IA AND M. ZAREI isotropy group of the action. We will denote this diagram by the 4-tuple(G , H , K − , K + ). The inclusion maps are an important element in the groupdiagram, as illustrated by the following simple example: (T , { e } , T , T )determines both S and S × S , where in the first case the inclusion mapsare to the first and second factors, respectively, of T , and in the second case,both inclusion maps are the same (cf. [39]). Now we prove Theorem A.2.3. Proof of Theorem A.
Let M n be a closed topological n -manifoldwith an (almost) effective topological G action of cohomogeneity one withprincipal isotropy H. By item (i) in [36, Theorem 4], the orbit space ishomeomorphic to either a closed interval or to a circle. Part (2) of Theo-rem A follows from item (ii) in [36, Theorem 4]. Therefore, we need onlyprove part (1) of Theorem A, where the orbit space M / G is homeomorphicto a closed interval [ − , +1]. The “if” statement in this case correspondsto part (iv) of [36, Theorem 4] (keeping in mind that one must add P as apossibility for K ± / H). Now we prove the “only if” statement.Let (G , H , K − , K + ) be a group diagram satisfying the hypotheses of part(1) of Theorem A. By the work of Mostert, we need only consider the casewhere at least one of K ± / H is the Poincar´e sphere P . In this case, n ≥ + / H = P . Since n ≥
5, thesingular orbit G / K + is at least one-dimensional. Observe now that the spaceX = G × K − C (K − / H) ∪ G / H G × K + C (K + / H)(2.1)is a finite polyhedron. Since the link of every point in the singular orbitG / K + is S n − ∗ P , the following result (see [50, p. 742]) implies that X isa topological manifold: Theorem 2.5 (Edwards) . A finite polyhedron P is a closed topological n -manifold if and only if the link of every vertex of P is simply connected if n ≥ , and the link of every point of P has the homology of the ( n − -sphere. (cid:3) Existence of Invariant Alexandrov metrics
In this section we point out that every closed cohomogeneity one topo-logical manifold admits an invariant Alexandrov metric. Let us first recallsome basic facts about Alexandrov spaces, all of which can be found in [6].A finite dimensional length space (X , d ) has curvature bounded from be-low by k if every point x ∈ X has a neighborhood U such that for anycollection of four different points ( x , x , x , x ) in U , the following condi-tion holds: ∠ x ,x ( k ) + ∠ x ,x ( k ) + ∠ x ,x ( k ) ≤ π. Here, ∠ x i ,x j ( k ), called the comparison angle , is the angle at x ( k ) in thegeodesic triangle in M k , the simply-connected 2-manifold with constant cur-vature k , with vertices ( x ( k ) , x i ( k ) , x j ( k )), which are the isometric imagesof ( x , x i , x j ). An Alexandrov space is a complete length space of curvature
OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 9 bounded below by k , for some k ∈ R . The isometry group Isom(X) of anAlexandrov space X is a Lie group (see [14]) and Isom(X) is compact if Xis compact. Alexandrov spaces of cohomogeneity one have been studied in[15].3.1. Proof of Corollary B.
Let M be a closed topological manifold with acohomogeneity one action of a compact Lie group G. If the action is equiv-alent to a smooth action, then it is well known that one can construct a G-invariant Riemannian metric on M. Since, M is compact, this Riemannianmetric has a lower sectional curvature bound and hence M is an Alexan-drov space. Suppose now that the G action is not equivalent to a smoothaction. In this case, M has a group diagram satisfying the hypotheses of[15, Proposition 5] and, by this result, M admits an invariant Alexandrovmetric. (cid:3) Tools and further definitions
In this section we review some standard results for cohomogeneity onesmooth manifolds in the context of topological manifolds. We will use thesetools in the proof of Theorem F.We first point out that all the propositions and lemmas used by Hoelscherin [25] to determine both the groups G that may act by cohomogeneity oneon a smooth closed manifold M and the fundamental group of M also holdfor topological manifolds. Indeed, the fact that M is a union of two mappingcylinders is a key point in the proofs of most statements in [25]. By Mostert’swork [36], this is also the case for a cohomogeneity one topological manifold.We collect the relevant results here for easy reference, focusing our attentionon the cases where at least one of K ± / H is the Poincar´e sphere.The following proposition determines when two different group diagramsyield the same manifold. Its proof follows as in [4, Theorem IV.8.2], afterobserving that a cohomogeneity one topological manifold decomposes as theunion of two mapping cylinders.
Proposition 4.1.
If a cohomogeneity one topological manifold is given bya group diagram (G , H , K − , K + ) , then any of the following operations onthe group diagram will result in a G -equivariantly homeomorphic topologicalmanifold:(1) Switching K − and K + ,(2) Conjugating each group in the diagram by the same element of G ,(3) Replacing K − with gK − g − for g ∈ N (H) .Conversely, the group diagrams for two G -equivariantly homeomorphic co-homogeneity one, closed topological manifolds must be mapped to each otherby some combination of these three operations. The following result of Parker [25, Proposition 1.8] is stated for smoothcohomogeneity one manifolds but the proof carries over to the topologicalcategory.
Proposition 4.2 (Parker) . A closed simply connected cohomogeneity onetopological manifold has no exceptional orbits.
The van Kampen Theorem applied to a closed cohomogeneity one mani-fold written as a union of two mapping cylinders yields the following result(cf. [25, Proposition 1.8]).
Proposition 4.3 (van Kampen Theorem) . Let M be the closed cohomo-geneity one topological manifold given by the group diagram (G , H , K − , K + ) with dim(K ± / H) ≥ . Then π (M) ∼ = π (G / H) / N − N + , where N ± = ker { π (G / H) → π (G / K ± ) } = Im { π (K ± / H) → π (G / H) } . In particular M is simply connected if and only if the images of K ± / H gen-erate π (G / H) under the natural inclusions. As the homogeneous spaces K ± / H are homeomorphic to either spheres orthe Poincar´e homology sphere, their fundamental groups are Z , the identityor the binary icosahedral group. Since these groups are finitely generated,the next lemma follows as in the proof of [25, Lemma 1.10]. Lemma 4.4.
Let M be the cohomogeneity one topological manifold given bythe group diagram (G , H , K − , K + ) with at least one of K ± / H homeomorphicto P . Denote H ± = H ∩ K ± , and let α i ± : [0 , → K ± be curves that generate π (K ± / H) , with α i ± (0) = 1 ∈ G . The manifold M is simply connected if andonly if(1) H is generated as a subgroup by H − and H + , and(2) α i − and α i + generate π (G / H ) . Recall that a cohomogeneity one action on a closed manifold M is non-primitive if for some diagram (G , H , K − , K + ) for M the isotropy groups K ± and H are contained in some proper subgroup L of G. Such a non-primitiveaction is well known to be equivalent to the usual G action on G × L M L ,where M L is the cohomogeneity one manifold given by the group diagram(L , H , K − , K + ).A cohomogeneity one action of G on a closed topological manifold M is reducible if there is a proper normal subgroup of G that still acts by coho-mogeneity one with the same orbits. Conversely, there is a natural way ofextending an arbitrary cohomogeneity one action to an action by a possiblylarger group. Such extensions, called normal extensions , are described asfollows (cf. [15, Propositions 11–13] and [25, Section 1.11]). Let M be a co-homogeneity one topological manifold with group diagram (G , H , K − , K +1 )and let L be a compact connected subgroup of N (H ) ∩ N (K − ) ∩ N (K +1 ).Notice that L ∩ H is normal in L and let G = L / (L ∩ H ). We then definean action by G × G on M orbitwise by( ˆ g , [ l ]) · g (G ) x = ˆ g g l − (G ) x on each orbit G / (G ) x for (G ) x = H or K ± . OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 11
Notice that every reducible action is a normal extension of its restrictedaction. Therefore it is natural to consider non-reducible actions in the clas-sification. We will use the following result on reducible actions (cf. [15,Proposition 11] and [25, Proposition 1.12]) in the proof of Theorem F.
Proposition 4.5.
Let M be a cohomogeneity one manifold given by thegroup diagram (G , H , K − , K + ) and suppose that G × G with Proj (H) =G . Then the subaction of G × on M is also by cohomogeneity one,with the same orbits, and with isotropy groups K ± = K ± ∩ (G × and H = H ∩ (G × . The following two propositions give restrictions on the groups that mayact by cohomogeneity one on a closed topological manifold. The next propo-sition can be found in [4] for locally smooth actions. Here we prove it in theslightly more general case of topological actions on topological manifolds.
Proposition 4.6.
If a compact connected Lie group G acts (almost) effec-tively on a topological n -manifold with principal orbits of dimension k , then k ≤ dim G ≤ k ( k + 1) / .Proof. Let G / H be a principal orbit. Since dim G / H = k , the left inequalityis immediate. To verify the right inequality, it suffices to know that Gacts almost effectively on principal orbits, since then we can equip G / Hwith a G-invariant Riemannian metric and obtain a homomorphism ϕ :G → Isom(G / H) with finite kernel K. It then follows that G / K ∼ = ϕ (G) ≤ Isom(G / H). Since K is finite,dim G = dim G / K ≤ dim Isom(G / H) ≤ k ( k + 1)2 , where the last inequality follows from a well-known theorem of Myers andSteenrod [38]. To finish the proof, let us show that G acts almost effectivelyon principal orbits. As mentioned in Section 2, all principal isotropy groupsare conjugate to each other and conjugate to a subgroup of the singularisotropy groups. As a result, G acts almost effectively on the principalorbits. (cid:3) An argument as in the proof of [25, Proposition 1.18] yields the followinglemma:
Lemma 4.7.
Let M be a closed, simply connected topological manifold withan (almost) effective cohomogeneity one action of a compact Lie group G .Suppose that the following conditions hold: • G = G × T m and G is semisimple; • G acts non-reducibly; • at least one of the homogeneous spaces K ± / H is the Poincar´e sphere.Then, G = 1 and m ≤ . Moreover, if m = 1 , then one of the homogeneousspaces K ± / H is a circle. It is well known that every compact connected Lie group has a finite coverof the form G ss × T k , where G ss is semisimple and simply connected andT k is a torus. The classification of compact simply connected semisimpleLie groups is also well known. We refer the reader to [25, Section 1.24] fora list of such groups and their subgroups needed in our classification. Wealso have the following proposition (cf. [25, Proposition 1.25]), which givesfurther restriction on the groups. Proposition 4.8.
Let M be the cohomogeneity one topological manifoldgiven by the group diagram (G , H , K − , K + ) , where G acts non-reducibly on M . Suppose that G is the product of groups G = Π i (SU(4)) × Π j (G ) × Π k (Sp(2)) × Π l (SU(3)) × Π m (S ) × (S ) n . Then dim H ≤ i + 8 j + 6 k + 4 l + m. We conclude this section with an observation on groups acting on thePoincar´e homology sphere.
Lemma 4.9.
Let G be a compact Lie group of dimension at most . If SU(2) × G acts transitively on P , then G acts trivially on P and theisotropy group of the SU(2) × G action is I ∗ × G .Proof. Assume first that G is connected and let K be the kernel of theaction. By Theorem 2.2, (SU(2) × G) / K is isomorphic to SU(2). Hence,dim G = dim K. Since K is a normal and connected subgroup of SU(2) × G,Proj (K) is a normal connected subgroup of SU(2). Thus Proj (K) is trivial,as dim G ≤
2. As a result, K = 1 × G and H = I ∗ × G, where H is the principalisotropy group.Suppose now that G is not connected. In this case, SU(1) × (G) isconnected and acts transitively on P as a restriction of the action of SU(2) × G. Therefore, I ∗ × (G) ⊆ H ⊆ I ∗ × G. Connectedness of the quotient P = (SU(2) × G) / H gives H = I ∗ × G. (cid:3) Proof of Theorem F
Let G be a compact, connected Lie group acting almost effectively, non-reducibly and with cohomogeneity one on a closed, simply connected topo-logical n -manifold M n , 5 ≤ n ≤
7. We assume that the action is non-smoothable. Hence, by Theorem C, at least one of K ± / H, say K + / H, ishomeomorphic to P , the Poincar´e homology sphere P . We analyze eachdimension separately. Dimension 5.
By Proposition 4.6, we have 4 ≤ dim G ≤
10. Hence, by [25,1.24], G is one of (S ) m × T n , SU(3) × T n or Spin(5). From Proposition 4.7, wesee that n ≤
1. Since dim H = dim G −
4, Proposition 4.8 gives the possiblegroups. These are, up to a finite cover: S × S , S × S , SU(3) and Spin(5).On the other hand, since K + / H = P , dim K + = 3 + dim H = dim G − OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 13
Therefore, G = S × S is the only possibility, since the other groups do nothave a subgroup of the given dimension.Now we determine the group diagrams for S × S . From Proposition 4.7we have K − / H = S , so H = { (1 , } , K +0 = S × − = { ( e ipθ , e iqθ ) } .Thus, K + = S × Z k and H = I ∗ × Z k by Lemma 4.9. Since I ∗ × Z k ⊆ K − ⊆ N S × S (K − ), the group K − must be 1 × S , which yields the followingdiagram: (S × S , I ∗ × Z k , I ∗ × S , S × Z k ) . Hence, M is equivalent to P ∗ S , which is homeomorphic to the doublesuspension of P . By the Double Suspension Theorem, M is homeomorphicto S and the action is the one described in Example 2.3. Dimension 6.
Proceeding as in the 5-dimensional case, we find that 5 ≤ dim G ≤
15 and dim H = dim G −
5. It follows from Propositions 4.8 and4.7 that G must be one of S × S , S × S × S , SU(3), SU(3) × S , Sp(2),Sp(2) × S or Spin(6). On the other hand, since K + / H = P , we must havethat dim K + = dim G −
2. This dimension restriction rules out all possiblegroups except S × S .Now we determine the possible diagrams for G = S × S . First, supposethat K + / H = P and K − / H = S l , l ≥
1, so K +0 = S × S and H = { ( e ipθ , e iqθ ) } . If l ≥
2, G / K − is simply connected, and consequently K + is connected by Proposition 4.3. Therefore, K + = S × S , which actstransitively on P . By Lemma 4.9, S acts trivially on P and H = I ∗ × S .For l = 2, K − has to be I ∗ × S , and we have the following diagram:(S × S , I ∗ × S , I ∗ × S , S × S ) . Therefore, M is equivalent to P ∗ S with the action described in Example2.3.For l = 3 there is no subgroup of S × S containing H = I ∗ × S suchthat K − / H = S . Accordingly, no finite extension of the maximal torus ofS × S contains H, so K − / H = S .The only remaining case to be considered is when K − / H is also thePoincar´e sphere, i.e. K − / H = P . We now show that the only possiblediagram that can occur is(S × S , I ∗ × S , S × S , S × S ) . First, notice that since K +0 = S × S acts transitively on P , Proposition4.6 implies that I ∗ × S ⊆ H. Since K − is a 4-dimensional subgroup of Gcontaining H, it has to be S × S as well. On the other hand, since M issimply connected, H must be generated by H + and H − as in Lemma 4.4, soI ∗ × S ⊆ H ⊆ S × S . The fact that I ∗ is a maximal subgroup of S impliesthat H = I ∗ × S and, as a result, K − = K + = S × S . On the other hand,the following action on Susp P × S gives rise to the same diagrams as the S × S -action on M:(S × S ) × (Susp P × S ) → Susp P × S (( g, h ) , ([ x, t ] , y )) ([ gx, t ] , hyh − ) . Therefore, M is homeomorphic to Susp P × S , which is in turn homeomor-phic to S × S by Theorem G. Dimension 7.
By Proposition 4.6 we know that 6 ≤ dim G ≤
21 anddim H = dim G −
6. As before, Propositions 4.8 and 4.7 give us the possi-ble acting groups: S × S , S × S × S , SU(3), S × S × S , SU(3) × S ,Sp(2), SU(3) × S , Sp(2) × S , G , SU(4), SU(4) × S and Spin(7). Sincedim K + = dim G −
3, we easily rule out most of the groups and the onlypossible groups remaining are S × S , S × S × S and S × S × S . Weanalyze each case separately. G = S × S . Assume first that K + / H = P and K − / H = S l , l ≥
2. ByProposition 4.3, K + is connected. Thus, we have K + = S × , ∆S , andH = I ∗ × , ∆I ∗ , respectively. On the other hand, since dim H = 0, andK − / H is simply connected, K − is also simply connected. A glance at thesubgroups of S × S shows that only S × S and its 3-dimensional subgroupsare simply connected. Since (S × S ) / H is not a sphere, K − is necessarily 3-dimensional. Therefore K − is one of S ×
1, 1 × S or ∆S . It is apparent that(S × / H is not a sphere, so we are left with the two other cases. We easilyrule out the the case K − = ∆S , since otherwise K − , being a subgroup of N (∆S ) = ± ∆S , has at most two components, while π (K − ) = I ∗ . Hence,K − = I ∗ × S , and we have the two following diagrams:(S × S , I ∗ × , I ∗ × S , S × × S , ∆I ∗ , I ∗ × S , ∆S ) , with the following actions, respectively:(S × S ) × ( P ∗ S ) → P ∗ S (( g, h ) , ([ x, y, t ] , y )) [ gx, hy, t ]and (S × S ) × ( P ∗ S ) → P ∗ S (( g, h ) , ([ x, y, t ] , y )) [ gx, gyh − , t ] . Now let K − / H = S , so K − = { ( e ipθ , e iqθ ) } . SinceI ∗ × ⊆ K − ⊆ N S × S (K − ) , K − must be 1 × S . Hence we have the diagram(S × S , I ∗ × Z k , I ∗ × S , S × Z k ) . OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 15
In this case, the action is non-primitive. In fact, for L = S × S ⊆ S × S ,we have the following diagram, which is the diagram of the cohomogeneityone action of S × S on P ∗ S already described in dimension 5:(S × S , I ∗ × Z k , I ∗ × S , S × Z k ) . Therefore, M is an S -bundle over S .Finally, suppose that K − / H = P . If K +0 = S ×
1, then there exists asubgroup of H, say ˜H, such that ˜H = I ∗ × +0 / ˜H = P . Now we havetwo possibilities: either I ∗ × ⊆ K − or I ∗ × * K − . Assume first thatI ∗ × ⊆ K − . Thus K − = S ×
1. On the other hand, by Proposition 4.4, His generated by H ∩ (S × ∗ ×
1. Therefore,we have the following group diagram:(S × S , I ∗ × , S × , S × . (5.1)On the other hand, Susp P × S admits a cohomogeneity one action givenby (S × S ) × (Susp P × S ) → Susp P × S (( g, h ) , ([ x, t ] , y )) ([ gx, t ] , hy ) , which gives diagram (5.1) above. Thus M is equivariantly homeomorphic toSusp P × S .Now assume that I ∗ × * K − . Hence K − has to be 1 × S and consequently1 × I ∗ ⊆ H. Therefore, the following diagram appears:(S × S , I ∗ × I ∗ , I ∗ × S , S × I ∗ ) . As a result, M is equivalent to P ∗ P with the action given by(S × S ) × ( P ∗ P ) → P ∗ P (( g, h ) , ([ x, y, t ])) [ gx, hy, t ] . Now let K +0 = ∆S , so that ∆I ∗ ⊆ H ⊂ K − . Notice that by the classifi-cation of transitive and almost effective actions on P , the group ∆S actson P in the natural way. As a result, K − = ∆S . On the other hand,by Proposition 4.4, H is generated by ∆S ∩ H, which is a finite diagonalsubgroup of S × S containing ∆I ∗ . Let ∆Γ = ∆S ∩ H. Therefore, I ∗ ⊆ Γ.Then Γ must be I ∗ , for I ∗ is a maximal subgroup of S . Hence H = ∆I ∗ andwe have the following diagram:(S × S , ∆I ∗ , ∆S , ∆S ) . Therefore, M is equivariantly homeomorphic to Susp P × S with theaction (S × S ) × (Susp P × S ) → Susp P × S (( g, h ) , ([ x, t ] , y )) ([ gx, t ] , gyh − ) . We conclude that M is homotopy equivalent to S × S . G = S × S × S . In this case dim H = 1 and K − / H = S by Proposition4.7. Therefore, K − would be a 2-torus subgroup of G, and K +0 ⊆ S × S × +0 = S × S ×
1, and consequently, I ∗ × S × ⊆ H.Since I ∗ × S × ⊆ K − ⊆ N (K − ) = N G (T ), K − has to be 1 × T . Therefore,we have the following diagram:(S × S × S , I ∗ × S × Z k , I ∗ × T , S × S × Z k ) . This action is a non-primitive action. Indeed, for L = S × T ⊆ S × S × S ,we have diagram(S × T , I ∗ × S × Z k , I ∗ × T , S × S × Z k ) , which is a non-effective extension of the cohomogeneity one almost effectiveaction of S × S on P ∗ S described in dimension 5. Thus M is a P ∗ S -fiber bundle on (S × S × S ) / (S × T ). Therefore, M is homeomorphic toan S -bundle over S . G = S × S × S . We show that no non-reducible diagram for this caseoccurs. In fact, we will show that all possible diagrams in this case reduceto the diagrams of the case G = S × S . First, note that dim H = 3 anddim K + = 6. Recall that Proj l , l = 1 , ,
3, denotes projection onto the l -thfactor of S × S × S . Since we assume that the action is non-reducible,it follows from Proposition 4.5 that Proj l (H ), l = 1 , ,
3, is not S . Onthe other hand, Proj l (H ) cannot be trivial. Otherwise, H would be a 3-dimensional subgroup of S × S and H must project onto one of the factors.This yields a reducible action, which contradicts the assumption that theaction is non-reducible. Thus, Proj l (H ) = S , for l = 1 , ,
3. An inspectionof the subgroups of S × S × S shows that none of the 6-dimensionalsubgroups of S × S × S contains H. Therefore, no non-reducible actioncan occur. (cid:3) Proof of Theorem G
To prove that Susp P × S is homeomorphic to S × S we will use aspecial instance of the classification of closed, oriented, simply connected6-dimensional topological manifolds with torsion free homology. This clas-sification follows from work of Wall [47], Jupp [27], and Zhubr [53]. We firstrecall the following theorem of Jupp (cf. [27, Theorem 1]). Theorem 6.1 (Jupp) . Orientation-preserving homeomorphism classes ofclosed, oriented, -connected -manifolds M with torsion free homology cor-respond bijectively with isomorphism classes of systems of invariants: • r = rank H (M , Z ) , a nonnegative integer; • H = H (M , Z ) , a finitely generated free abelian group; OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 17 • µ : H ⊕ H ⊕ H → Z , a symmetric trilinear form given by the cupproduct evaluated on the orientation class; • p (M) ∈ H (M , Z ) , the first Pontryagin class; • w (M) ∈ H (M , Z ) , the second Stiefel–Whitney class; • ∆(M) ∈ H (M , Z ) , the Kirby–Siebenmann class.The systems of invariants must satisfy the equation µ (2 x + W, x + W, x + W ) ≡ ( p + 24 T )(2 x + W ) (mod 48) for all x ∈ H , where W ∈ H , T ∈ Hom Z ( H, Z ) reduce mod 2 to w , ∆ . Sucha manifold has a smooth (or PL) structure if and only if ∆(M) = 0 , and thesmooth structure is unique. We point out that for closed, oriented, simply connected 6-dimensionaltopological manifolds the first Pontryagin class is always integral (see [27]).Okonek and Van de Ven [40, pp. 302–303] have summarized the classificationin the special case where r = 0 and H = Z . This is the case that is relevantto us, since for Susp P × S we have r = 0 and H = Z . We now recall theseresults. Proposition 6.2 ([40]) . Let M be as in Theorem 6.1 with r = 0 and H = Z .The system of invariants introduced in Theorem 6.1 can be identified with -tuples ( ¯ W , ¯ T , d, p ) ∈ Z × Z × Z × Z , where the degree d corresponds tothe cubic form µ . Such a -tuple is admissible if and only if (6.1) d (2 x + W ) ≡ ( p + 24 T )(2 x + W ) (mod 48) , for every integer x . Definition 6.3 ([40]) . Two admissible 4-tuples ( ¯
W , ¯ T , d, p ) and ( ¯ W ′ , ¯ T ′ , d ′ , p ′ )are equivalent if and only if ¯ W = ¯ W ′ , ¯ T = ¯ T ′ and ( d ′ , p ′ ) = ± ( d, p ). Proposition 6.4 ([40]) . The assignment X → ( ¯ W , ¯ T , d, p ) . induces a - correspondence between oriented homeomorphism classes ofclosed, oriented, simply connected -dimensional topological manifolds withtorsion free homology and equivalence classes of admissible systems of in-variants, where ( ¯ W , ¯ T , d, p ) is a normalized -tuple, i. e. d ≥ , and p ≥ if d = 0 . Definition 6.5 ([40]) . Two normalized 4-tuples ( ¯
W , ¯ T , d, p ), ( ¯ W ′ , ¯ T ′ , d ′ , p ′ )are weakly equivalent if and only if d ′ = d , ¯ W ′ = ¯ W , p + 24 T ≡ p ′ + 24 T ′ (mod 48) if d ≡ p ≡ p ′ (mod 24) if d ≡ Proposition 6.6 ([40]) . The assignment X → ( ¯ W , ¯ T , d, p ) . induces a - correspondence between homotopy classes of simply connected,closed, oriented, -dimensional topological manifolds with torsion free ho-mology and weak equivalence classes of admissible systems of invariants. Now we use the above results to prove that Susp P × S is homeomorphicto S × S . Let ( ¯ W , ¯ T , d, p ) be the admissible 4- tuple of M = Susp P × S asin Proposition 6.2. Since (0 , , ,
0) is the admissible 4-tuple of S × S , and Mis homotopy equivalent to S × S , Proposition 6.6 implies that d = 0, ¯ W = 0,and p + 24 T ≡ p ≡ p = 24 k , for k = 0 , ± , ± , . . . Hence, T ≡ p/
24 (mod 2); therefore, it suffices to compute the first Pontryaginclass of M. If p = 0, then M is homeomorphic to S × S by Proposition6.4; if p ≡ p ≡
24 (mod 48), M isnon-smoothable by Theorem 6.1.In the remainder of this section, we compute the first Pontryagin class p of Susp P × S and show that p = 0. Note that rational Pontryaginclasses are defined for topological manifolds and, more generally, for rationalhomology manifolds. One can also define Hirzebruch l -classes so that l = p (see [22, 28, 32, 33, 46, 51]). The l -classes are multiplicative, i.e. givenany two rational homology manifolds X and Y, one has l i (X × Y) = X p + q = i l p (X) l q (Y) . Thus in our case, since Susp P is a rational homology manifold, we have l (Susp P × S ) = l (Susp P ) + l ( S )= l (Susp P ) . Therefore, to find p (Susp P × S ), it suffices to find l (Susp P ).First observe that Susp P ∼ = S / I ∗ , where I ∗ acts in the obvious wayon a round S by orientation preserving diffeomorphisms. This observationallows us to use a formula of Zagier [52] to compute the Hirzebruch L -class of the quotient space X / G of a closed oriented smooth manifold X bythe orientation preserving diffeomorphism action of a finite group G. SinceSusp P is 4-dimensional, the top dimensional component of L is l and,by [52], l equals the signature. Thus, to compute l (Susp P ), we needonly compute Sign(Susp P ). We do this using results of Atiyah and Singer[2] and of Atiyah and Bott [1], which we briefly outline in the followingparagraphs.Let X be a closed, oriented, smooth manifold and G a finite group actingon X by orientation-preserving diffeomorphisms. Let π : X → X / G OHOMOGENEITY ONE TOPOLOGICAL MANIFOLDS REVISITED 19 be the projection map of X onto the orbit space X / G. As mentioned above,the top dimensional component of L ( X/ G) is Sign(X / G). By the Atiyah-Singer G-equivariant signature theorem (see [2, Section 6] or [23]), the sig-nature of X / G is given bySign(X / G) = 1 | G | X g ∈ G Sign( g, X) . (6.2)In our particular case, where G = I ∗ acts on X = S with only two isolatedfixed points, Sign( g, X) is given by a signature formula of Atiyah and Bott[1, Theorem 6.27], which we state as Theorem 6.7 below. Before quotingthe theorem, we recall some notation.Let f : X → X be an isometry of a compact, oriented even-dimensionalRiemannian manifold X and p be a fixed point of f . Consider the differential df p : T p X → T p X . Because f is an isometry of X , df p will be an isometry of T p X. Hence, onemay decompose T p X into a direct sum of orthogonal 2-planes T p X = E ⊕ E ⊕ ... ⊕ E n , which are stable under df p . Let ( e k , e ′ k ) be an orthogonal basis of E k . Wemay choose ( e k , e ′ k ) so that v p ( e ∧ e ′ ∧ ... ∧ e n ∧ e ′ n ) = 1 , where v is the volume form of X. Relative to such a basis df P is then givenby rotations by angles θ k in E k . That is, df p e k = cos θ k e k + sin θ k e ′ k df p e ′ k = − sin θ k e k + cos θ k e ′ k The resulting set of angles { θ k } is called a coherent system for df p . Theorem 6.7 (Atiyah and Bott) . Let f : X n → X n be an isometry ofthe compact oriented even dimensional Riemannian manifold X . Assumefurther that f has only isolated fixed points { p } , and let θ pk be a system ofcoherent angles for df p . Then the signature of f is given by Sign( f, X) = X p i − n Y k cot( θ pk / . We now use Theorem 6.7 to compute Sign( g, X ) for each g ∈ I ∗ andrecover Sign( S / I ∗ ) via equation (6.2). For non-trivial g ∈ G , the fixedpoint set X g has two elements, say { p, q } . Let { α, β } be the coherent systemfor p . Then the coherent system for q will be {− α, β } , so Sign( g, S ) = 0.On the other hand, Sign( e, S ) = Sign( S ) = 0 and Sign( − e, S ) = 0 byTheorem 6.7. As a result Sign( S / I ∗ ) = 0. Hence the top component of L ( S / I ∗ ) is zero. Since the top component of L ( S / I ∗ ) is 3 p ( S / I ∗ ) =3 p ( S / I ∗ × S ), we conclude that the first Pontryagin class of Susp P × S is zero. Therefore, Susp P × S is homeomorphic to S × S . (cid:3) References [1] Atiyah, M.F. and Bott, R.,
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Institut f¨ur Algebra und Geometrie, Karlsruher Institutf¨ur Technologie (KIT), Karlsruhe, Germany.
E-mail address : [email protected] (M. Zarei) Department of Pure Mathematics, Faculty of Mathematical Sci-ences, Tarbiat Modares University, Tehran, Iran and Institut f¨ur Algebraund Geometrie, Karlsruher Institut f¨ur Technologie (KIT), Karlsruhe, Ger-many.
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