Cohomogeneity one Alexandrov spaces in low dimensions

We classify closed, simply-connected cohomogeneity-one Alexandrov spaces in dimensions $5$, $6$ and $7$. We show that every closed, simply-connected smooth $n$-orbifold, $2\leq n\leq 7$ with a cohomogeneity one action is equivariantly homeomorphic to a smooth good orbifold of cohomogeneity one.


Introduction
Alexandrov spaces (with curvature bounded from below) are complete length spaces with a lower curvature bound in the triangle comparison sense; they generalize Riemannian manifolds with a uniform lower sectional curvature bound. Instances of Alexandrov spaces include Riemannian orbifolds (with a lower sectional curvature bound), orbit spaces of isometric actions of compact Lie groups on Riemannian manifolds with sectional curvature bounded below, or Gromov-Hausdorff limits of sequences of n-dimensional Riemannian manifolds with a uniform lower bound on the sectional curvature.
In dimensions two and three, the basic topological properties of Alexandrov spaces are fairly well-understood. Indeed, two-dimensional Alexandrov spaces are topological two-manifolds, possibly with boundary (see [8,Corollary 10.10.3]); closed (i.e. compact and without boundary) threedimensional Alexandrov spaces are either topological three-manifolds or are homeomorphic to quotients of smooth three-manifolds by orientation reversing involutions with isolated fixed points, and closed four-dimensional Alexandrov spaces are homeomorphic to orbifolds (see [16]). In higher dimensions, however, similar general results are lacking and considering spaces with large isometry groups provides a systematic way of studying Alexandrov spaces. This yields manageable families of spaces with a reasonably simple structure but flexible enough to generate interesting examples on which to test conjectures or carry out geometric constructions. This framework has been successfully used in the smooth category to construct, for instance, Riemannian manifolds satisfying given geometric conditions, such as positive Ricci or sectional curvature (see [12,20,21,22]).
One of the measures for the size of an isometric action of a compact Lie group G on an Alexandrov space X is its cohomogeneity, defined as the dimension of the orbit space X/G. This quotient space, when equipped with the orbital distance metric, is itself an Alexandrov space with the same lower curvature bound as X. From the point of view of cohomogeneity, transitive actions are the largest one can have. These actions preclude any topological or metric singularities: by the work of Berestovskiȋ [5], homogeneous Alexandrov spaces are isometric to Riemannian manifolds. The next simplest case to consider is when the orbit space is one-dimensional, i.e. when the action is of cohomogeneity one. Alexandrov spaces of cohomogeneity one were first studied in [17], where the authors obtained a structure result and classified these spaces (up to equivariant homeomorphism) in dimensions 4 and below. Simple instances of these spaces are, for example, spherical suspensions of homogeneous spaces X with sectional curvature bounded below by 1, equipped with the canonical suspension action of the transitive action on X.
It was shown in [17,Proposition 5] that the orbit space of an isometric cohomogeneity one Gaction on a closed, simply-connected Alexandrov space X is homeomorphic to a closed interval [−1, 1] and there exist compact Lie subgroups H and K ± of G such that H ⊆ K ± ⊆ G and K ± /H are positively curved homogeneous spaces. The group H is the principal isotropy group of the action and the groups K ± are isotropy groups of points in the orbits corresponding to the boundary points ±1 of the orbit space. The groups K ± are called non-principal isotropy groups and the orbits G/K ± are called non-principal orbits. We collect these groups in the quadruple (G, H, K − , K + ), called the group diagram of the action. The space X is the union of two bundles whose fibers are cones over the positively curved homogeneous spaces K ± /H. Conversely, any diagram (G, H, K − , K + ), with K ± /H positively curved homogeneous spaces, gives rise to a cohomogeneity one Alexandrov space. In the present article we complete the classification of these spaces in dimensions 5, 6 and 7 (assuming simply-connectedness), identify which of these spaces are smooth orbifolds, and show that the underlying space of such an orbifold is equivariantly homeomorphic to a good orbifold.
Closed, smooth manifolds of cohomogeneity one have been classified by Mostert [32,31] and Neumann [33] in dimensions 2 and 3, and by Parker [35] in dimension 4, without assuming any restrictions on the fundamental group. In dimensions 5, 6 and 7, Hoelscher [25] obtained the equivariant classification of closed smooth cohomogeneity one manifolds assuming simply-connectedness. It is well-known that these manifolds admit invariant Riemannian metrics and are therefore Alexandrov spaces of cohomogeneity one. In the topological category, the corresponding classification results in dimensions at most 7 follow from combining the smooth classification with the classification of closed, simply-connected cohomogeneity one topological manifolds with a non-smooth cohomogeneity one action in dimensions at most 7, obtained in [18]. It was also shown in [18] that closed, simply-connected cohomogeneity one topological manifolds decompose as double cone bundles whose fibers are cones over spheres or the Poincaré homology sphere, and hence they admit invariant Alexandrov metrics. Our first main result completes the equivariant classification of closed, simply-connected Alexandrov spaces in dimensions 5, 6 and 7: Theorem A. Let X be a closed, simply-connected Alexandrov space of dimension 5, 6 or 7 with an (almost) effective cohomogeneity one isometric action of a compact connected Lie group. If the action is not equivalent to a smooth action on a smooth manifold, then it is given by one of the diagrams in Table 4.2 if dim X = 5, Table 4.3 if dim X = 6, or Tables 4.4-4.8 if dim X = 7.
We point out that the diagrams (G, H, K − , K + ) in Tables 4.2-4.8 contain, as particular cases, the diagrams of non-smoothable cohomogeneity one actions on closed, simply-connected topological manifolds in [18]; in this special situation the positively curved homogeneous spaces K ± /H are either spheres or the Poincaré homology sphere. Compared to the smooth and topological cases, the number of closed, simply-connected cohomogeneity one Alexandrov spaces that are not manifolds increases substantially, due to the fact that at least one of the positively curved homogeneous spaces K ± /H is no longer a sphere or the Poincaré homology sphere. In many cases, we can identify the spaces in Theorem A as joins, suspensions, products or bundles of familiar spaces. Moreover, many of the spaces in Theorem A are equivariantly homeomorphic to smooth cohomogeneity one orbifolds. Indeed, they admit a double cone bundle decomposition, where the cones are taken over spherical homogeneous spaces; this structure characterizes closed, smooth orbifolds of cohomogeneity one whose orbit space is a closed interval (see [19]). A natural question, then, is whether there exists a good orbifold structure on the underlying topological space |Q| of a given orbifold group diagram, that is, whether |Q| is equivariantly homeomorphic to a quotient of a cohomogeneity one smooth manifold. Our second main result gives a positive answer to this question in dimensions at most 7: Theorem B. Every closed, simply-connected smooth orbifold of dimension at most 7 with an (almost) effective cohomogeneity one smooth action is equivariantly homeomorphic to a good cohomogeneity one smooth orbifold.
As in the smooth and topological cases, the proof of Theorem A follows from a case-by-case analysis of the possible group actions. Using dimension restrictions, one first determines the possible groups that can act. One then considers each group action individually, taking into account the fact that the groups must satisfy restrictions imposed by the fact that the homogeneous spaces K ± /H are positively curved. In this way, one obtains all the possible diagrams (G, H, K − , K + ), which determine the equivariant type of the Alexandrov space. Recognition results for specific types of actions help us identify the topological type of the space. To prove Theorem B, we first compute the orbifold fundamental group in each case. We then find a manifold with a cohomogeneity one action and a commuting action of the orbifold fundamental group whose quotient induces the diagram of the cohomogeneity one orbifold under consideration.
Our article is divided as follows. In Section 2 we collect background material on cohomogeneity one Alexandrov spaces and prove some results we will use in the proof of Theorem A. The proof of this theorem is contained in Section 3. Finally, in Section 4 we recall some basic facts on orbifolds and prove Theorem B.
Acknowledgements. The authors would like to thank Wilderich Tuschmann at the Karlsruher Institut für Technologie (KIT), Alexander Lytchak and Christian Lange at the Universität Köln, and Burkhard Wilking at the Universität Münster for their hospitality and for helpful conversations.

Preliminaries
In this section, we collect some background material which we will use in the proof of Theorem A.
2.1. Group actions. Let X be a topological space and let x be a point in X. Given a topological (left) action G × X → X of a Lie group G, we let 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 that G(x) ≈ G/G x . We will denote the orbit space of the action by X/G and let π : X → X/G be the orbit projection map. The (ineffective) kernel of the action is the subgroup K = x∈X G x . The action is effective if K is the trivial subgroup {e} of G; the action is almost effective if K is finite.
We will say that two G-spaces are equivalent if they are equivariantly homeomorphic. From now on, we will suppose that G is compact and connected, and assume that the reader is familiar with the basic notions of compact transformation groups (see, for example, Bredon [7]). We will assume all spaces to be connected, unless stated otherwise.

Alexandrov spaces.
A finite (Hausdorff) dimensional length space (X, d) has curvature bounded below by k if every point x ∈ X has a neighborhood U such that, for any collection 3 of four different points (x 0 , x 1 , x 2 , x 3 ) in U , the following condition holds: Here, ∠ x i x j (k), called the comparison angle, is the angle at x 0 (k) in the geodesic triangle in M 2 k , the simply-connected Riemannian 2-manifold with constant curvature k, with vertices (x 0 (k), x i (k), x j (k)), which are the isometric images of (x 0 , x i , x j ). An Alexandrov space is a complete length space with finite Hausdorff dimension and curvature bounded below by k for some k ∈ R. Recall that the Hausdorff dimension of an Alexandrov space is an integer and is equal to its topological dimension. The space of directions of a general Alexandrov space X n of dimension n at a point x is, by definition, the completion of the space of geodesic directions at x. We will denote it by Σ x X n . It is a compact Alexandrov space of dimension n − 1 with curvature bounded below by 1. We refer the reader to [8,9] for the basic results on Alexandrov geometry. We will say that an Alexandrov space is closed if it is compact and has no boundary.
2.3. Group actions on Alexandrov spaces. Let X be an n-dimensional Alexandrov space. Fukaya and Yamaguchi proved in [14, Theorem 1.1] that Isom(X), the isometry group of X, is a Lie group. Moreover, Isom(X) is compact, if X is compact and connected (see [10,p. 370,Satz I] or [29,Corollary 4.10 and its proof in pp. 46-50]). As in the Riemannian case, the maximal dimension of Isom(X) is n(n + 1)/2 and, if equality holds, X must be isometric to a Riemannian manifold (see [15,Theorems 3.1 and 4.1]).
As for locally smooth actions (see [7, Ch. IV, Section 3]), for an isometric action of a compact Lie group G on an Alexandrov space X there also exists a maximal orbit type G/H (see [15,Theorem 2.2]). This orbit type is the principal orbit type and orbits of this type are called principal orbits. A non-principal orbit is exceptional if it has the same dimension as a principal orbit.
The structure of the space of directions in the presence of an isometric action is given by the following proposition. . Let X be an Alexandrov space with an isometric G-action and fix x ∈ X with dim(G/G x ) > 0. Let S x ⊆ Σ x X be the unit tangent space to the orbit G(x) ≃ G/G x , and let S ⊥ x = {v ∈ Σ x X : ∠(v, w) = π/2 for all w ∈ S x } be the set of normal directions to S x . Then the following hold: (1) The set S ⊥ x is a compact, totally geodesic Alexandrov subspace of Σ x X with curvature bounded below by 1, and the space of directions Σ x X is isometric to the join S x * S ⊥ x with the standard join metric.
(2) Either S ⊥ x is connected or it contains exactly two points at distance π. 2.4. Alexandrov spaces of cohomogeneity one. In this subsection we collect basic facts on cohomogeneity one Alexandrov spaces and prove some preliminary results that we will use in the proof of Theorem A. For cohomogeneity one actions on smooth or topological manifolds, we refer the reader to [25] or [18], respectively. Definition 2.2. Let X be a connected n-dimensional Alexandrov space with an isometric action of a compact connected Lie group G. The action is of cohomogeneity one if the orbit space is onedimensional or, equivalently, if there exists an orbit of dimension n − 1. A connected Alexandrov space with an isometric action of cohomogeneity one is a cohomogeneity one Alexandrov space.
Cohomogeneity one Alexandrov spaces were first studied in [17]. Recall that the orbit space X/G of an Alexandrov space X by an isometric action of a group G with closed orbits is again an Alexandrov space (see [8,Proposition 10.2.4]). Since one-dimensional Alexandrov spaces are topological manifolds, the orbit space of a cohomogeneity one Alexandrov space is homeomorphic to a connected 1-manifold (possibly with boundary). When the orbit space is homeomorphic to [−1, 1], we denote the isotropy groups corresponding to a point in the orbit mapped to ±1 by K ± . 4 By the Isotropy Lemma (see [15,Lemma 2.1]) and the fact that principal orbits are open and dense, the orbits that project to the interior (−1, 1) of the orbit space all have the same isotropy group H (up to conjugacy) and H is a subgroup of K ± . The subgroup H is the principal isotropy group of the action and the corresponding orbits are the principal orbits. Let us now show that H is a proper subgroup of K ± . It suffices to show that if dim K ± = dim H, then K ± = H. Observe first that, in this case, S ⊥ = S 0 with a transitive action of K ± with isotropy H. Hence K ± /H = S 0 , which shows that K ± = H. We call the orbits mapped to ±1 non-principal orbits.
Let X be a closed cohomogeneity one Alexandrov G-space. Since the orbit space X/G must be a compact one-manifold, it must be either a circle or a closed interval. When X/G is a circle, X is equivariantly homeomorphic to a fiber bundle over S 1 with fiber a principal orbit G/H. In particular, X is a smooth manifold (see [17,Theorem A]). Since we are interested in non-manifold Alexandrov spaces, we will focus our attention on the case where X/G is a compact interval.
A cohomogeneity one G-action on a closed Alexandrov space whose orbit space is an interval determines a group diagram (G, H, where K ± are isotropy subgroups at the non-principal orbits corresponding to the endpoints of the interval, and H is the principal isotropy group of the action. The following theorem determines the structure of closed cohomogeneity-one Alexandrov spaces with orbit space an interval.
. Let X be a closed Alexandrov space with an effective isometric G-action of cohomogeneity one with principal isotropy H and orbit space homeomorphic to [−1, 1]. Then X is the union of two fiber bundles over the two singular orbits whose fibers are cones over positively curved homogeneous spaces, that is, The group diagram of the action is given by (G, H, K − , K + ), where K ± /H are positively curved homogeneous spaces. Conversely, a group diagram (G, H, K − , K + ), where K ± /H are positively curved homogeneous spaces, determines a cohomogeneity one Alexandrov space.
We will use the following proposition to identify equivalent actions.
Proposition 2.4 ([17, Proposition 9]). If a cohomogeneity one Alexandrov space is given by a group diagram (G, H, K − , K + ), then any of the following operations on the group diagram will result in an equivalent Alexandrov space: (1) Switching K − and K + , (2) Conjugating each group in the diagram by the same element of G, Conversely, the group diagrams for two equivalent cohomogeneity one, closed Alexandrov space must be mapped to each other by some combination of these three operations.
Let G be a compact connected Lie group acting on a closed Alexandrov space X with cohomogeneity one and let π : X → X/G = [0, 1] be the projection map. A minimizing geodesic γ : [0, d] → X between non-principal orbits has the following properties (see [15,Lemma 2.1]): • it goes through all principal orbits, • for all t ∈ (0, d), H = G c(t) ⊂ G c(0) , G c(d) , and • the direction of γ is horizontal.
We set K − = G c(0) and K + = G c(d) . We call such a geodesic a normal geodesic. Definition 2.5. We say that the cohomogeneity one Alexandrov space X is non-primitive if it has some group diagram representation (G, H, K − , K + ) for which there is a proper connected closed subgroup L ⊂ G with K ± ⊂ L. It then follows that (L, H, K − , K + ) is a group diagram which determines some cohomogeneity one Alexandrov space Y . Proposition 2.6 ([17, p. 96]). Take a non-primitive cohomogeneity one Alexandrov space X with L and Y as in Definition 2.5. Then X is equivalent to (G × Y )/L, where L acts on G × Y by l · (g, y) = (gl −1 , ly). Hence, there is a fiber bundle Definition 2.7. A cohomogeneity one action of a Lie group G on an Alexandrov space X is called reducible if there is a proper closed normal subgroup of G that acts on X with the same orbits.
We now recall the following results which describe the reduction or extension of certain cohomogeneity one actions (cf. [25,Section 1.11] and [17,Section 2]). These results show why it is natural to consider only non-reducible actions. Proposition 11]). Let X be the cohomogeneity one Alexandrov space given by the group diagram (G, H, K − , K + ) and suppose that G = G 1 × G 2 with Proj 2 (H) = G 2 . Then the subaction of G 1 × 1 on X is also of cohomogeneity one, has the same orbits as the action of G, and has isotropy groups K ± 1 = K ± ∩ (G 1 × 1) and H 1 = H ∩ (G 1 × 1). For the next proposition we will need the concept of a normal extension, which we now recall. Definition 2.9. Let X be a cohomogeneity one Alexandrov space with group diagram (G 1 , H 1 , K − , K + ) and let L be a compact, connected subgroup of N ( Observe that the subgroup L ∩ H 1 is normal in L and define G 2 := L/(L ∩ H 1 ). We can then define an action of G 1 × G 2 on X orbitwise by letting Such an extension is called a normal extension of G 1 .
Proposition 2.10 ([17, Proposition 12]). A normal extension of G 1 describes a cohomogeneity one action of G := G 1 × G 2 on X with the same orbits as G 1 and with group diagram Proposition 2.11 ([17,Proposition 13]). For X as in Proposition 2.8, the action by G = G 1 × G 2 occurs as the normal extension of the reduced action of G 1 × 1 on X.
By the above propositions, it is natural then to consider only non-reducible actions in the classification.
2.5. Further tools. The following proposition, whose proof is as in [25,Proposition 1.25], yields bounds on the dimension of a Lie group acting by cohomogeneity one in terms of the dimension of a principal isotropy subgroup. Proposition 2.12. Let X be a closed Alexandrov space of cohomogeneity one with group diagram (G, H, K − , K + ). Suppose that G acts non-reducibly on X and that G is the product of groups Then dim(H) ≤ 10i + 8j + 6k + 4l + m. 6 We now state some useful results on the fundamental group of cohomogeneity one Alexandrov spaces. Their proofs follow as in the manifold case (see [25,Section 1.6] and [18,Section 4] . Let X be the closed cohomogeneity one Alexandrov space given by the group diagram (G, H, Corollary 2.14 ( [18,Corollary 4.4]). Let X be the closed simply-connected cohomogeneity one Alexandrov space given by the group diagram (G, H, K − , K + ), with dim(K ± /H) ≥ 1, and K − /H = S l , for l ≥ 2. Then G/K + is simply-connected and, if G is connected, then K + is also connected. . Let X be the closed cohomogeneity one Alexandrov space given by the group diagram (G, H, K − , K + ). Denote H ± = H ∩ K ± 0 , and let α i ± : [0, 1] → K ± 0 be curves that generate π 1 (K ± /H), with α i ± (0) = 1 ∈ G. The space X is simply-connected if and only if (1) H is generated as a subgroup by H − and H + , and (2) α i − and α i + generate π 1 (G/H 0 ). We will use the following results on transitive actions. Lemma 2.16 (cf. [18,Lemma 4.11]). Let G 1 be a compact, connected, simply-connected, simple Lie group of dimension n. Assume that G 1 is, up to a finite cover, the only Lie group that acts transitively and (almost) effectively on a manifold M with isotropy group H. Let G 2 be a compact, connected Lie group of dimension at most n − 1. If G 1 × G 2 acts transitively on M , then the following hold: (1) The G 2 factor acts trivially on M and (2) the isotropy group K of the (G 1 × G 2 )-action is H × G 2 .
Proof. Let L ⊆ G 1 ×G 2 be the kernel of the action of G 1 ×G 2 on M . Then (G 1 ×G 2 )/L is isomorphic to G 1 . Hence, dim G 2 = dim L. Since L is a normal and connected subgroup of G 1 × G 2 , Proj 1 (L) is a normal connected subgroup of G 1 . Thus Proj 1 (L) is trivial, since dim G 2 ≤ n − 1. As a result, L = 1 × G 2 and K = H × G 2 . The following two results give restrictions on the groups that may act by cohomogeneity one on a closed Alexandrov space. The next proposition can be found in [25,Proposition 1.19] for smooth actions. It was proven in [18] in the slightly more general case of topological actions on topological manifolds. The proof for Alexandrov spaces follows as in the topological case [18,Proposition 4.7], taking into account that, by the Principal Orbit Theorem for Alexandrov spaces [15,Theorem 2.2], all principal isotropy groups are conjugate to each other and conjugate to a subgroup of non-principal isotropy groups.
• G = G 1 × T m and G 1 is semisimple; • G acts non-reducibly; • at least one of the homogeneous spaces K ± /H is other than standard spheres.
Then, G 1 = 1 and m ≤ 1. Moreover, if m = 1, then one of the homogeneous spaces K ± /H, say K − /H, is a circle and K − 0 = H 0 · S − , where S − is a circle group with Proj 2 (S − ) = T 1 and K + 0 ⊂ G 1 × 1. 2.6. Special actions and recognition results. In this subsection we list some special types of cohomogeneity one actions and prove some recognition results that will allow us to identify such actions (cf. [25, 1.21]). Definition 2.20 (Product action). Let G 1 and G 2 be Lie groups such that G 1 acts on an Alexandrov space X with cohomogeneity one and G 2 acts on a homogeneous space G 2 /L transitively. We call the natural action of G 1 × G 2 on X × G 2 /L given by the product action of G 1 × G 2 .
Proposition 2.21. Suppose that G 1 acts on an Alexandrov space X with cohomogeneity one and with group diagram (G 1 , H, K − , K + ), and G 2 acts transitively on the homogeneous space G 2 /L. Then the product action of G 1 × G 2 on X × G 2 /L is of cohomogeneity one with group diagram Conversely, a cohomogeneity one action of G 1 × G 2 with the above group diagram, and G 1 /K ± positively curved homogeneous spaces, is equivalent to a product action of G 1 × G 2 on X × G 2 /L, where X is the cohomogeneity one Alexandrov space determined by the diagram (G 1 , H, K − , K + ).
Proof. It is clear that the product action of G 1 × G 2 on X × G 2 /L is of cohomogeneity one. Now we prove that its group diagram is as in (2.1). Let γ be a normal geodesic between the non-principal orbits G 1 /K ± in X giving the group diagram (G 1 , H, K − , K + ). If we fix a G 2 -invariant metric on G 2 /L, then, in the product metric on X × G 2 /L, the curveγ = (γ, 1) is a shortest geodesic between non-principal orbits. The resulting diagram is as claimed. The converse follows from Proposition 2.4.
Definition 2.22 (Join action). Let G 1 and G 2 be two Lie groups which act on Alexandrov spaces X 1 and X 2 , respectively. The action of G 1 × G 2 on X 1 * X 2 is called join action, if G 1 × G 2 acts on X 1 * X 2 naturally, i.e.
Proposition 2.23. If two Lie groups G 1 and G 2 act transitively on positively curved homogeneous spaces M 1 and M 2 with isotropy groups H 1 and H 2 , respectively, then the join action of G = G 1 ×G 2 on M 1 * M 2 is of cohomogeneity one with the following diagram: Conversely, a cohomogeneity one action of G 1 × G 2 with the above group diagram, and G i /H i positively curved homogeneous spaces, for i = 1, 2, is equivalent to the join action of G on (G 1 /H 1 ) * (G 2 /H 2 ). 8 Proof. Let x ∈ M 1 and y ∈ M 2 be such that H 1 = (G 1 ) x and H 2 = (G 2 ) y . The curve is a shortest geodesic between [x, y, 0] and [x, y, π/2] which goes through all orbits. Furthermore, G γ(0) = H 1 × G 2 , G γ(π/2) = G 1 × H 2 , and t ∈ (0, π/2), G γ(t) = H 1 × H 2 . Therefore, the action is of cohomogeneity one with the given diagram. By Proposition 2.4, the converse is immediate.
Definition 2.24 (Suspension action). Let G be a Lie group which acts on an Alexandrov space X. The action of G on Susp(X) is called suspension action, if G acts on Susp(X) as follows: is a shortest geodesic between [x, 0] and [x, π] which goes through all orbits. Furthermore, G γ(0) = G, G γ(π) = G, and for t ∈ (0, π), G γ(t) = H. Therefore, the action is of cohomogeneity one with given diagram. By Proposition 2.4, the inverse is clear.
Proposition 2.26 (Spin action). Let G be a compact, simply-connected Lie group which acts almost effectively and by cohomogeneity one on a closed, simply-connected Alexandrov space X n with group diagram (G, H, K − , K + ). If dim G = n(n − 1)/2, then G is isomorphic to Spin(n) and the action is equivalent to the cohomogeneity one action of Spin(n) on Susp(RP n−1 ), which is the suspension of the transitive action of Spin(n) on RP n−1 .
Proof. The proof of this proposition is analogous to Hoelscher's proof in [25,Proposition 1.20] with slight changes. Namely, since in our case K ± is not a sphere, H 0 = H. As the only proper subgroup of Spin(n) containing Spin(n − 1) is N Spin(n) (Spin(n − 1)), we have H = N Spin(n) (Spin(n − 1)) and K ± /H = RP n−1 .

2.7.
Transitive actions on spheres. We conclude this section by recalling the well known classification of almost effective transitive actions on spheres (see [3] and the references therein). We will use this classification throughout our work. . Suppose that a compact, connected Lie group G acts almost effectively and transitively on the sphere S n−1 (n ≥ 2). Then the G-action on S n−1 is equivalent to the following linear action of G on S n−1 via the standard representation ι : G → SO(n) with an isotropy subgroup H.

Proof of Theorem A
3.1. Possible groups. We first list the Lie groups that can act (almost) effectively and by cohomogeneity one on an Alexandrov space of dimension 5, 6 or 7. This list is obtained as in the manifold case, and we refer the reader to [25, Section 1.24] for more details.
Let G be a compact connected Lie group acting (almost) effectively and by cohomogeneity one on an n-dimensional Alexandrov space X n . It is well-known that every compact and connected Lie group has a finite cover of the form G ss × T k , where G ss is semisimple and simply-connected, and T k is a torus. The classification of simply-connected semisimple Lie groups is also well-known and all the possibilities are listed in Table 3.1 for dimensions 21 and less.
If an arbitrary compact connected Lie group G acts on an Alexandrov space X, then every cover G of G still acts on X, although less effectively. Hence, allowing for a finite ineffective kernel, and because G will always have dimension 21 or less, we can assume that G is a product of groups from Table 3.1 with a torus T k .  In Table 3.2 we list the proper, connected, non-trivial closed subgroups of the groups in Table 3.1, in dimensions at most 15, and of T 2 ; these are the dimensions that will be relevant in our case. These subgroups are well-known (see, for example, [13]

3.2.
Possible normal spaces of directions. As stated in Theorem 2.3, for a cohomogeneity one action with group diagram (G, H, K − , K + ), the homogeneous spaces K ± /H are positively curved. The classification of simply-connected positively curved homogeneous spaces has been carried out by Berger [6], Wallach [38], Aloff and Wallach [2], Berard-Bergery [4] and Wilking [40] (for a Group Subgroups   [41]). Combining this with the classification of homogeneous space forms due to Wolf [42], and the fact that in even dimensions there can be at most Z 2 quotients, by Synge's theorem, it follows that the positively curved homogeneous spaces in dimensions 5 and below are (diffeomorphic to) S 0 , S 1 , S 2 , RP 2 , the three-dimensional spherical space forms, S 4 , RP 4 , CP 2 (noting that CP 2 admits no Z 2 quotient) and, in dimension 5, the five-dimensional spherical space forms. In dimension 6, there appear S 6 , RP 6 , CP 3 , CP 3 /Z 2 and, finally, the Wallach manifold W 6 = SU(3)/T 2 and its Z 2 quotient. We collect this information in Table 3.3.
Dimension Space Let X be a closed Alexandrov space of cohomogeneity one. If both K ± /H are spheres, then X is equivalent to a smooth manifold. These manifolds and their actions have been classified by Mostert [32] and Neumann [33] in dimensions 2 and 3, Parker [35] in dimension 4, and Hoelscher [25] in dimensions 5, 6 and 7 (assuming X is simply-connected). If both K ± /H are integral homology spheres, then X is equivalent to a topological manifold and K ± /H must be either a sphere or the Poincaré homology sphere P 3 (see [18]). These manifolds and their actions have been classified in [18] up to dimension 7, assuming, as in the manifold case, simply-connectedness in dimensions 5, 6 and 7. From now on we will assume that at least one of the homogeneous spaces K ± /H is not a sphere, i.e. that the action is not equivalent to a smooth action on a smooth manifold.
3.3. Classification in dimension 5. To find the group diagrams of cohomogeneity one actions on closed, simply-connected Alexandrov spaces in dimension 5, we first determine the acting groups. By Proposition 2.18, 4 ≤ dim G ≤ 10. Hence, by Table 3.1, G has the form (S 3 ) m × T n , SU(3) × T n or Spin (5). From Proposition 2.19, we have n ≤ 1. Since dim H = dim G − 4, Proposition 2.12 gives the possible groups. These are, up to a finite cover: Now we examine the action of each group case by case.
In this case, dim H = 0, so H 0 = {1}. By Proposition 2.19, and without loss of generality, we can assume that , with x ∈ Im(H), q = 0 and (p, q) = 1. Now we want to determine K + /H. Since we have assumed that the action is non-smoothable, K + /H is not a sphere. Hence, the possible dimensions for K + /H are 2, 3 or 4. Since, by Proposition 2.17, K + 0 acts transitively on K + /H , it cannot be 1-dimensional. Further, by Proposition 2.19, , e 2πi k qs ) | 1 ≤ s ≤ k, 1 ≤ l ≤ m}. By Proposition 2.17, we have then that We now look for conditions on the parameters p, q, m, k. By Proposition 2.13, and the long exact sequences of homotopy groups of the fiber bundles we have (k, q) = q, i.e. q|k. We can also assume that H ∩ (1 × S 1 ) = 1 to have a more effective action. This condition gives, in particular, that (p, k) = 1. Therefore, the diagram is given by where (p, k) = 1 and q|(m, k).
We have dim H = 2. Since the only connected 2-dimensional subgroup of G is its maximal torus, we have that H 0 = T 2 . Therefore, K ± 0 , which contains T 2 , must be S 3 × S 1 or S 1 × S 3 . In particular, K ± /H is 2-dimensional. Since at least one of the positively curved homogeneous spaces K ± /H is not a sphere, we may assume, without loss of generality, that K + /H = RP 2 . The other homogeneous space K − /H can be S 2 or RP 2 .
First assume that K − /H = S 2 . Then by Proposition 2.13, K + is connected. Let K + = S 3 × S 1 . Recall that S 3 is, up to a finite cover, the only Lie group that acts (almost) effectively and transitively on RP 2 . Then by Proposition 2. 16 which corresponds to a join action. By Proposition 2.23 X is equivariantly homeomorphic to RP 2 * S 2 .
Therefore K ± are both connected and we obtain the diagram This action is non-primitive with L = S 3 × S 1 as in Definition 2.5. Hence, by Proposition 2.6, X is equivariantly homeomorphic to a Susp(RP 2 )-bundle over S 2 .
Assume now that . As before, the assumption that X is simply-connected implies, by Lemma 2.15 . Therefore we get the following diagram: This action is a join action and X is equivariantly homeomorphic to RP 2 * RP 2 . G = SU(3). In this case dim H = 4. By Table 3.2, one can see that the only 4-dimensional subgroup of SU(3) is U(2). Therefore, H = H 0 = U(2), as U(2) is a maximal subgroup of SU(3). Since X is simplyconnected, the action does not have any exceptional orbits. Hence, K ± must be SU (3). Thus the diagram is and, by Proposition 2.25, X is equivalent to Susp(CP 2 ). G = Spin(5). Since dim G = 10, by Proposition 2.26, the group diagram is (Spin(5), N Spin(5) (Spin(4)), Spin(5), Spin(5)), (3.7) and X is equivariantly homeomorphic to Susp(RP 4 ).
3.4. Classification in dimension 6. Proceeding as in dimension 5, we see that 5 ≤ dim G ≤ 15 and dim H = dim G − 5. It follows from Propositions 2.12 and 2.19 that G is one of the following Lie groups: (2), then dim H = 5. Since Sp(2) does not have a subgroup of dimension 5, we can rule it out. We now carry out the classification for the remaining groups in the list.
Notational convention. The binary dihedral group D * 2m of order 4m, m ≥ 3, is a finite subgroup of S 3 (see [42], Section 2.6). Throughout the rest of the paper, we consider it as the following subgroup: If, in the right-hand side of (3.8), we assume that m = 1, then e π/mi , j = Z 4 . Therefore, we use the notation D * 2m for m ≥ 3 (the binary dihedral group as in [42]), and, when m = 1, D * 2m will correspond to the cyclic subgroup j of S 3 generated by j.
In this case the principal isotropy group H is 1-dimensional. Thus H 0 = T 1 ⊆ S 3 × S 3 . After conjugation, we can assume that H 0 = {(e ipθ , e iqθ ) | θ ∈ R} with (p, q) = 1. Exploring the subgroups of G and the homogeneous spaces with positive curvature, we see that the normal space of directions to the singular orbits has to be a sphere, a real projective plane or S 3 /Γ with Γ = {1}.
Hence we have the following diagram: By Proposition 2.23, X is equivariantly homeomorphic to RP 2 * S 3 . Let This action is non-primitive with L = S 3 × S 1 . Therefore, by Proposition 2.6, X is equivariantly homeomorphic to the total space of an (RP 2 * S 1 )-bundle over S 2 . Let , and we get the following diagram: This action is equivalent to the following action on Susp(RP 2 ) × S 3 : In this case, since S 3 is, up to a finite cover, the only Lie group which acts transitively and almost effectively on By Proposition 2.23, X is equivariantly homeomorphic to RP 2 * (S 3 /Γ). Now let Γ = Z k . According to Theorem 2.27, S 1 × S 3 acts on S 3 /Z k in the following way: , which yields p = 0. Therefore we have the following diagram: Therefore, this case does not occur.
We now repeat the above procedure for K + 0 = ∆S 3 . In this case H 0 = ∆S 1 and We consider the different possibilities for K − /H, namely, S l , l ≥ 1, RP 2 , and S 3 /Γ with Γ = {1}.
If K − /H = S l , l ≥ 1, then, as before, l = 1, 3 only. First, suppose that K − /H = S 3 . Therefore, K + is connected and H = ∆S 1 ∪ (j, j)∆S 1 . Since K − 0 is 4-dimensional, after exchanging the factors of G if 14 necessary, we can assume that , and the following diagram is obtained This action is equivalent to the following action: Thus we have the following diagrams: Note that this action is equivalent to the following action: After exchanging the factors of G, if necessary, we can assume that 0 , this cannot happen. Therefore Γ = Z k , and the action of S 3 × S 1 on S 3 /Z k is given by: This action is equivalent to the action given by Thus X is equivariantly homeomorphic to RP 2 * RP 3 .
Since the connected 4-dimensional subgroups of S 3 × S 3 are S 3 × S 1 and S 1 × S 3 , we can assume, without loss of generality, that K + 0 = S 3 × S 1 . The possibilities for K − /H are S l , l ≥ 1, RP 2 , and S 3 /Λ, where Λ is a non-trivial finite subgroup of S 3 . The case where K − /H = RP 2 has been treated above, so we only examine the cases where K − /H is a sphere or a 3-dimensional spherical space form.
Therefore, we have the following diagram: Thus we have the following diagram: Assume now that K − /H = S l , l ≥ 2. Hence by Corollary 2.14, K + is connected. First assume that Γ = Z k . As a result H = Γ × S 1 . For l = 2, the only possibility for K − 0 is 1 × S 3 . Then we obtain the following diagram: This action is equivalent to the join action on (S 3 /Γ) * S 2 .
For l ≥ 3, there are no subgroups of G such that K − /H = S l ; therefore we need not consider these cases. Now let Γ = Z k . Then the isotropy subgroup of the transitive action of Thus we have the following diagrams corresponding to p = 0 and p = 1, respectively: The first action is equivalent to the join action on (S 3 /Z k ) * S 2 . The second one is the join action on RP 3 * S 2 given by For l ≥ 3, there are no subgroups of G such that K − /H = S l . Assume now that K − /H = S 3 /Λ with Λ a non-trivial subgroup of S 3 . Therefore, Note that according to the classification of the transitive actions on 3-dimensional space forms, q = 0, which gives that the circles in the second component of K ± 0 are the same, so Consequently, for Γ = Z k , we have the following diagram: This action is equivalent to the product action on Susp(S 3 /Γ) × S 2 . If Γ = Z k , the following diagram is obtained: For p = 0, this action is equivalent to the product action on Susp(S 3 /Z k ) × S 2 , and for p = 0, it is nonprimitive. In particular, in the preceding diagram, if Z k = Z 2 and p = 1, then the action is as follows: Hence H 0 = ∆S 1 , and both Γ and Λ are cyclic subgroups of S 3 , say Z k and Z l , respectively. Then we have Hence H + and H − are subgroups of both K + 0 and K − 0 , which gives that H = H + , H − ⊆ K ± 0 . It follows then from Proposition 2.17 that Thus H − = H = H + and, in particular, Γ = Λ. The diagram is then given by However, S 3 × S 3 does not act transitively on a 4-dimensional homogeneous space with positive curvature (see [41]). Therefore, Without loss of generality, we can assume that K + 0 = S 3 × S 1 . Thus K + /H = RP 2 . By the classification of the transitive actions on spheres, S 3 , up to a finite cover, is the only Lie group which acts transitively and almost effectively on RP 2 . Therefore, K + 0 ∩ H = N S 3 (S 1 ) × S 1 × 1, and we obtain the following diagram:  (1) is a maximal connected subgroup of Sp(2). Thus dim K + /H = 4, and therefore K + /H = RP 4 (note that the other positively curved homogeneous space in dimension 4 is CP 2 , which does not admit an Sp(2)-transitive action (see [41,Table B])). Hence we get the following diagram By Proposition 2.23, X is equivalent to RP 4 * S 1 . G = Spin (6). In this case, since dim G = 15 = (6)(6 − 1)/2, by Proposition 2.26, we obtain the diagram (Spin (6), N Spin(6) (Spin(5)), Spin (6), Spin (6)). (3.32) and X is equivariantly homeomorphic to Susp(RP 5 ).
In this case, dim H = 0. Having looked at the classification of homogeneous spaces with positive curvature, and the subgroups of S 3 × S 3 , one can see that the only homogeneous spaces with positive curvature that can happen as the normal space of directions of singular orbits are 3-dimensional spherical space forms.
Assume that K + /H = S 3 /Γ, with Γ a nontrivial finite subgroup of S 3 . Then K + is 3-dimensional and, as a result, (3.33) and X is equivariantly homeomorphic to the total space of an (S 3 /(Γ * S 1 ))-bundle over S 2 .
If q = 0, then Thus we get the following diagram: This action is equivalent to the product action of S 3 × S 3 on X 4 × S 3 , where X 4 is the 4-dimensional Alexandrov space with the following diagram (see [17]): Indeed, X 4 is equivariantly homeomorphic to CP 2 /Z k (for more details see Subsection 4.1, Diagram (4.1)).
For Γ = D * 2m , we have K − = N S 3 (S 1 ) × 1, and we obtain the following diagram: Similarly, this action is equivalent to the product action of S 3 × S 3 on X 4 × S 3 , where X 4 is given by . Again, X 4 is equivariantly homeomorphic to CP 2 /Z m (for more details see Subsection 4.1, Diagram (4.2)).
If pq = 0, then, since where z ∈ x ⊥ ∩ Im(H) ∩ S 3 and w ∈ y ⊥ ∩ Im(H) ∩ S 3 , we have Γ = Z k . Also, without loss of generality, we may assume that K − 0 = {(e ipθ , e iqθ )}. Therefore, we get the following diagram: where (k, q) = (q, m). Now, assume that K − /H = S 3 . As a result, K + is connected and H = Γ × 1. On the other hand, K − 0 is a 3-dimensional subgroup containing Γ × 1. Therefore, there are two possibilities: then we obtain the following diagram: By Proposition 2.23, X is equivariantly homeomorphic to S 3 * (S 3 /Γ). Now let K − 0 = ∆ g0 S 3 . Since 1 × S 3 ⊆ N (H) 0 , by Proposition 2.4 we can conjugate K − by (1, g −1 0 ) without changing the spaces. Moreover, K − ⊆ N (∆ g0 S 3 ) = ±∆ g0 S 3 , so we can assume that g 0 = 1. Now, since K − /H is simply-connected, the number of connected components of K − and H are the same. Since H = 1, and K − has at most two components, we conclude that Γ = Z 2 . Thus, we get the following diagram: This action is equivalent to the following action on RP 3 * S 3 : This action is equivalent to the product action on Susp(S 3 /Γ) × S 3 . Now let K − 0 = 1 × S 3 . In this case, Γ × 1 = K + 0 ∩ H, and K − 0 ∩ H = Λ × 1, so by Lemma 2.15, H = Γ × Λ. Hence, we get the following diagram: , and Γ = 1, then K − has to be ±∆ g0 S 3 , and Γ = Z 2 . Also, the classification of transitive actions on spheres gives us that K − 0 ∩ H = ∆ g0 Λ. Therefore H = ±∆ g0 Λ, and the following diagram is obtained: ). According to Proposition 2.4 and Equation (3.41), we can assume that g 0 = 1. This action is equivalent to the following action and X is then equivariantly homeomorphic to RP 3 * (S 3 /Λ): Now assume that K + 0 = ∆ g0 S 3 . Thus H ∩ K + 0 = ∆ g0 Γ. As before, K − /H can be a circle, a 3-sphere, or a 3-dimensional spherical space form.
Assume that K + = ∆ g0 S 3 . Therefore H = ∆ g0 Γ. Let q = 0. Then K − 0 = S 1 × 1 and 2m , respectively. By conjugating the subgroups by (1, g −1 0 ), we have the following diagrams: If p = 0, we have, similarly, Observe that these two diagrams are the same as Diagrams (3.42) and (3.43) up to exchanging the factors of G. Now assume that pq = 0. Then N ( where k|(p − q), and if k is even, then p, q are odd, and we get where, k|(p + q), and if k is even, then p, q are odd.
If K − = {(e ipθ , e iqθ )} ∪ {(je ipθ , je iqθ )}, similar arguments as above give rise to the following diagrams with the same conditions, respectively: 2m . By the same argument, we obtain the following diagrams: Observe that the last two diagrams are the same as Diagrams (3.50) and (3.51) up to exchanging the factors of G.
In the following diagrams, p is odd and q is even, so k has to be odd: Now assume that K − /H = S 3 . Therefore, K + = K + 0 = ∆ g0 S 3 and H = ∆ g0 Γ. Further, K − 0 is equal to The number of connected components of K − is at most 2, for K − ⊆ N (K − 0 ) = ±∆ g1 S 3 . Thus H = (−1, 1) . But then H is not a subgroup of K + . Therefore, this case cannot occur. Now, let K − /H = S 3 /Λ with Λ a non-trivial subgroup of S 3 . Again, we have three possibilities for K − 0 : where z ∈ C(Γ). If Γ = Z k , then by (3.60), z = ±1 and in particular K − = ∆ g0 S 3 . Hence, after conjugating all subgroups by (1, g −1 0 ), we obtain an equivalent diagram given by If Γ = Z k , then we first conjugate all subgroups by (1, g −1 0 ). Then, since (1, z) ∈ N (H) 0 by (3.60), we can conjugate K − by (1, z) to obtain diagram (3.61).
First assume that K + /H = RP 2 . Therefore, dim K + = 3. The possibilities for K + 0 are S 3 × 1 × 1, 1 × S 3 × 1 and ∆ g0 S 3 × 1. Let Therefore, we obtain the following diagram: By Proposition 2.5, this action is a non-primitive action with L = S 3 × S 1 × S 1 and X is equivariantly homeomorphic to the total space of an (RP 2 * S 1 )-bundle over S 2 . Note that the action of S 3 × S 1 × S 1 on RP 2 * S 1 is in fact the normal extension of the action of S 3 × S 1 on RP 2 * S 1 .
Let K + 0 = 1×S 3 ×1. This case only differs from the previous one by an isomorphism of G which exchanges the factors. Therefore the analysis is analogous to the one in the preceding case. Since the action is non-reducible, The long exact sequence of homotopy groups corresponding to the fiber bundle . By Proposition 2.13, π 1 (G/K − ) = Z 2 as X is simply-connected. Thus c = 1. On the other hand, Z k ⊆ N (K + 0 ), which gives k|2p, and since we can assume H ∩ (1 × 1 × S 1 ) = 1 to have a more effective action, k = 1, 2. Therefore, we obtain the following diagram: where T 2 = {(e iφ , e iφ e ipθ , e iθ ) | φ, θ ∈ R}, and k = 1, 2. Now let K + /H = S 3 /Γ with Γ finite and non-trivial. Therefore, k e pθi , e θi , 1) | 1 ≤ l ≤ k}. Since X is simply-connected, by Proposition 2.13, and the exact sequences of homotopy groups related to the fiber bundles Therefore, c|k and the following diagram is obtained: for some integers a ′ , c ′ . If a ′ = 0, then we have the following diagram: By Proposition 2.21, this action is a product action and X is equivariantly homeomorphic to (S 3 /Γ * S 1 )×S 2 . If a ′ = 0, then N (K − ) = {(e ia ′ θ , e iφ , e ic ′ θ )} ∪ {(e ia ′ θ , je iφ , e ic ′ θ )} which implies that Γ = Z k . Thus, this case cannot happen.
In this case, dim H = 3. Since the action is non-reducible, Proj i (H) S 3 , i = 1, 2, 3. Therefore, H 0 must be a maximal torus of G. Further, by considering the subgroups of G containing H, we only have RP 2 and S 2 as the normal spaces of directions of singular orbits.
Assume, without loss of generality, that K + /H = RP 2 . Then there are two possibilities for K − /H, namely, K − /H = S 2 or K − /H = RP 2 .
Let K − /H = S 2 . Therefore, K + is connected and, after exchanging the factors of G if necessary, we can assume that Also, since K − /H is simply-connected, π 0 (K − ) = π 0 (H) = Z 2 , and their components intersect each other. Hence K − = N S 3 (S 1 ) × S 3 × S 1 , and we get the following diagram: By Proposition 2.21, this action is a product action and X is equivariantly homeomorphic to (RP 2 * S 2 ) × S 2 .
, and the following diagram is obtained:

By Proposition 2.21, this action is a product action and X is equivariantly homeomorphic to Susp(RP
By Proposition 2.21, this action is a product action and X is equivariantly homeomorphic to (RP 2 * RP 2 )×S 2 . The space X is equivariantly homeomorphic to Susp(W 6 ) or Susp(W 6 /Z 2 ), respectively, where W 6 = SU(3)/T 2 is the Wallach manifold.
Suppose now that K + 0 = SU(3) × S 1 and K − 0 = U(2) × S 3 . In this case, K − /H can be either S 2 or RP 2 . Therefore, we have the following diagrams, respectively, and X is equivariantly homeomorphic to CP 2 * S 2 or CP 2 * RP 2 , respectively.

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Assume first that Therefore, K ± are connected, and we get the following diagram: By Proposition 2.21, this action is a product action and X is equivariantly homeomorphic to Susp(RP 2 )×S 4 .
This concludes the proof of Theorem A. 25

Proof of Theorem B
In this section we prove Theorem B, i.e. we show that the underlying space of a closed, simply-connected cohomogeneity one smooth orbifold in dimension at most 7 is equivariantly homeomorphic to a good orbifold. First, we recall basic definitions and facts about orbifolds (for more details see, for example, [1], [11]). Definition 4.1. An orbifold chart on a topological space X is a quadruple (Ũ , G, U, π), where • U is an open subset of X, •Ũ is open in R n and G is a finite group of diffeomorphisms ofŨ , • π :Ũ → U is a map which can be factored as π =πp, where p :Ũ →Ũ /G is the orbit map and π :Ũ /G → U is a homeomorphism.
For i = 1, 2, suppose that (Ũ i , G i , U i , π i ) are orbifold charts on X. The charts are compatible if given pointsũ i ∈Ũ i with π 1 (ũ 1 ) = π 2 (ũ 2 ), there is a diffeomorphism h from a neighborhood ofũ 1 inŨ 1 onto a neighborhood ofũ 2 inŨ 2 so that π 1 = π 2 h on this neighborhood. An orbifold atlas on X is a collection (Ũ i , G i , U i , π i ) of compatible orbifold charts which cover X.
be orbifold atlases over a given topological space X. We say that they define the same orbifold structure on X if the union atlas (Ũ i , G i , U i , π i ) i∈I1∪I2 satisfies the compatibility condition in Definition 4.1.  Definition 4.4. Let q ∈ |Q| and (Ũ , G, U, π) be any local chart around q = π(x). We define the local group at q as G q = {g ∈ G | gx = x}. This group is uniquely determined up to conjugacy in G.
The notion of local group is used to define the singular set of the orbifold. Definition 4.5. For an orbifold Q, we define its singular set as We call R Q = Q \ S Q the regular set of Q.    . Any connected orbifold Q admits a simply-connected orbifold covering π :Q → Q. This has the following universal property: if q ∈ |Q| − S Q is a base point for Q andq is a base point forQ which projects to q, then for any other covering orbifold π ′ :Q ′ → Q with base pointq ′ , there is a lift σ :Q →Q ′ of π ′ to a covering map ofQ ′ . Definition 4.10. The orbifold fundamental group π orb 1 (Q) of an orbifold Q is the group of deck transformations of the universal orbifold cover π :Q → Q.
The following proposition gives a necessary and sufficient condition for an orbifold to be good in terms of its local groups (see [11,Page 7]). For each q ∈ |Q|, let G q denote the local group at q. We can identify G q with the orbifold fundamental group of a neighborhood of the formŨ q /G q whereŨ q is a ball in some linear representation. We say that G q is the local fundamental group at q. The inclusion of the neighborhood into Q induces a homomorphism G q → π orb 1 (Q). Proposition 4.11 ([11, Proposition 1.18]). An orbifold Q is good if and only if each local group injects into the orbifold fundamental group, i.e. for each q ∈ |Q|, the homomorphism G q → π orb 1 (Q) is injective. Definition 4.12. In an orbifold Q, a stratum of type (H) is the subspace of |Q| consisting of all points with local group isomorphic to H. . Let Q (2) denote the complement of the strata of codimension > 2 of an orbifold Q. Then π orb 1 (Q) = π orb 1 (Q (2) ). We state the proof of the following theorem as in [39], since it has an algorithm to recover the fundamental group of an orbifold from the fundamental group of the regular part.
Theorem 4.14 ([39, Theorem 5.5]). Let Q be a connected orbifold. Then a presentation for the orbifold fundamental group can be constructed using the topology, stratification, and the orders of points in codimension 2 strata.
Proof (See [39]). LetQ be the differential subspace of Q consisting of codimension 0 and codimension 1 strata. Fix a base point q in the codimension 0 stratum. Let G be the (topological) fundamental group of Q with respect to q.
(1) For each codimension 1 stratum S i , and for each homotopy class µ of paths starting at q and ending in S i attach a generator β i,µ to G with relation β 2 i,µ = 1.
(2) For each codimension 2 stratum T j not in the closure of a codimension 1 stratum, let α j be an element of G represented by a loop starting at q and going around T j . Then add the relation α k j = 1 to G where k is the order of any point in T j .
(3) For each codimension 2 stratum R in the closure of a codimension 1 stratum, for each pair of codimension 1 strata S i , S ′ i with R in their closures, and for each pair β i,µ , β i ′ ,µ ′ (where µ = µ ′ ) as constructed in Item (1) above, add the relation (β i,µ β i ′ ,µ ′ ) k = 1, where 2k is the order of any point in R. The resulting group is the orbifold fundamental group of Q.
The structure of closed cohomogeneity one smooth orbifolds is given by the following theorem. . Let Q be a closed, connected, smooth orbifold with an (almost) effective smooth action of a compact, connected Lie group G with principal isotropy group H. If the action is of cohomogeneity one, then the orbit space Q/G is homeomorphic to a circle or to a closed interval and the following statements hold.
(i) If the orbit space is a circle, then Q is equivariantly diffeomorphic to a G/H-bundle over a circle with structure group N (H)/H, where N (H) is the normalizer of H in G. In particular, Q is a manifold and its fundamental group is infinite.
(ii) If the orbit space is homeomorphic to an interval, say [−1, 1], then: (a) there are two non-principal orbits, π −1 (±1) = G/K ± where π : Q → Q/G is the natural projection and K ± is the isotropy group of the G-action at a point in π −1 (±1).
(b) The orbifold singular set of Q is either empty, a non-principal orbit or both non-principal orbits.

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(c) The orbifold Q is equivariantly diffeomorphic to the union of two orbifiber bundles over the two non-principal orbits whose fibers are cones over spherical space forms, that is, where S ± denotes the round sphere of dimension (dim Q − dim G/K ± − 1) and Γ ± is a finite group acting freely and by isometries on S ± . The action is determined by a group diagram (G, H, K − , K + ) with group inclusions H ≤ K ± ≤ G and where K ± /H are spherical space forms S ± /Γ ± .
(d) Conversely, a group diagram (G, H, K − , K + ) with group inclusions H ≤ K ± ≤ G and where K ± /H are spherical space forms, determines a cohomogeneity one orbifold as in part (c).
Note that when the normal space of directions to a singular orbit is S 1 , i.e. when the orbit is of codimension 2, there are different orbifold structures on the underlying topological space corresponding to the diagram (G, H, K − , K + ). Namely, the local group at a point in this singular orbit can be trivial, or a finite cyclic group. For example, consider the group diagram (S 1 , 1, S 1 , S 1 ) of a topological action of S 1 on S 2 . There are at least three orbifold structures on S 2 : the usual smooth structure, the teardrop structure, and the rugby ball structure.
When there is no orbit of codimension 2, or we choose the local group at a point on the orbit of codimension 2 to be trivial, then we can obtain the orbifold fundamental group of the orbifold Q given by the group diagram (G, H, K − , K + ) as follows: Proposition 4.16. Let Q be a cohomogeneity one orbifold given by the group diagram (G, H, K − , K + ).
(i) If the normal spaces of directions at both singular orbits are not spheres, then π orb 1 (Q) = π 1 (G/H).
(ii) If there is an orbit, say G/K − , such that K − /H is a sphere, and the local group at a point in this orbit is trivial, then π orb 1 (Q) = π 1 (G/K − ). In particular, if G is simply-connected, then Q is a bad orbifold.
Proof. We first prove part (i). If the normal spaces of directions at both singular orbits are not spheres, then the regular subset of the orbifold coincides with the union of the regular orbits. Since by assumption, the codimension of each of the singular orbits is at least 3, we have Q (2) = R Q . Thus, by Proposition 4.13, π orb 1 (Q) = π 1 (R Q ). Since R Q = G/H × (−1, 1), we have π orb 1 (Q) = π 1 (G/H). Now we prove part (ii). In this case the regular part of the orbifold consists of the space of regular orbits and the singular orbit with sphere as its normal space of directions. The singular orbit with non-sphere normal space of directions is the other stratum of the orbifold, whose codimension is at least 3. Therefore, Q (2) = R Q = G × K − D − . By the long exact sequence of homotopy groups of the disk bundle 4.1. Cohomogeneity one orbifolds with exactly one fixed point. In this subsection we prove that if the group acting on Q is simply-connected and the action has exactly one fixed point, then there exists a good orbifold structure on the underlying topological space. Let G be a compact simply-connected Lie group which acts almost effectively, non-reducibly and with cohomogeneity one on a smooth simply-connected orbifold Q with group diagram (G, H, K, G). Assume that K G so that the action of G on Q has only one fixed point. Since the normal space of directions is a spherical space form, G/H = S n−1 /Γ, for Γ a finite subgroup of SO(n). On the other hand, since G/H is a principal orbit, G acts on S n−1 /Γ almost effectively. Therefore, we can lift the almost effective action of G on S n−1 /Γ to an almost effective and transitive action of G on S n−1 . These actions are well known and are given in Theorem 2.27.
Note that in the group diagram of the G-action on Q, the group K is a proper subgroup of G which cannot be a finite extension of H either, because there are no exceptional orbits. Thus we can rule out the actions (2.2)-(2.5) in Theorem 2.27, for in these cases H is a maximal connected subgroup of G. Further, since we assume that the action is non-reducible, we rule out cases (2.6) and (2.8). in Theorem 2.27. Thus, we need only consider the actions (2.7), (2.9) and (2.10) in Theorem 2.27.
For each of the corresponding groups we will find the possible cohomogeneity one diagrams and explore the existence of a good orbifold structure on the underlying topological space determined by the diagram. Our strategy is as follows. Observe first that a cohomogeneity one action on an orbifold Q lifts to a cohomogeneity one action on its universal orbicover. Therefore, if Q is a good cohomogeneity one orbifold with finite orbifold fundamental group, then its universal orbicoverQ is a compact cohomogeneity one smooth manifold and, provided that dimQ ≤ 7, it must be one of those listed in [25]. In our case we will compute the orbifold fundamental group of Q. To do so, we need to pay especial attention to the local groups of Q. Then we will show that the underlying topological space |Q| corresponds to some diagram given by the quotient of a cohomogeneity one action on a manifold. G = Sp(n). This case corresponds to action (2.7) in Theorem 2.27. We need the following Lemma: . Recall now that H is a finite extension of H 0 = Sp(n − 1) ⊂ Sp(n). Using the fact that H K, we get the following diagrams: . , x n ], λ ∈ Z k , respectively. The action of Sp(n) is of cohomogeneity one (see [25,Proposition 1.23]) and commutes with the Z k -action. Therefore, Sp(n) acts on CP 2n /Z k with cohomogeneity one with group diagram (4.1) above. Hence by Proposition 2.4, Q is equivariantly homeomorphic to CP 2n /Z k . Note that in this case one of the normal space of directions is S 4n−1 /Z k and the other one is S 1 . Therefore, we have a codimension 2 stratum, namely, the orbit G/K. If we choose the local group on this orbit to be trivial, then by Proposition 4.16, π orb 1 (Q) = π 1 (G/K) = 0. Therefore, CP 2n /Z k admits a bad orbifold structure. If we choose the local group at points on G/K to be Z k , then CP 2n is the universal orbicover of CP 2n /Z k , giving that Q is a good orbifold.
Diagram By the definition of Sp(n), the action of Sp(n) and the action of D * 2m on CP 2n commute. Thus we have a cohomogeneity one action of Sp(n) on CP 2n /D * 2m with the same diagram as (4.2). By Proposition 2.4, Q is equivariantly homeomorphic to CP 2n /D * 2m . Again, we can choose two different orbifold structures on CP 2n /D * 2m . First we let the local group at points on the orbit G/K be trivial. Then by Proposition 4.16, π orb 1 (Q) = π 1 (G/K) = Z 2 . Since D * 2m cannot be embedded in Z 2 , CP 2n /D * 2m admits a bad orbifold structure by Proposition 4.11. Now, assume that the local group at the points on the orbit G/K is Z m . Then CP 2n is the universal orbicover of CP 2n /D * 2m .
Clearly, these two actions commute. As a result, we have a cohomogeneity one action on CP n /Z k with group diagram (4.5). Therefore, by Proposition 2.4, Q is equivariantly homeomorphic to CP n /Z k . Since the orbit SU(n)/S(U(n − 1)U (1)) is of codimension 2, we can choose different orbifold structures on CP n /Z k . If we let the local group at this orbit be trivial, then by Proposition 4.16, π orb 1 (Q) = 0. Hence CP n /Z k is a simply-connected orbifold. We now choose the local group to be Z k . Then CP n is the universal orbicover of CP n /Z k , that is, in this case CP n /Z k is a good orbifold. G = Spin (9). This case corresponds to action (2.9) in Theorem 2.27. We have the following diagram: (Spin (9), N Spin(9) (Spin (7)), Spin(8), Spin (9)). (4.6) Since the codimension of the non-principal orbits of this action is not 2, we have only one cohomogeneity one orbifold structure on the underlying topological space. The local groups of both orbits are Z 2 and, by Proposition 4.16, π orb 1 (Q) = π 1 (G/H) = Z 2 . Now, we consider the following action of Spin(9) on the Cayley plane CaP 2 (for more details see [27]). Let J be the set of all 3 × 3 Hermitian matrices over the Cayley number field Ca. A matrix A ∈ J has the form where ξ i ∈ R and u i ∈ Ca, for i = 1, 2, 3. Let Then, the set {E 1 , E 2 , E 3 , F ei 1 , F ei 2 , F ei 3 | i = 0, 1, . . . , 7} constitutes an R-basis of J, where {e 1 , . . . , e 7 } is the standard basis of Ca. The Jordan product is defined on J by An R-isomorphism ϕ : J → J is called an automorphism of J, if it preserves the Jordan product, i.e.
for all X, Y ∈ J. It is well-known that the group of automorphisms of J is the exceptional Lie group F 4 . The Cayley projective plane CaP 2 , defined by is identified with the left coset space F 4 /Spin (9), where Spin (9) contains and Spin (8) contains Through the inclusion Spin(9) ⊆ F 4 , Spin(9) acts on CaP 2 with cohomogeneity one (see, for example [27,Example 1]). The orbits of the action are given by for 0 ≤ s ≤ 1. In particular, the following hold: The isotropy group at E 2 is Spin(8).
• A s , for 0 < s < 1, is the principal orbit which is a 15-dimensional sphere. The isotropy group at Then σ ∈ Z(Spin(9)) and σ 2 = 1 (see, for example, [43,Section 2.9]). Thus σ is an involution commuting with the Spin(9) action on CaP 2 described above. This gives a cohomogeneity one action of Spin(9) on CaP 2 /Z 2 with group diagram (4.6). Therefore, by Proposition 2.4, Q is equivariantly homeomorphic to CaP 2 /Z 2 . Since Q admits only one orbifold structure compatible with the cohomogeneity one action, Q is indeed equivariantly diffeomorphic to CaP 2 /Z 2 , i.e. Q is a good orbifold.

4.2.
Orbifold structures on non-primitive cohomogeneity one spaces. We now inquire whether or not one can endow an underlying space of a non-primitive diagram whose primitive part admits a good structure, with a good structure. The next proposition shows that it is indeed the case under certain mild restrictions.  Proof. By Proposition 2.6, |Q| is equivariantly homeomorphic to G× L ((S m /Γ 1 ) * (S n /Γ 2 )). Define the action of Γ 1 × Γ 2 on S n1 * S n2 as the join action, and apply Proposition 4.18 withL = L 1 × L 2 .
We have a similar situation when there is a suspended action instead of the join action. Proof. By Proposition 2.6, |Q| is equivariantly homeomorphic to G × L Susp(S n /Γ). Similarly, define the action of Γ on Susp(S n ) as a suspended action and apply Proposition 4.18 withL = L.

Proof of Theorem B.
In what follows we exploit the classification of cohomogeneity one actions on smooth manifolds in low dimensions (see [25]) to find possible smooth coverings for our orbifolds. Since the cohomogeneity one action on an orbifold lifts to a cohomogeneity one action on the orbifold universal covering, taking as departure point the information that we get from the group diagram about the orbifold fundamental group and the local groups at non-principal points, we look for a finite group acting on the cohomogeneity one smooth manifolds commuting with the acting group. In the cases that we have a representative of the group diagram and the action, we describe such actions; otherwise we use the following general argument. Let (G, H, K − , K + ) be a group diagram for a cohomogeneity one action on a smooth manifold M , and let Γ ⊆ N = NG(H) ∩ NG(K − ) ∩ NG(K + ) be a finite subgroup ofG. The group Γ acts on M orbitwise, i.e. γ · gL = gγ −1 L, for L = H, K − , K + . This action is well-defined and commutes with theG-action. Let ρ :G → G be a group covering. If, in addition, {Φ g | g ∈ ker ρ} ⊆ Γ, where Φ g is the action map, then G acts on M/Γ with cohomogeneity one (cf. [44,Proposition 3.2]). If c(t) is a normal geodesic of the cohomogeneity one action on M used to determine a group diagram, then π • c is a normal geodesic for M/Γ. Thus we can find the group diagram of M/Γ using the group diagram of M .
Remark 4.21. In the sequel, we let N = NG(H) ∩ NG(K − ) ∩ NG(K + ), where (G, H, K − , K + ) is the group diagram of the smooth manifold under consideration. We also refer to the orbitwise action defined above only as the orbitwise action, without mentioning the explicit action. Now, we prove Theorem B case by case. Remark 4.22. Note that if the action is a join, suspension, product, or a non-primitive action (as in Corollaries 4.18,4.19,and 4.20), or has a fixed point, then it is already clear what the good structure should be. Therefore, we only explore a good structure on the spaces corresponding to actions not equivalent to these actions.
Dimensions 2 and 3. By Perelman's Conical Neighborhood Theorem, and the fact that the only closed 1-dimensional Alexandrov space is a circle, any 2-dimensional Alexandrov space is a topological manifold. Therefore, the 2-sphere is the only closed, 2-dimensional simply-connected Alexandrov space. In dimension 3, the 3-sphere and Susp(RP 2 ) are the only closed, simply-connected Alexandrov spaces of cohomogeneity one (see [17]). Clearly, Susp(RP 2 ) has a good orbifold structure.
Dimension 4. In dimension 4, every space of directions is homeomorphic either to Susp(RP 2 ) or to a spherical space form (see [15] or [23]). Hence every 4-dimensional Alexandrov space is homeomorphic to an orbifold. In this dimension, however, not every closed orbifold is good. For example, the so-called weighted complex projective spaces are bad 4-dimensional orbifolds (see [1, p. 27]). Closed, 4-dimensional Alexandrov spaces of cohomogeneity one were equivariantly classified in [17]. In Table 4.1 we have collected the diagrams corresponding to the simply connected spaces. Observe that every action in the table is a join, suspension or one-fixed-point action. Therefore, by Remark 4.22 above, the underlying space of each diagram admits a good structure.
Dimension 5. The only diagram that we should consider in dimension 5 is where K − 0 = {(e ipθ , e iqθ ) | θ ∈ R} and q|(m, k) and (p, k) = 1. In this case, the local groups at the points on G/K + are Z m , and we assume the local groups at the points on G/K − to be trivial. Thus, by Proposition 4.16, π orb 1 (Q) = π 1 (G/K − ) = Z m . Now consider the following diagram coming from the smooth classification (see [25, Case 1 5 A]): (4.8) where (p, k) = 1. The explicit description of this action is as follows (see [25,Diagram (Q 5 C ) in p. 172]), where we consider S 5 as the unit sphere in H × C: z), (x, w)) → (gxz p , z k w).
Since in Diagram (4.7), (p, k) = 1, we can choose the same p, k in Diagram (4.8). Let be a group covering of S 3 × S 1 . Since q|(m, k), it is easy to see that {Φ g | g ∈ ker ρ} ⊆ Z m × 1. Therefore, S 3 × S 1 acts on S 5 /Z m as follows: where (g, h) = ρ(g,h). The isotropy groups of this action are precisely the ones in Diagram 4.7. Therefore, Q is equivariantly diffeomorphic to S 5 /Z m . 33 Dimension 6. In this dimension we have six cases to consider.
the local group at the points on (S 3 × S 3 )/∆S 3 is Z 2 and we let the local group at the points on the other non-principal orbit be trivial. Then, by Proposition 4.16, π orb 1 (Q) = π 1 (G/K − ) = Z 2 . Consider the counterpart diagram in the smooth classification ([25, Case 2 6 C], for n = 1): This action is equivalent to the natural action of G = SO(4) ⊆ SO(5) on M = SO(5)/(SO(2)SO (3)). Let for 0 ≤ t ≤ π/2, and let L = SO(2)SO(3). Then c(t)L is a normal geodesic in M (see, for example, [24, p. 88]) and the corresponding isotropy groups are as follows: where the latter is isomorphic to SO(3). Let Then σ 2 = 1 and σ ∈ N G (G c(t)L ), for all t. Hence σ acts on M orbitwise, fixing G(c(π/2)L), and commutes with the G-action. As a result, G acts on M/Z 2 with as isotropy groups. Therefore Q is equivariantly diffeomorphic to M/Z 2 .
Case 6.2. This case corresponds to diagram Here, depending on the local group that we choose for the codimension 2 orbit, we have different manifold diagrams.
Assume first that the local group at the points of the codimension 2 orbit is Z 2 . Then Thus, by Theorem 4.14, π orb 1 (Q) = Z 2 ⊕ Z 2 , since the local group of the codimension 2 stratum is Z 2 . Now consider the diagram and the manifold in the Case 6.1. Let σ be as in (4.9) and let Let τ act on M by τ · xc(t)L = τ xc(t)τ L, where x is an element of SO (5). Note that τ c(t)τ L = c(t)L if and only if t = 0. That is, σ fixes the orbit G(c(π/2)L) and τ fixes G(c(0)L). The action of Z 2 ⊕ Z 2 on M commutes with the action of G. Thus G acts on M/(Z 2 ⊕ Z 2 ) with the same diagram as (4.10). That is, Q is equivariantly diffeomorphic to M/(Z 2 ⊕ Z 2 ). Now let the local group of the codimension 2 orbit be trivial. Then by Proposition 4.16, Here, we consider the following diagram in the smooth classification ([?, Table F [30]). Consequently, SO(4) acts on CP 3 /Z 2 with the same diagram as (4.10). Therefore, Q is equivariantly diffeomorphic to CP 3 /Z 2 .
Note that this argument in particular shows that (SO(5)/SO(2)SO(3))/(Z 2 ⊕ Z 2 ) is homeomorphic to CP 3 /Z 2 , since they are both the underlying spaces of the same diagram. Case 6.3. This case corresponds to diagram If we let the local group of the codimension 2 orbit be trivial, then by Proposition 4.16, π orb 1 (Q) = Z 2 . The local group of the other singular orbit is D * 2m , which gives rise to a bad structure by Proposition 4.11. Now assume that the local group of the codimension 2 orbit is Z m . The action of S 3 × S 3 on S 2 × CP 2 given by yields the smooth diagram we need, i.e.
This action commutes with the cohomogeneity one action of S 3 ≈ 1 0 0 SU (2) on CP 2 and fixes only the orbit with isotropy group S 3 . If we take the action of D * 2m on S 2 to be trivial, then the actions of S 3 × S 3 and D * 2m commute. Therefore, S 3 × S 3 acts on S 2 × (CP 2 /D * 2m ) with the same diagram as (4.11).
The group Z k fixes both a point, (z : 0 : 0), and CP 1 = {(0 : z 1 : z 2 )} (cf. [7, Chapter VII, Section 3]), which are indeed the non-principal orbits of the cohomogeneity one action of S 3 × S 1 . Further, the action of Z k commutes with the action of S 3 × S 1 . Therefore, S 3 × S 1 acts on CP 2 /Z k with the same diagram as (4.13). As a result, the underlying space, |Q|, of Diagram 4.12, is equivariantly homeomorphic to the total space of a (CP 2 /Z k )-bundle over S 2 . Thus by Proposition 4.18, |Q| admits a good structure. In fact, if we choose the local group at the codimension 2 orbit to be Z k , then by Theorem 4.14, π orb 1 (Q) = Z k , and the good structure is just the natural good structure on (S 3 × S 3 ) × (SU(2)×S 1 ) (CP 2 /Z k ).
Note that if we let the local group at the codimension 2 orbit be trivial, then by Proposition 4.16, π orb 1 (Q) = 0 since K − = T 2 is connected. In this case, Q is a bad orbifold.
Let Z k = 1 × Z k ⊆ S 3 × S 1 act on S 4 via (1, h) · x = diag(1, Ψ(1, h))x. This action commutes with the (S 3 × S 1 )-action and fixes the non-principal orbits, which are two points. Therefore, S 3 × S 1 acts on S 4 /Z k with the same diagram as (4.15). This means that the underlying space, |Q|, of Diagram (4.14) is equivariantly homeomorphic to the total space of an (S 4 /Z k )-bundle over S 2 . Thus, by Proposition 4.18, Q is a good orbifold and π orb 1 (Q) = Z k , by Proposition 4.16.
Dimension 7. Unlike dimension 6, in which there exists a diffeomorphism type classification of closed, smooth, simply-connected cohomogeneity one manifolds, there is no such classification in dimension 7. Therefore, here we do not give an explicit description of smooth universal covers of our good orbifolds. In fact, in most cases, we only use the smooth equivariant classification and show that a finite subgroup of N G (H) ∩ N G (K − ) ∩ N G (K + ) acts on the smooth manifold given by the diagram (G, H, K − , K + ), commutes with the given G-action, and has the desired fixed point set. as its primitive diagram. This primitive diagram corresponds to an (S 3 × S 1 )-action on S 5 /Z q . Let the local group at the orbit of codimension 2 be trivial. Then, by Proposition 4.16, π orb 1 (Q) = π 1 (G/K − ) = Z m/(m,q) . Thus Q is good if and only if (m, q) = 1, as the local group Z m should inject into the orbifold fundamental group by Proposition 4.11. Indeed, if (m, q) = 1 = (q, k), we can choose Diagram (4.21) with the same parameters as (4.19). Let Then Γ acts orbitwise on S 5 /Z q and commutes with the (S 3 × S 1 )-action on S 5 /Z q . Therefore, S 3 × S 1 acts on (S 5 /Z q )/Z m with the same diagram as (4.19). By Proposition 4.18, Q is a good orbifold. Now suppose that the local group at the orbit of codimension 2 is Z k . Let Hence Γ acts orbitwise on the manifold given by (G, H, K − , K + ) and commutes with the G-action. Thus G acts on (S 5 /Z q )/(Z m Z k ) with the same diagram as (4.19). By Proposition 4.18, Q admits a good structure.
Case 7.2. The diagram for this case is (S 3 × S 3 , ∆Z k , S 1 × Z k , ∆S 3 ). (4.23) Let the local group of the codimension 2 orbit be trivial. Then, by Proposition 4.16, π orb 1 (Q) = Z k . Consider the diagram (S 3 × S 3 , Z n , {(e ipθ , e iqθ )}, Z n ∆S 3 ), (4.24) of the smooth classification (see [25, Case 1 7 B, P 7 D ]), with n = 1 and p, q arbitrary, or n = 2 and p even or q even (but not both). In our case we assume n = 1 and q = 0. Recall that N = N G (H) ∩ N G (K − ) ∩ N G (K + ), where (G, H, K − , K + ) is the group diagram of the smooth manifold we are currently considering. For Γ = ∆Z k ⊆ N , we let Γ act on M orbitwise. Then we get an action on M/Γ giving rise to Diagram (4.23). Therefore, Q admits a good structure. Note that if the local group of the orbit of codimension 2 is Z l , where (l, k) = 1, then by Proposition 4.16, Q is simply-connected and, therefore, bad. with k|(p − q) and, if k is even, then p and q are odd. Suppose first that the local group at the codimension 2 orbit is trivial. Then by Proposition 4.16, π orb 1 (Q) = 0, i.e. Q is simply-connected and is therefore bad. Now let the local group be Z k . Then π 1 (R Q ) = π 1 (G/H) = Z k , which gives, in particular, that π orb 1 (Q) = Z k by Proposition 4.14. Consider the smooth diagram (4.24) above with the same p, q as diagram (4.26). Let Γ = ∆Z k ⊆ N ∩ {(e ipθ , e iqθ )} act orbitwise on M . Then S 3 × S 3 acts on M/Γ giving the same diagram as (4.26). Hence, Q is a good orbifold. with k|(p + q) and, if k is even, then p and q are odd. The same argument as in the preceding case gives the result.
Case 7.6. In this case we consider the following diagram: (S 3 × S 3 , ∆D * 2m , {(e ipθ , e iqθ )} ∪ {(je ipθ , je iqθ )}, ∆S 3 ), (4.28) where m|(p − q) and, if m is even, then p and q are odd. Consider Diagram (4.24) with n = 1. If the local group at the codimension 2 orbit is trivial, then, by Proposition 4.16, π orb 1 (Q) = Z 2 , and hence Q is a bad orbifold since D * 2m does not inject into Z 2 . Assume then that the local group at the codimension 2 orbit is Z m . Choose Γ = ∆D * 2m = ((e πip m , e πiq m ), (j, j)) ⊆ N , and act on M orbitwise. Then we have a S 3 × S 3 -action on M/Γ with the same diagram as (4.27). Thus Q is a good orbifold. with m|(p + q) and, if m is even, then p and q are odd. The same argument as above gives the result.