Abstract
We obtain an equivariant classification for orientable, closed, fourdimensional Alexandrov spaces admitting an isometric torus action. This generalizes the equivariant classification of Orlik and Raymond of closed fourdimensional manifolds with torus actions. Moreover, we show that such Alexandrov spaces are equivariantly homeomorphic to 4dimensional Riemannian orbifolds with isometric \({T^{2}}\)actions. We also obtain a partial homeomorphism classification.
1 Introduction
In in the presence of a continuous action by a fixed Lie group G on a space X there are usually two kinds of classifications. By homeomorphism classification, we mean characterizing all homeomorphism types of the spaces which admit such Gactions. By equivariant classification, we mean determining all possible actions of G in terms of “enough data”. This means being able to decide when two sets of data of the actions on two spaces give rise to an equivariant homeomorphism between them.
The study of \({T^{2}}\)actions on closed (i.e., compact and without boundary) 4manifolds was initiated by the work of Orlik and Raymond in [40], where they obtained an equivariant classification for such actions on orientable manifolds under the added condition that the action does not have exceptional orbits. In particular, they showed that the smooth classification is equivalent to the topological classification, i.e., a continuous effective action on a closed topological 4manifold is equivariantly homeomorphic to a smooth effective action on a closed smooth 4manifold. In a followup paper [41], Orlik and Raymond extended the equivariant classification for the case when the action does admit exceptional orbits. The classification of the homeomorphism type of smooth closed oriented 4manifolds with an effective \({T^{2}}\)action was carried out in [40,41,42,43]. We provide this classification in Sect 2.4. For the nonorientable case, both the equivariant classification problem and the homeomorphism type classification problem were addressed by Kim in [34].
Alexandrov spaces (with curvature bounded form below) are complete length spaces with a lower curvature bound in the sense of triangle comparison. By Toponogov’s Theorem, they generalize complete Riemannian manifolds with a uniform lower curvature bound. Riemannian orbifolds (with a lower sectional curvature bound), orbit spaces of isometric actions of compact Lie groups on Riemannian manifolds with sectional curvature bounded from below, and Gromov–Hausdorff limits of sequences of ndimensional Riemannian manifolds with a uniform lower bound on their sectional curvature are examples of Alexandrov spaces.
Twodimensional Alexandrov spaces are topological twomanifolds, possibly with boundary (see [4, Corollary 10.10.3]). Closed threedimensional Alexandrov spaces are either topological threemanifolds, or are homeomorphic to quotients of smooth threemanifolds by orientation reversing involutions with isolated fixed points (see [10] and cf. [27]). In higher dimensions though, similar general results do not exist. Considering Alexandrov spaces with a large amount of symmetry provides a systematic way of studying them. This framework has been successfully used in the smooth category to study, for instance, Riemannian manifolds with prescribed lower curvature conditions, such as positive Ricci or positive sectional curvature (see for example [20, 22,23,24,25, 49]). In particular, many authors studied such spaces in the presence of a circle or a torus action either in low dimensions or in general (see for example [11, 13, 19, 21, 29, 32, 35, 47]).
Since the group of isometries of a compact Alexandrov space is a compact Lie group [7], one may assume the existence of an isometric compact Lie group action and consider such spaces in analogy with the Riemannian case. This allows us to bring the theory of compact transformation groups into the context of Alexandrov spaces (see for example [2, 9, 12, 15, 16, 27, 28, 39, 50]).
In particular, there exists a (nonunique) maximal dimension torus acting by isometries on X. When the space has curvature bounded below by 1, Harvey and Searle showed in [27] that the dimension of any maximal torus contained in the isometry group is bounded above by \(\lfloor (n+1)/2\rfloor \). In particular, for a compact 4dimensional Alexandrov space with curvature bounded below by 1, if we have an effective \(T^k\)action by isometries, then \(k\leqslant 2\).
In the present article, we study the equivariant classification of effective, isometric \({T^{2}}\)actions on closed, orientable Alexandrov 4spaces. We show that the techniques used in the study of smooth effective \({T^{2}}\)actions on smooth 4manifolds extend to this metric setting. Furthermore, this family of Alexandrov spaces contains a subfamily of spaces having maximal Abelian symmetry (see [27]).
Our main results are the following: Given a closed, orientable Alexandrov 4space X with an effective, isometric \({T^{2}}\)action, first we find a set of invariants classifying X up to equivariant homeomorphism. More precisely, we show that given another closed orientable Alexandrov 4space Y with an effective, isometric \({T^{2}}\)action, if Y has the same set of invariants as X, then Y is equivariantly homeomorphic to X.
Theorem A
Let X be a closed, orientable Alexandrov 4space admitting an effective, isometric \({T^{2}}\)action. Then the set of inequivalent (up to equivariant homeomorphism) effective, isometric \({T^{2}}\)actions on X is in onetoone correspondence with the set of unordered tuples of isotropic and topological invariants
We give the precise definitions and value constraints of the invariants stated in Theorem 1 in Sect. 5.
Recall that a topologically regular Alexandrov space is an Alexandrov space with every tangent cone homeomorphic to an Euclidean space. In particular, we recover the following result of GalazGarcía [8] for any topologically regular Alexandrov 4space, when comparing with the equivariant classification of effective \({T^{2}}\)actions on orientable 4manifolds.
Corollary B
Let X be a closed, orientable and topologically regular Alexandrov 4space with an effective, isometric \({T^{2}}\)action. Then X is equivariantly homeomorphic to a smooth 4manifold and the \({T^{2}}\)action is equivalent to a smooth action.
Second, we show that the set of invariants in Theorem 1 is complete, that is, for each “legal” set of invariants we show that there exists a closed, orientable Alexandrov 4space X with an effective, isometric \({T^{2}}\)action, such that it has the given set of invariants. In fact, we show that it is possible to realize such an Alexandrov 4space as a closed Riemannian 4orbifold, thus obtaining the following result.
Theorem C
Let X be a closed, orientable Alexandrov 4space admitting an effective, isometric \(T^2\)action. Then X is equivariantly homeomorphic to the underlying topological space of a closed 4orbifold.
Theorem 1 strengthens the results of [27, Theorem D] in dimension 4, where the authors show that positively curved Alexandrov 4spaces with \({T^{2}}\)action are equivariantly homeomorphic to 4dimensional \({T^{2}}\)orbifolds. In fact, our theorem shows that this is always the case, regardless of who is the lower curvature bound on the Alexandrov space.
Due to Theorem 1, the homeomorphism classification for 4dimensional Alexandrov spaces with an effective isometric \({T^{2}}\)action is contained in the homeomorphism classification of orbifolds with an effective \({T^{2}}\)action. Such orbifolds were studied by Haefliger and Salem in [26], but they did not consider the homeomorphism classification. We recall that a 2korbifold with an effective \(T^k\)action is called a torus orbifold. These orbifolds were studied in [17, 48], for the case when the orbifold is simply connected and rationally elliptic. A simply connected topological space X is called rationally elliptic if it satisfies \({{\,\mathrm{dim}\,}}_{\mathbb {Q}}(H^*(X,{\mathbb {Q}}))<\infty \) and \({{\,\mathrm{dim}\,}}_{\mathbb {Q}}(\pi _*(X)\otimes {\mathbb {Q}})<\infty \).
Regarding the homeomorphism type of closed, orientable Alexandrov 4spaces with effective and isometric \({T^{2}}\)actions we provide a basic topological recognition result. More precisely we show that any closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action can be decomposed as an “equivariant connected sum” along subspaces homeomorphic to \(S^2\times S^1\), where one of the pieces is an OrlikRaymond 4manifold and the remaining pieces are “simple” Alexandrov 4spaces. We refer the reader to Sect. 7 for the details. From this basic decomposition it follows that the topological classification could be answered, by identifying the homeomorphism type of these “simple" Alexandrov pieces. The identification of such “simple" pieces is beyond the scope of the present work.
Remark 1.1
The proof of Theorem 1 is mainly topological. The synthetic curvature condition is used implicitly to describe small balls around points (see Proposition 2.7), and to study the local isotropy action (see Theorem 2.8). Moreover, for Theorem 1 we explicitly use the characterization of the spaces of directions.
The organization of the article is as follows. In Sect. 2 we recall the structural properties of closed Alexandrov 4spaces and the basic aspects and different notions of orientability of Alexandrov spaces. We also recall the main features of the theory of transformation groups on Alexandrov spaces. We conclude the section with a short summary of the classification of \({T^{2}}\)actions on 3 and 4dimensional manifolds. In Sect. 3 we describe the set of topologically singular points. In Sect. 4 we describe the local structure of the orbit space of a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action, around the different orbit types. We complete the analysis of Sect. 4 by providing the topological structure and distribution of the different orbits in the orbit space. In Sect. 5 we define the isotropy and topological invariants appearing in Theorem 1, as well as the proof of this result. In Sect. 6 we show the completeness of the invariants in Theorem 1 by constructing for every given set of invariants a closed, orientable Alexandrov 4space with a \({T^{2}}\)action. Finally in Sect. 7 we define the equivariant gluings along \(S^2\times S^1\) and prove the aforementioned topological recognition result.
2 Preliminaries
2.1 FourDimensional Alexandrov Spaces
We give a brief account of the main structural properties of closed fourdimensional Alexandrov spaces. For simplicity, throughout the article we refer to fourdimensional Alexandrov spaces as Alexandrov 4spaces. We refer the reader to [3, 4] for a general introduction to the theory.
Let X be a closed (i.e., compact and without boundary) Alexandrov 4space. The space of directions \(\Sigma _xX\) at each point \(x\in X\) is a closed Alexandrov 3space with \({{\,\mathrm{curv}\,}}\ge 1\). It is a consequence of the classification of closed positively curved Alexandrov 3spaces ([10, Theorem 1.1]) that the possible spaces of directions of a closed Alexandrov 4space are homeomorphic to \({\mathbb {S}}^3\), \({{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2)\) or a closed spherical 3manifold with nontrivial fundamental group (see [10, Corollary 2.3], see also [27, Proposition 3.6 and Corollary 3.7]). Namely
Proposition 2.1
[10, Corollary 2.3] The space of directions of a 4dimensional Alexandrov space without boundary is homeomorphic to \({{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2)\) or to a spherical threemanifold.
A point whose space of directions is homeomorphic to \({\mathbb {S}}^3\) is called topologically regular. Otherwise, the point is said to be topologically singular. The set of topologically regular points of X is an open and dense subset of X. The local structure of X is determined by Perelman’s Conical Neighborhood Theorem [45] which states that every point \(x\in X\) has a neighborhood pointedhomeomorphic to the cone over \(\Sigma _xX\).
2.2 Orientable Alexandrov Spaces
In this section, we recall the notion of orientability for Alexandrov spaces (see [36] for more details).
An Alexandrov space is called locally orientable if every point has an orientable neighborhood and locally nonorientable otherwise. Equivalently, we say that X is locally orientable if the space of directions at each point is orientable. That is \(H^n(X,X\smallsetminus \{p\}, {\mathbb {Z}}) \cong H^{n1}(\Sigma _p X;{\mathbb {Z}})\cong {\mathbb {Z}}\). Otherwise X is locally nonorientable ( [46, p. 124]).
Now we recall the global definition of orientability, following [36]:
Theorem 2.2
[36, Theorem 1.8] Let X be a compact ndimensional Alexandrov space. Then the following conditions are equivalent:

(a)
The manifold part \(X_{top}\) of X is orientable as an nmanifold;

(b)
\(H_n(X; {\mathbb {Z}})\cong {\mathbb {Z}}\);

(c)
\(H^n(X; {\mathbb {Z}})\cong {\mathbb {Z}}\) and the canonical morphism \(H^n(X, X\smallsetminus \{x\}; {\mathbb {Z}})\rightarrow H^n(X; {\mathbb {Z}}) \) is an isomorphism for any \(x\in X\).
Definition 2.3
[36, Definition 1.10] A compact Alexandrov space is called orientable if it satisfies one of the conditions (a)–(c) listed in Theorem 2.2.
Remark 2.4
For a compact Alexandrov space of dimension at least 2, the equivalence between items (a) and (c) in Theorem 2.2 it is also proven by Lemma 3.3 in [27].
Remark 2.5
It follows directly from Proposition 2.1 that a closed Alexandrov 4space is locally orientable if and only if no space of directions is homeomorphic to \({{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2)\).
2.3 Group Actions
Let X be a finitedimensional Alexandrov space. Fukaya and Yamaguchi showed in [7] that, as in the Riemannian case, the isometry group of X is a Lie group. Moreover, if X is compact then its isometry group is also compact. We consider isometric actions \(G\times X \rightarrow X\) of a compact Lie group G on X. The orbit of a point \(x\in X\) is denoted by \(G(x) \cong G/G_x\). Here, \(G_x= \{ g\in G \mid \ gx= x \}\) is the isotropy subgroup of x in G. The ineffective kernel of the action is defined as the closed subgroup of G given by \(\cap _{x\in X} G_x\). If the ineffective kernel is trivial, we say that the action is effective. In this article, we assume all actions considered to be effective unless stated otherwise. Given a subset \(A\subset X\) we denote its image under the canonical projection \(\pi : X\rightarrow X/G\) by \(A^*\). In particular, we denote the orbit space by \(X^*=X/G\).
The following generalization of the Principal Orbit Theorem for Alexandrov spaces was obtained in [9, Theorem 2.2].
Theorem 2.6
(Principal Orbit Theorem) Let G be a compact Lie group acting isometrically on an ndimensional Alexandrov space X. Then there is a unique maximal orbit type and the orbits with maximal orbit type, the socalled principal orbits, form an open and dense subset of X.
The following result stated in [12, Proposition 4] gives a description of the tangent and normal spaces to the orbits of a Lie group action on an Alexandrov space.
Proposition 2.7
Let X be an Alexandrov space admitting an isometric Gaction and fix \(x\in X\) with \(\dim G/G_x > 0\). If \(S_x \subset \Sigma _xX\) is the unit tangent space to the orbit \(G(x) = G/G_x\), and
then the following hold:

(1)
The set \(S^{\perp }_x\) is a compact, totally geodesic Alexandrov subspace of \(\Sigma _xX\) with curvature bounded below by 1, and the space of directions \(\Sigma _xX\) is isometric to the join \(S_x*S^{\perp }_x\) with the standard join metric.

(2)
Either \(S^{\perp }_x\) is connected or it contains exactly two points at distance \(\pi \).
We finally recall a crucial tool in the theory of transformation groups, the socalled Slice Theorem. This result provides a canonical identification of a small invariant neighborhoods of the orbit of through a given point x with a fiber bundle over the orbit whose structure group is \(G_x\). The fiber is the socalled slice, and for an Alexandrov space a slice can be identified with the cone over the space of unit normal directions to the orbit.
Theorem 2.8
(Slice Theorem, [27, Theorem B]) Consider a compact Lie group G acting by isometries on an Alexandrov space X. Then for all \(x\in X\), there is some \(\varepsilon _0 > 0\) such that for all \(\varepsilon < \varepsilon _0\) there is an equivariant homeomorphism
where \(K(S^{\perp }_x)\) is the cone over \(S^{\perp }_x\).
2.4 Torus Actions on 4Manifolds
We conclude the preliminaries section with a brief review of the equivariant and homeomorphism classifications due to Orlik, Raymond, Pao and Kim of closed 4manifolds admitting effective torus actions for the sake of completeness. Observe that closed 4manifolds with any Riemannian metric are Alexandrov spaces, since each Riemannian metric has a global lower bound for its sectional curvature. Thus, by taking an equivariant Riemannian metric, the classification of closed orientable 4manifolds with a torus actions, is a particular case of the main results of the present work.
Let us begin by recalling the equivariant classification of closed, orientable 4manifolds with an effective torus action.
Theorem 2.9
[40, 41] Let M be a closed, orientable 4manifold admitting a \({T^{2}}\)action by homeomorphisms. Then the set of inequivalent (up to equivariant homeomorphism) effective \({T^{2}}\)actions on M is in onetoone correspondence with the set of unordered tuples of isotropy and topological invariants
To avoid repetition, the definition of the invariants appearing in the previous theorem, and their admissible values, are defined in Sect. 5 for the more general case of isometric, effective torus actions on Alexandrov 4spaces. Let us only mention here that each invariant of the form \(\{a_{l},b_{l}\}\) corresponds to the invariant \(\big ( (a_{l},b_{l}), \pm 1\big )\) as formulated in Sect. 5. This is due to the fact that there are no topologically singular points in the manifold case.
Although it is not needed for the proofs of the results stated in Sect. 1, for the sake of completeness, we recall here that Kim obtained the equivariant classification for closed nonorientable 4manifolds admitting an effective torus action [34].
Different orbit types correspond to the possible isotropy groups of the torus action. An orbit is called a principal orbit or Porbit if the isotropy group of a point (and hence for any point) contained in the orbit is trivial. The set of points in M whose orbits are Porbits is denoted by P. Theorem 2.6, i.e., the Principal Orbit Theorem, states that P is an open and dense subset of M. Orbits with nontrivial finite isotropy acting by preserving the local orientation are called exceptional orbits or Eorbits. The subset of M of points on Eorbits is called E. For the case of 4dimensional manifolds, The Slice Theorem 2.8 implies that each Eorbit is isolated from other Eorbits, since we have a free linear action of a finite subgroup of \({T^{2}}\) on the circle \(S^1\) (see [40, p. 534] for the list of possible finite subgroups). The orbit of a point whose isotropy is isomorphic to the circle is called a circular orbit or Corbit. We denote the set of points of M contained in Corbits by C. Finally, the set of points which are fixed points of the action (that is, having isotropy subgroup equal to \({T^{2}}\)) are denoted by F.
The orbit space \(M^*\) is a 2manifold (possibly with boundary) in which the interior points are made up by \(P^*\cup E^*\). Moreover, \(E^*\) is a finite set. The boundary, which may be empty, is equal to \(F^*\cup C^*\).
For the case in which \(F\ne \emptyset \) some families of 4manifolds appear naturally as “building blocks” of 4manifolds admitting effective \({T^{2}}\) actions, and are presented next. Let M be a closed 4manifold admitting an effective \({T^{2}}\)action. We say that M is of type:

(i)
R(m, n), if \(\gcd (m,n)=1\), and the orbit space \(M^*\) is homeomorphic to an annulus where the inner circle boundary is composed of orbits with isotropy
$$\begin{aligned} G(m,n) =\{ (\varphi , \theta )\ \mid \ m\varphi + n\theta =0, \ \mathrm {gcd}(m,n)=1\}. \end{aligned}$$The outer circle boundary consists of two arcs joined by their endpoints where one of them consists of orbits with G(0, 1) isotropy, while the other consists of orbits with G(1, 0) isotropy.

(ii)
\(T(m,n;m',n')\), if \(\gcd (m,n)=\gcd (m',n')= 1\), \(mnm'n'=\pm 1\), and the orbit space \(M^*\) is homeomorphic to an annulus where the inner circle boundary is composed of two arcs joined at their endpoints: one of them has G(m, n) isotropy while the other has \(G(m',n')\) isotropy. The outer circle boundary consists of two arcs joined by their endpoints where the orbits in one of them has G(0, 1) isotropy, while the other arc consists of orbits with G(1, 0) isotropy.

(iii)
L(n; p, q; m), if \(\gcd (n,p,q)=1\), and the orbit space is homeomorphic to a 2disk with the center point corresponding to an Eorbit with (nonnormalized) Seifert invariants (n; p, q; m) (see Sect. 5) and with the boundary consisting of two arcs joined by their corresponding endpoints. One arc consists of orbits with isotropy G(m, n) while the other arc consists to orbits with isotropy G(p, q).
The homeomorphism type of these manifolds was completely described by Pao [42], and they are the following (see [42, Theorem III.3]):

(i)
For each m, n with \(\gcd (m,n)=1\),
$$\begin{aligned} R(m,n) = {\left\{ \begin{array}{ll} S^2\times S^2 \# S^3 \times S^1 &{} \text {if}\ mn\equiv 0 \ \ (\text {mod 2}),\\ {\mathbb {C}}P^2\# \overline{\mathbb {C}\mathbb {P}^2} \# S^3\times S^1 &{} \text {if}\ mn\equiv 1 \ \ (\text {mod 2}). \end{array}\right. } \end{aligned}$$ 
(ii)
Observe that since \(mnm'n'=\pm 1\) then at least two of the four integers \(m,n,m', n'\) must be odd and one is even. Take \(t=0\) if there are two even integers, and \(t=1\) otherwise. Then (see [42, Theorem IV.1]),
$$\begin{aligned} T(m,n;m',n') = {\left\{ \begin{array}{ll} S^2\times S^2 \# S^2\times S^2 \# S^3 \times S^1 &{} \text {if}\ t=0,\\ {\mathbb {C}}P^2\# \overline{\mathbb {C}\mathbb {P}^2}\# {\mathbb {C}}P^2\# \overline{\mathbb {C}\mathbb {P}^2}\# S^3\times S^1 &{} \text {if}\ t=1. \end{array}\right. } \end{aligned}$$ 
(iii)
For every m, n, p, q with \(\gcd (n,p,q)=1\), there are two homeomorphism types (see [42, Theorem V.1]):
$$\begin{aligned} L(n;p,q;m) = {\left\{ \begin{array}{ll} L(n;0,1;1) &{} \text {or}\\ L(n;1,1;1). &{} \end{array}\right. } \end{aligned}$$We point out that when n is odd, then L(n; 0, 1; 1) is homeomorphic to L(n; 1, 1; 1). But for the case when n is even, the spaces L(n; 0, 1; 1) and L(n; 1, 1; 1) do not have the same homotopy type (see [42, Theorem V.2]). Both of these manifolds result from a \({T^{2}}\)equivariant surgery of type (2, 3) on \(S^1\times S^3\). By performing a surgery of type (2, 3) on one element in the fundamental group of the manifold, we can produce at most two different manifolds (see [42, Corollary V.8]).
With these definitions in hand, we now summarize the whole topological classification of closed, orientable 4manifolds with an effective \({T^{2}}\)action.
Theorem 2.10
[40,41,42,43] Let M be a closed, orientable 4manifold admitting a \({T^{2}}\)action by homeomorphisms. Then the following hold true:

(1)
If M is simply connected, then \(E=\emptyset \), \(F\ne \emptyset \) and M is equivariantly homeomorphic to an equivariant connected sum of a finite number of copies of \(\mathbb {C}\mathbb {P}^2\), \(\overline{\mathbb {C}\mathbb {P}^2}\), \(\mathbb {C}\mathbb {P}^2\# \overline{\mathbb {C}\mathbb {P}^2}\), \(S^2\times S^2\) and \(S^4\). This connected sum decomposition is not necessarily unique (see [40, Sect. 5]).

(2)
Assume that \(F\cup C = \emptyset \). Then M admits a fibering over \(S^1\) where the fiber is a 3dimensional Seifert manifold (see [41, Sect. 2]).

(3)
If M is not simply connected and \(F\ne \emptyset \) then M is equivariantly homeomorphic to an equivariant connected sum of the simply connected manifolds appearing in item (1) and elementary 4manifolds of types L, R and T. The decomposition as a connected sum is possibly nonunique (see [41, Sect. 3]).

(4)
Suppose \(F=\emptyset \) and \(C\ne \emptyset \), then the following hold (see [43]):

(a)
If there are Corbits with isotropies \(G(m_1,n_1)\) and \(G(m_2,n_2)\) such that \(m_1n_2m_2n_1=\pm 1\), then M is an equivariant connected sum of copies of \(S^4\), \(S^2\times S^2\), \({\mathbb {C}}P^2\), \(\overline{{\mathbb {C}}P^2}\), \(S^3\times S^1\) and manifolds of type L.
In the remaining cases, that is, when there are no mutually orthogonal Corbits, the topological classification is not yet complete. However, the following properties hold:

(b)
If M and the action have the following set of invariants
$$\begin{aligned} \left\{ o,g,s, , \left\{ \left\langle p_i,q_i\right\rangle \right\} _{i=1}^s ; \big \{ (\alpha _j;\gamma _{j,1},\gamma _{j,2}) \big \}_{j=1}^k \right\} , \end{aligned}$$then, for the second Stiefel–Whitney class \(\omega _2(M)\) of M, we have \(\omega _2(M)\ne 0\) if and only if there exist integers i and j with \(2\le i\) and \(j\le s\) such that
$$\begin{aligned} p_i\equiv p_j\equiv q_i\equiv 1&\ \ (\text {mod 2}),\\ q_j\equiv 0&\ \ (\text {mod 2}). \end{aligned}$$ 
(c)
The integers \(2g+s\), k, \(\alpha _1, \ldots \alpha _k\) and \(m=\gcd (p_1,\ldots ,p_s)\) determine the fundamental group of M (see [43, Sect. 3]).

(a)
It is worth remarking that the case \(F\cup C = \emptyset \) was treated in [41] as a particular case of [6], where the authors classify holomorphic Seifert fiberings through socalled Bieberbach classes (see also the work of Cišang in [5]). We refer the reader to [6] and [41, Sect. 2] for a detailed exposition.
In view of the nonuniqueness of the connected sum decompositions in items (1) and (3) in Theorem 2.10 above, Pao defined the socalled normal decomposition for 4manifolds which admit an effective \({T^{2}}\)action with fixed points, and proved that this decomposition is unique. We say that a connected sum decomposition of such a 4manifold M into summands \(S^4\), \(S^2\times S^2\), \(S^3\times S^1\), \({\mathbb {C}}P^2\), \(\overline{{\mathbb {C}}P^2}\), and manifolds of type L, is a normal decomposition if the number of copies of \(S^4\), \({\mathbb {C}}P^2\), \(\overline{{\mathbb {C}}P^2}\) and the manifolds of type L, denoted by \(N(S^4)\), \(N({\mathbb {C}}P^2)\), \(N(\overline{{\mathbb {C}}P^2})\), N(L(n; 0, 1; 1)), and N(L(n; 1, 1; 1)) respectively, satisfy the following conditions:

(i)
\(N(S^4)=1\),

(ii)
\(N(\overline{{\mathbb {C}}P^2})=0\) or \(N(\overline{{\mathbb {C}}P^2})= N({\mathbb {C}}P^2)=1\),

(iii)
if \(N({\mathbb {C}}P^2)\ne 0\) then \(N(L(n;1,1;1))=0\) for all \(n=2,3,\ldots ,\)

(iv)
\(N(L(n;1,1;1))=0\) for all \(n=3,5,7,\ldots \).
Theorem 2.11
[42, Theorem II.4.2] Let M be a closed, orientable 4manifold admitting a \({T^{2}}\)action by homeomorphisms with \(F\ne \emptyset \). Then M has a unique normal decomposition.
2.5 Cohomogeneity one 3Manifolds
Throughout this article the classification of cohomogeneity one closed 3manifolds due to Mostert [37] and Neumann [38] is used and we review it here as well for the convenience of the reader.
Let G be a compact, connected Lie group acting by diffeomorphisms on a closed smooth 3manifold M with cohomogeneity one. Then the orbit space \(M^*\) is homeomorphic to either a circle \(S^1\), or to a closed bounded interval \(I=[1,1]\). In the first case, there is a single orbit type with corresponding isotropy H. Thus, M is a fiber bundle over \(S^1\) with fiber G/H and structure group N(H)/H. In this case, M can be classified by the components of N(H)/H.
In the case that the orbit space \(M^*\) is homeomorphic to I, the two endpoints of I correspond to the only two nonprincipal orbit types (which could be the same), with isotropy subgroups \(K^\), \(K^+\), while the interior corresponds to principal orbit types with isotropy H. Moreover, there is a group diagram made up of inclusions
where \(K^{\pm }/H\) are isometric to positively curved \(r_i\)spheres, with \(0\leqslant r_i \leqslant 2\). Using the Slice Theorem, it is possible to prove that M is equivariantly diffeomorphic to the union of two fiber bundles \( G\times _{K^+}D^{r_1+1}\) and \( G\times _{K^+}D^{r_2+1}\), where \(D^{r_i+1}\) is the closed unit disk in \({\mathbb {R}}^{r_i+1}\). Here the union is made via an equivariant diffeomorphism between the boundaries of \(G\times _{K^{\pm }}D^{r_i+1}\), which are diffeomorphic to a principal orbit G/H. Therefore, M can be classified via this construction by the components of the double quotient \(N_1 \backslash N(H) / N_2 \), where \(N_1= N(H)\cap N(K^{})\) and \(N_2= N(H)\cap N(K^{+})\).
Using these structure results, Mostert and Neumann obtained that either \(G={T^{2}}\) with \(H=\{e\}\), or \(G=\mathrm {SO}(3)\) with \(H=\mathrm {O}(2)\), or \(H= \mathrm {SO}(2)\). In Table 1 we collect the possibilities for the homeomorphism type of M, when \(M^*\) is a closed interval, together with the corresponding orbit structures. Here, by orbit structure we mean the ordered tuple \((H,K^,K^+)\). We denote the Klein bottle as \(\mathrm {Kl}\), the nonorientable \(S^2\) bundle over \(S^1\) by \(S^2{\tilde{\times }} S^1\), a lens space as L(p, q), the closed Möbius band as \(\mathrm {Mb}\), and A denotes the manifold obtained by gluing \(\mathrm {Mb}\times S^1\) to \(S^1\times \mathrm {Mb}\) via the identity map along the boundary \(S^1\times S^1\).
GalazGarcia and Zarei proved that for a topological manifold of dimension at most 4, any continuous group action by a Lie group is equivariantly diffeomorphic to a smooth action on a smooth manifold.
Theorem 2.12
[14, Corollary E] Consider a topological manifold M with an (effective) topological action of a compact Lie group G of cohomogeneity one. If M has dimension at most 4, then M is equivariantly homeomorphic to a smooth manifold N with an effective smooth action by G.
Therefore for dimension 3, Table 1 gives a complete list of continuous actions of \({T^{2}}\) on topological 3manifolds.
3 Local Structure of Topological Singular Points
In general, the set of topologically singular points \( Sing _X\) of an Alexandrov space X may be wildly arranged in X. However, in the presence of an isometric \({T^{2}}\)action the set \( Sing _X\) acquires more structure. We begin by pointing out that since \({T^{2}}\) acts by isometries, then \( Sing _X\) is an invariant subset of X under the action of \({T^{2}}\).
Observe that \( Sing _X\) is invariant under the isometric action of \(T^2\). We split the set \( Sing _X\) into the disjoint union of the following subsets:
3.1 Orbits of Points in \( Sing _X\)
As mentioned before, the canonical projection \(\pi :X \rightarrow X^*\) is a submetry. Therefore a small neighborhood \(B_{\varepsilon }(x^{*})\subset X^*\) centered at \(x^*\in X^*\) is homeomorphic to \(\pi (B_{\varepsilon }(x)) = B_{\varepsilon }(x)^*\), where \(\pi (x)=x^*\). The Conical Neighborhood Theorem of Perelman (see [44, Theorem 0.1], [33, Theorem 6.8]) together with Proposition 2.1 imply that \(B_{\varepsilon }(x)\) is either homeomorphic to a cone over the suspension of the real projective plane \(K({{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2))\), or a cone over a spherical manifold. Furthermore, Theorem 2.8 yields that \(B_{\varepsilon }(x)^*\) is homeomorphic to \(K(S^{\perp }_x/G_x)\).
To further investigate the local structure of \(X^*\) at orbits of points in \( Sing _X\) we split the analysis using the dimension of the orbits.
Proposition 3.1
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action. Then \( Sing _X= Sing ^0_X\), that is, the set of topologically singular points of X consists only of fixed points.
Proof
Assume that there exists \(x\in Sing _{X}^{2}\). By Proposition 2.7, we have \(\Sigma _x X = S_x *S_x^\perp \). Since \(\dim G(x) = 2\), \(S_x = S^1\), and thus the space of normal directions \(S_x^\perp \) is a positively curved Alexandrov 1space. Thus \(S_x^\perp \) is homeomorphic to \(S^1\). But this implies that the space of directions \(\Sigma _x X = S^1*S^1 = S^3\). Thus x is a regular point, which is a contradiction.
Assume there exists \(x\in Sing _{X}^{1}\). By Proposition 2.7, we have \(\Sigma _x X = S_x *S^\perp _x\). Since \(\dim G(x) = 1\), \(S_x\) consists of two points and hence the space of normal directions \(S_x^\perp \) is a positively curved Alexandrov 2space. Thus \(S_x^\perp \) is either homeomorphic to \(S^2\) or \({\mathbb {R}}P^2\). Therefore, we have that \(\Sigma _xX\) is either homeomorphic to \({{\,\mathrm{Susp}\,}}(S^2) = S^3\) or \({{\,\mathrm{Susp}\,}}( {\mathbb {R}}P^2)\). The first case means that x is a topologically regular point, which is a contradiction. Moreover, the case where \(\Sigma _xX = {{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2)\) contradicts the orientability of X. Hence, we conclude that \( Sing _{X}^{1}=\emptyset \). Recall that the isotropy subgroups, are closed subgroups of \({T^{2}}\). Since for \(x\in Sing _{X}^0\) we have an homeomorphism between \({T^{2}}(x)\) and \({T^{2}}/{T^{2}}_x\), we conclude that \({T^{2}}_x\) is a twodimensional subgroup. This implies that the isotropy subgroup is \({T^{2}}\), and x is a fixed point. \(\square \)
In the following lemma, we describe the normal space of directions for each orbit type of the effective action by isometries of \({T^{2}}\) on X.
Lemma 3.2
Let X be a closed, orientable Alexandrov 4space admitting an effective isometric \(T^2\)action. Then the normal space of directions to an orbit must be one of the following:

(a)
For a twodimensional orbit, the normal space of directions is homeomorphic to \(S^1\);

(b)
For a onedimensional orbit, the normal space of directions is homeomorphic to \(S^2\); and

(c)
For a zerodimensional orbit, the space of directions is homeomorphic to a 3dimensional spherical manifold.
Proof
We begin by proving (3.2). Let \(x\in X\) be an element contained in a twodimensional orbit. Thus its normal space of directions is a closed positively curved 1dimensional Alexandrov space by Proposition 2.7; i.e., it is homeomorphic to the circle \(S^1\). Consider now \(x\in X\) contained in a onedimensional orbit. Observe that by Proposition 3.1, it is topologically regular. Moreover, the tangent sphere to the orbit consists of two points. Thus by Proposition 2.7, its normal space of directions is a closed 2dimensional Alexandrov space of curvature bounded below by 1. Hence \(S^{\perp }_{x}\) is \(S^2\).
We consider \(x\in X\) in a zerodimensional orbit. Then, x is a fixed point and Proposition 2.1 gives that \(\Sigma _x X\) is either homeomorphic to a spherical 3manifold, or to \({{\,\mathrm{Susp}\,}}({\mathbb {R}}P^2)\). The orientability of X then implies that \(\Sigma _xX\) is a spherical 3manifold. \(\square \)
4 Local Structure of the Orbit Space
In this section, we give the local characterization of the action by describing the structure the orientable 4dimensional Alexandrov space \(X^*\) around the orbits of an isometric effective action of \({T^{2}}\) . We stress that most of the contents of this section are known (see for example, [40, 41]), but for the sake of completeness we review them here. From Lemma 3.2, it follows that an orientable 4dimensional Alexandrov space X with an isometric effective \({T^{2}}\)action is in fact a rational cohomology manifold. Note that \(X\setminus Sing ^0_X\) is an orientable manifold by Theorem 2.2. In particular, \(X\setminus Sing _X^0\) is an integral cohomology manifold. Such spaces were treated in [40, Appendix]. However, thanks to the Slice Theorem (Theorem 2.8), we have a clear picture of the local structure of an orientable 4dimensional Alexandrov space with an effective isometric \({T^{2}}\)action and its orbit space. This enables us to give a more precise description of the action.
The possible isotropy groups correspond to closed subgroups of \({T^{2}}\). In order to set notation, let us assume that \({T^{2}}\) is parametrized by \((\varphi ,\theta )\) with \(0\le \varphi \le 2\pi \) and \(0\le \theta \le 2\pi \). Then the closed subgroups of \({T^{2}}\) consist of \(\{1\}\), \({\mathbb {Z}}_n\), \({\mathbb {Z}}_n\times {\mathbb {Z}}_m\), the subgroups of the form
\(G(m,n)\times {\mathbb {Z}}_p\) and \({T^{2}}\) itself. Note that G(m, n) is isomorphic to the circle group \(\mathrm {SO}(2)\).
We write, \(G={T^{2}}\), and assume it acts isometrically and effectively on a closed, oriented Alexandrov 4space X. We list all the possible isotropy groups, slices and the local picture of the orbit space.
Proposition 4.1
Let X be an orientable closed Alexandrov 4space with an effective isometric \(T^2\)action. Then the possible isotropy groups are either the trivial one, \(T^2\), or of the form \({\mathbb {Z}}_p\), or G(m, n).
Proof
We only need to show that \(G(m,n)\times {\mathbb {Z}}_p\) and \({\mathbb {Z}}_n\times {\mathbb {Z}}_m\) cannot be the isotropy group of any \(x\in X\). We consider two cases: When x is not a topologically singular point, then the analysis done by Orlik and Raymond in [40, 41] applies. That is, for a topological regular point \(x\in X\) the possible isotropy groups are the trivial subgroup, \(T^2\), or one of the form \({\mathbb {Z}}_p\), or G(m, n) with \(\gcd (m,n) = 1\). From Proposition 3.1, we know that for the case when x is a topological singular point, it is a fixed point which concludes the proof. \(\square \)
4.1 Local Structure of \(X^{*}\) Around Points in \( Sing _{X}\)
Given a point \(x\in Sing _X\) we describe a small neighborhood in \(X^*\) of its orbit \(x^*\in X^*\), and the isotropies of the orbits contained in this neighborhood. First, recall that x is a fixed point of the \({T^{2}}\)action on X, and \(\Sigma _x X\) is a spherical space form by Lemma 3.2, Part (3.2). Moreover, it is not \( S^3\) as x is a singular point. Since \({T^{2}}\) acts on \(\Sigma _x X\) isometrically and effectively, the action is a continuous action of cohomogeneity one on a spherical space form.
Moreover, since the space of directions \(\Sigma _x X\) is a positively curved Alexandrov space, by the generalized Bonnet–Myers Theorem for positively curved Alexandrov spaces [27, Theorem 2.10], we conclude that \(\Sigma _x X\) has finite fundamental group. Since the fundamental group surjects onto the fundamental group of \(\Sigma _x X/{T^{2}}\), the orbit space \(\Sigma _xX /{T^{2}}\) is a closed interval.
By the classification of cohomogeneity one actions on 3dimensional manifolds [14, 38] and by the fact that \(\Sigma _x X\) is a spherical space form it follows that \(\Sigma _xX\) is a lens space L(q, p), and the orbit space \(\Sigma _xX/{T^{2}}\) has group diagram of the form \(({T^{2}}, G(1,0),G(p,q),\{1\})\), for some appropriate splitting of \({T^{2}}\) (see Table 1). Furthermore, it follows from [38] that \((p,q)\ne (1,0)\) and \((p,q)\ne (0,1)\). Otherwise we would have that \(\Sigma _xX\) is a 3sphere or \(S^2\times S^1\), respectively, which yields a contradiction, since \(x\in Sing _X\) and \(\Sigma _x X\) has finite fundamental group. Therefore, for a general group diagram of the form \(({T^{2}}, G(a,b), G(c,d), \{1\})\) this implies that we have \(adbc \ne 0\) and \(adbc \ne \pm 1\). Figure 1 depicts the local structure of \(X^*\) around \(x\in Sing ^0_X\).
Moreover, since X is compact, we conclude that the set \( Sing _X\) consists of a finite set of fixed points.
Lemma 4.2
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action. Then \( Sing _X\) consist of a finite number of fixed points.
4.2 Fixed Points
Assume that \(x\in X\smallsetminus Sing _X\) is a point with isotropy group \(G_x={T^{2}}\), i.e., x is a fixed point. To describe the local picture of the orbit space around \(x^*\), we need to examine the effective action of \({T^{2}}\) on the normal space of directions to the orbit, that is \(S_x^\perp = S^3\). A priori the action of \({T^{2}}\) on \(S^3\) is only continuous, but since it is of cohomogeneity one, by Theorem 2.12, there is an equivariant homeomorphism to the standard smooth \({T^{2}}\)action on \(S^3\). We recall that the standard action of \({T^{2}}\) on \(S^3\) is as follows:
where we regard \(S^3\subseteq {\mathbb {C}}^2\), and assume \(\mathrm {gcd}(m, n)=1\), and \(\mathrm {gcd}(p, q)=1\). Note that this action is effective if and only if \(mqnp=\pm 1\). The group diagram associated to the cohomogeneity one action of \({T^{2}}\) on \(S^3\) is \(({T^{2}}, G(m, n), G(p, q), \{1\})\). Then we have the local picture of \(X^*\) around \(x^*\) as in Fig 2(a).
4.3 OneDimensional Orbits
Assume that \(x\in X\smallsetminus Sing _X\) is a point with a onedimensional orbit. In this case, by Lemma 3.2, Part (3.2), we have that \(S_x^\perp \cong S^2\). Furthermore, the isotropy group \(G_x\) is also 1dimensional. To obtain the local structure of \(X^*\) around \(x^*\), namely, \(K((S_x^\perp )^*)\), the classification of the effective continuous actions of 1dimensional compact Lie groups on \(S^2\) is needed. By Theorem 2.12 this action can be assumed to be a smooth action. From [31, Sect. 2.3] we get that this action of \(G_x\) on \(S^2\) is equivalent to the action of \(\mathrm {SO}(2)\), by rotating \(S^2\) with respect to the northsouth poles. The group diagram of this cohomogeneity one action is \((G(m,n), G(m, n), G(m, n), \{1\})\) and the orbit space \((S_x^\perp )^*\) is a closed interval. Therefore, \(x^*\) is a boundary point of \(X^*\) and the local structure of \(X^*\) around such orbits is as in Fig. 2b.
4.4 Finite Isotropy Groups
Let \(x\in X\smallsetminus Sing _X\) and \(G_x\) be its isotropy group. Assume that \(G_x\) is finite. Then, the orbit G(x) is 2dimensional, and the normal space of directions to this orbit, \(S_x^{\perp }\), is 1dimensional. Therefore, \(S_x^{\perp }\cong S^1\).
To describe the slice and local picture of \(X^*\) around \(x^*\), we need to examine the effective actions of finite groups on \(S^1\), preserving the orientation. Since \(G_x\) is a closed subgroup (as it is finite) of the Lie group \({T^{2}}\), it is a compact Lie group by Cartan’s theorem. Since we are assuming it preserves the orientation of \(S^1\), it is conjugate to a closed subgroup of \(\mathrm {SO}(2)\) (see [18, Sect. 4.1]). Thus we conclude that \(G_x\) is a cyclic group acting by rotations on \(S^1\).
Hence, the local picture of the orbit space \(X^*\) around \(x^*\) is a 2disk contained in the interior of \(X^*\). All of the points contained in this disk have trivial isotropy except for \(x^*\), which has \(G_x\) as isotropy. Therefore, \(x^*\) is an isolated interior point.
4.5 Orbit Types
We have different orbit types according to the possible admissible isotropy groups of the action. In the following, we recall those that already appear in the manifold case, and define a new orbit type to account for the presence of topologically singular points.
As in the manifold case, an orbit with trivial isotropy group is called a principal orbit or Porbit. We denote the set of points in X whose orbits are Porbits by P. By Theorem 2.6, P is an open and dense subset of X consisting entirely of topologically regular points. Points having orbits with nontrivial finite isotropy are topologically regular points, and therefore, the concept of local orientation is well defined: An orbit with nontrivial finite isotropy acting by preserving the local orientation is called an exceptional orbit or Eorbit. The subset of X of points lying on Eorbits is denoted by E. Each Eorbit is isolated from other Eorbits. The orbit of a topologically regular point whose isotropy is isomorphic to the circle is called a circular orbit or Corbit. We denote the set of points on Corbits by C. Note that C consists entirely of topologically regular points. The set of topologically regular points which are fixed points of the action (that is, having isotropy \({T^{2}}\)) is denoted by RF. By definition RF consists only of topologically regular points. Finally we relabel the set of topologically singular points (which by our previous analysis are fixed points of the action) by SF.
We collect all of this information in Table 2.
In the following proposition, we summarize our previous discussions and state immediate consequences obtained by putting together the analysis of the local structure of \(X^*\) around orbits of topologically singular points and the analysis at topologically regular points obtained in [40, 41].
Proposition 4.3
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action. Then the following hold:

(1)
The orbit space \(X^*\) is a 2manifold (possibly) with boundary.

(2)
The interior of \(X^*\) consists of Porbits and a finite number of Eorbits.

(3)
Each connected component of the boundary of \(X^*\) consists of C, RF and SForbits.
5 Equivariant Classification
As done for smooth \({T^{2}}\)actions on 4manifolds (see [40, 41]) and \(\mathrm {SO}(2)\)actions on 3manifolds (see [39]), we may endow the orbit space \(X^*\) with a set of “weights" that encode its topology, the isotropy information of the action, and the topological regularity of the points in the orbit. In this section we prove that this set of weights characterizes the action. We list these invariants:

(i)
For a closed, oriented Alexandrov 4space X with an effective \({T^{2}}\)action by isometries we have by Proposition 4.3 that the orbit space \(X^*\) is an oriented 2manifold of a certain genus possibly with boundary. The integer \(g\ge 0\) denotes the genus of \(X^*\).

(ii)
Fix an orientation for \({T^{2}}\). Recall that the 2manifold \(X^*\) inherits an orientation from X via the orbit projection map and vice versa. We denote such an orientation of \(X^*\) by the symbol \(\varepsilon \). Hence, \(\varepsilon \) has two different possible values.

(iii)
The integer \(s\ge 0\) denotes the number of boundary components of \(X^*\) which consist only of Corbits. To each connected component of this type we associate the weight \(\left\langle p_i,q_i\right\rangle \) corresponding to the isotropy subgroup \(G(p_i,q_i)\) of the corresponding circular orbits.

(iv)
The integer \(t\geqslant 0\) is the number of boundary components of \(X^*\) which have nonempty intersection with \(RF^*\cup SF^*\). We label these boundary components from 1 to t. On the \(\ell \)th boundary component of this type, there are \(r_\ell \geqslant 2\) fixed points, and the orbits between two such fixed points are circular orbits with isotropy \(G(a_{\ell ,w},b_{\ell ,w})\). We set the weight \(f_{\ell ,w}\) to be the determinant of the isotropy groups \(G(a_{\ell ,w},b_{\ell ,w})\) and \(G(a_{\ell ,w+1},b_{\ell ,w+1})\), i.e., \(f_{\ell ,w} =a_{\ell ,w}b_{\ell ,w+1}a_{\ell ,w+1}b_{\ell ,w}\), for \(1\le w\le r_{\ell }1\), and \(f_{\ell ,r_{\ell }} =a_{\ell ,r_{\ell }}b_{\ell ,1}a_{\ell ,1}b_{\ell ,r_{\ell }}\). Therefore, for a fixed topologically regular point we have \(f_{\ell ,w} = \pm 1\), and for a fixed topologically singular point we have \(f_{\ell ,w} \ne \pm 1 \), and \(f_{\ell ,w} \ne 0\). Hence, we associate to the \(\ell \)th boundary component the sequence of weights
$$\begin{aligned} \big ( (a_\ell ,b_\ell ),f_\ell \big )= \big (&(a_{\ell ,1},b_{\ell ,1}),f_{\ell ,1},(a_{\ell ,2},b_{\ell ,2}),\ldots ,\\&(a_{\ell ,(r_\ell 1)},b_{\ell ,(r_\ell 1)}), f_{\ell ,(r_\ell 1)}, (a_{\ell ,r_\ell },b_{\ell ,r_\ell }),f_{\ell ,r_\ell } \big ).\nonumber \end{aligned}$$(5.1)If \(r_{\ell }=2\), for some \(1\le \ell \le t\), then we require that \(f_{\ell , 1}=f_{\ell , 2}\). This implies that for \(r_{\ell }=2\), both fixed points are topologically singular, or they both are topologically regular.

(v)
Let \(k\geqslant 0\) denote the number of exceptional orbits. To each exceptional orbit we associate the weight \((\alpha _l;\gamma _{l,1},\gamma _{l,2})\), the socalled oriented Seifert invariants of the orbit (see [41, p. 93] for the precise definition).

(vi)
In the case that \(C\cup RF\cup SF=\emptyset \) we further associate another invariant. Consider \(E^*=\{x_i^*\}_{i=1}^k\), and let \(D_i^*\) be disjoint closed 2disks in \(X^*\) centered at \(x_i^*\) such that \(D_i^*\setminus \{x_i^*\}\subset P^*\) for all \(1\le i \le k\). We arbitrarily choose a Porbit \(x_0^*\in X^*\setminus \bigcup _{i=1}^k D_i^*\) and let \(D_0^*\) be a small 2disk centered at \(x_0^*\) fully contained in \(P^*\). Let \(\chi :\bigcup _{i=0}^k \partial D_i^*\rightarrow X\) be a crosssection to the restriction of the action to \(\bigcup _{i=0}^k \partial D_i^*\). By standard Obstruction Theory this crosssection can be extended to \(X^*\setminus \bigcup _{i=0}^k \mathrm {int}(D_i^*)\) and the obstruction to extending it to \(X^*\setminus \bigcup _{i=1}^k \mathrm {int}(D_i^*)\) is an element
$$\begin{aligned} (b_1,b_2)\in H^2\left( X^*, \bigcup _{i=1}^k D_i^*; {\mathbb {Z}}\oplus {\mathbb {Z}} \right) \cong {\mathbb {Z}}\oplus {\mathbb {Z}}. \end{aligned}$$We refer the reader to [41, 1.3] for a more detailed exposition of this invariant.
In sum, to every orbit space \(X^*\) of an effective and isometric \({T^{2}}\)action on X we associate the following set of invariants,
In the following we show that, as in the smooth case, this set of weights classifies not only the space X up to equivariant homeomorphism, but also the action up to orientation. Some remarks are in order.
Remark 5.1
If one of the sets C, \(RF\cup SF\) or E is empty, we denote it on the set of invariants by –.
Remark 5.2
Observe that the set of invariants (5.2) reduces to the set of invariants given by Orlik and Raymond [40, 41] in the case that X is homeomorphic to a topological manifold. Indeed, in this case, \(SF=\emptyset \), and \(f_{\ell ,w}=\pm 1\) for all values of \(\ell \) and w, making the invariant \(f_{\ell ,w}\) superfluous (Fig. 3).
Remark 5.3
A boundary component of \(X^*\) may contain an odd number of topologically singular fixed points. In particular, it may contain a single topologically singular fixed point, as long as the total number of fixed points is at least three (see Example 7.3, with \((m_1,n_1) = (1,0)\), \((m_2,n_2) = (0,1)\), and \((m_3,n_3) = (m,1)\), for \(m\ne 0, \pm 1\)). This stands in contrast with the case of circle actions on 3dimensional closed Alexandrov spaces (cf. [39, Lemma 3.3] )
Let us now recall the definition of isomorphism between weighted orbit spaces.
Definition 5.4
Let X and Y be two closed, oriented Alexandrov 4spaces carrying effective and isometric \(T^2\)actions. We say that the orbit spaces \(X^*\) and \(Y^*\) are isomorphic if there exists an orientation preserving homeomorphism \(f:X^*\rightarrow Y^*\), such that for each \(x\in X^*\), the weight associated to \(f(x^*)\) is the same as the weight associated to \(x^*\).
5.1 CrossSectioning Theorems
As in the manifold case, a key step in achieving the equivariant classification of effective and isometric \({T^{2}}\)actions, is to show the existence of a crosssection to the quotient map \(\pi : X\rightarrow X^*\) in the absence of exceptional orbits.
This has been done in [40, Appendix] for the more general family of 4dimensional cohomology manifolds with a continuous \({T^{2}}\)action. As pointed out in Sect. 4, an orientable 4dimensional Alexandrov space X with an effective and isometric \({T^{2}}\)action is a cohomology manifold (see also [36, Corollary 8.7]). However, we present a more detailed proof here given that we have a clearer picture of the local structure of X.
We first consider the simpler case when the orbit space is homeomorphic to a 2disk and proceed in a similar fashion to the proof of Theorem 1.10 in [40] for smooth closed 4manifolds with a smooth \({T^{2}}\)action, by splitting the proof on the existence of the crosssection into several lemmas. More precisely, we show how we can define a crosssection on simple subsets (in terms of the topology and isotropy information) and then show it is possible to extend any given crosssection on such subsets to a full crosssection.
Lemma 5.5
(cf. [40, Lemma 1.6]) Let X be a compact Hausdorff space with an effective continuous \(T^2\)action. Assume that the orbit space is homeomorphic to \(I\times I\), where the orbits on the arc \([0,1]\times \{0\}\) all have isotropy G(m, n). Then there exist a crosssection \(\rho :X^*\rightarrow X\). Moreover, assume that over a connected subset \(A^*\) of \(J^*= \{0\}\times I \cup I\times \{1\}\cup \times \{1\}\times I\) we have defined a crosssection. Then this crosssection can be extended to a crosssection over all of \(X^{*}\).
Proof
Let G(a, b) be a circle subgroup of \(T^2\) such that \(an  bm = 1\), i.e., such that \(T^2 = G(m,n)\times G(a,b)\). Consider \(Y = X/G(m,n)\). Then G(a, b) acts on Y freely, and we have a principal circle fibration \(Y\rightarrow X^*\), with fiber G(a, b). Since \(X^*\) is contractible then Y is equivariantly homomorphic to \(X^*\times G(a,b)\) and there exist a crosssection \(\rho _1:X^*\rightarrow Y\). Next, we observe that over \(X^*\setminus I\times \{0\}\), we have a principal \(G(m,n)\times G(a,b)\)bundle. Again, since \(X^*\setminus I\times \{0\} \) is contractible, we conclude that over \(X^*\setminus I\times \{0\} \) the bundle is trivial, i.e., the total space is \((X^*\setminus I\times \{0\} )\times G(m,n)\times G(a,b)\), and in particular we have a crosssection \(\rho _2:Y \setminus (I\times \{0\} \times G(a,b) ) \rightarrow (X^*\setminus I\times \{0\} )\times G(m,n)\times G(a,b)\subset X\). We can extend this crosssection to all of Y since over \(I\times \{0\} \times G(a,b)\) we have only the fixed points of the G(m, n)action on X. Thus, combining the crosssections \(\rho _1:X^*\rightarrow Y\) and \(\rho _2:Y\rightarrow X\) we obtain the desired crosssection \(\rho :X^*\rightarrow X\). We also conclude that X is equivariantly homeomorphic to \(X^*\times G(m,n)\times G(a,b)/\sim \), where \(((t,0),\zeta _1,\xi )\sim ((t,0),\zeta _2,\xi )\) for \(t\in I\), \(\zeta _1,\zeta _2\in G(m,n)\), and \(\xi \in G(a,b)\).
Assume that we are given a crosssection \(\sigma :A^*\rightarrow X\). Since X is equivariantly homeomorphic to \(X^*\times G(m,n)\times G(a,b)/\sim \), we have a crosssection \(\sigma :A^*\rightarrow X^*\times G(m,n)\times G(a,b)/\sim \). We prove that we can extend this crosssection to all of \(X^*\). We fix \(Z = X^*\times G(m,n)/\sim \), where \(((t,0),\zeta _1)\sim ((t,0),\zeta _2)\) for \(t\in I\) and \(\zeta _1, \zeta _2\in G(m,n)\). Observe that Z is the orbit space of the G(a, b)action on \(X^*\times G(m,n)\times G(a,b)/\sim \). Thus, we have two projection maps \(\pi _1 :X^*\times G(m,n)\times G(a,b)/\sim = X \rightarrow Z\) and \(\pi _2 :Z\rightarrow X^*\). Moreover, the crosssection \(\sigma :A^*\rightarrow X^*\times G(m,n)\times G(a,b)/\sim \) is of the form \(\sigma (a^*) = [a^*,\zeta (a^*), \xi (a^*)]\). Observe that we have a crosssection \(\sigma _2:A^*\rightarrow Z\) for \(\pi _2\) defined as \(\sigma _2(a^*) = [a^*,\zeta (a^*)]\). We fix \(A_1 = \sigma _2(A^*)\subset Z\), and \(A= \sigma (A^*)\subset X^*\times G(m,n)\times G(a,b)/\sim \). We obtain a crosssection \(\sigma _1:A_1\rightarrow A\) for \(\pi _1\), given by \(\sigma _1[a^*,\zeta (a^*)] = [a^*,\zeta (a^*),\xi (a^*)]\).
We show first that we can extend \(\sigma _2\) to \(J^*\). We assume that \(A^*\) is a proper subset of \(J^*\). After removing the fixed points \(\{(0,0), (0,1)\}\) from \(J^*\), and possibly \(A^*\), we get two spaces \(J_1^*\subset J^*\) and \(B^*\subset A^*\) which have the same homotopy type. Moreover, we have a circle fibration \(\pi _2:\pi _2^{1}(J_1^*)\rightarrow J_1^*\), since the action of G(m, n) over \(\pi _2^{1}(J_1^*)\) is free. From the fact that \(J_1^*\) has the same homotopy type as \(B^*\), we conclude that \(H^2(J_1^*,B^*;{\mathbb {Z}})=0\), and thus the obstruction to extend \(\sigma _2\) from \(B^*\) to \(J_1^*\) vanishes (see [30, pp. 415–419]). Therefore, we can extend the map \(\sigma _2:B^*\rightarrow Z\) to \(J^*_1\). By adding back the fixed two fixed points to \(J_1^*\), we get a crosssection \(\sigma _2:J^*\rightarrow Z\) extending the crosssection \(\sigma _2:A^*\rightarrow A_1\).
We show that we can extend this crosssection \(\sigma _2:J^*\rightarrow Z\) to a crosssection \(\sigma _2:X^*\rightarrow Z\). Consider the point \(p^*= (1/2,0)\in X^*\). Observe that \(X^*\) is homeomorphic to the cone of \(J^*\) with vertex \(p^*\), denoted by \(C(p^*,J^*)\). For \(x^*\in J^*\) and \(1\geqslant t>0\) we set \(\sigma _2(x^*,t) = [(x^*,t),\zeta (x^*)]\in Z\), and at the vertex, \(\sigma _2(p^*) = [p^*,\zeta ]\) for any \(\zeta \in G(m,n)\).
Now we prove we can extend the crosssection \(\sigma _1:A_1\rightarrow X\) to a crosssection \(\sigma _1:Z\rightarrow X\). Recall that \(\pi _1:X\rightarrow Z\) is a fibration with fiber a circle, since the action of G(a, b) on X is free. We point out that \(A_1 = A^*\times G(m,n)/\sim \) has the same homotopy type as \(Z = X^*\times G(m,n)/\sim \). From this it follows that \(H^2(Z,A_1;{\mathbb {Z}}) =0\), and thus the only obstruction to extending the given crosssection \(\sigma _1:A_1\rightarrow X^*\times G(m,n)\times G(a,b)/\sim \) to the whole of Z vanishes (see [30, pp. 415–419]). Composing \(\sigma _2\) with \(\sigma _1\) we obtain a crosssection \(\sigma = \sigma _1\circ \sigma _2 :X^*\rightarrow X\) extending the given crosssection \(\sigma :A^*\rightarrow X\). \(\square \)
Remark 5.6
Observe that in the previous proof, we can assume that the corner point \((0,0)\in X^*\) is a fixed point of the \(T^2\) action on X, and we still recover the same conclusion: given a crosssection on \(A^*\), we can extend this crosssection to \(X^*\).
Lemma 5.7
(cf. [40, Lemma 1.8]) Let X be a compact Hausdorff space with an effective continuous action of \({T^{2}}\), and assume that \(X^*= [1, 1] \times I\), where the orbits in \([1, 0)\times \{0\}\) have isotropy subgroup G(m, n), the orbits in \((0, 1] \times \{0\}\) have isotropy subgroup G(p, q) such that \(mpnq \ne 0\), the point \((0, 0)\in X^*\) is a fixed point, and all other points correspond to principal orbits. Then there is a crosssection to this action. Moreover any crosssection on a connected subset A of the arc \( J^*= \{1\} \times I \cup [1,1]\times \{1\} \cup \{1\} \times I\) may be extended to a crosssection over \(X^*\).
Proof
We split the orbit space \(X^*\) into two components: \(X_1^*= [1,0]\times I\), and \(X_2^*= [0,1]\times I\). Assume without loss of generality that \(A^*\cap X_1\ne \emptyset \). We then apply Remark 5.6 to obtain a crosssection \(\sigma _1:X_1^*\rightarrow X\) extending the given crosssection \(\sigma :A^*\cap X_1^*\rightarrow X_1\). We then apply Remark 5.6 to the crosssection \(\sigma _2:\{0\}\times I \cup A^*\cap X_2^*\rightarrow X\), where \(\sigma _2(0,t) = \sigma _1(0,t)\) for \(t\in I\), and \(\sigma _2(a^*) = \sigma (a^*)\) for \(a\in A^*\cap X_2^*\). Thus we obtain a continuous crosssection \(\sigma :X^*\rightarrow X\) by setting \(\sigma (x^*)= \sigma _1(x^*)\) for \(x^*\in X_1^*\), and \(\sigma (x^*)= \sigma _2(x^*)\) for \(x^*\in X_2^*\) \(\square \)
Another lemma that we need to prove the existence of the crosssection is as follows:
Lemma 5.8
[40, Lemma 1.7] Suppose that \(\pi :X\rightarrow X^{*}\) is the orbit map of a \({T^{2}}\)action on a compact Hausdorff space and assume that \(X^{*} = \mathrm {S}^{1} \times I\), where \(\mathrm {S}^{1} \times \{0\} \) corresponds to orbits with isotropy group G(m, n) with \(\mathrm {gcd}(m, n)= 1\) and all other points to principal orbits. Then the map \(\pi \) has a crosssection. Moreover any crosssection on \(\mathrm {S}^{1} \times \{0\}\) may be extended to a crosssection over \(X^{*}\).
Remark 5.9
We observe that the hypotheses of both lemmas only require the space to be Hausdorff, a condition which is satisfied by Alexandrov space since they are metric spaces.
Proposition 5.10
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \(T^2\)action. If the orbit space \(X^*\) is homeomorphic to a 2disk and \(E=\emptyset \), then there exists a crosssection.
Proof
We follow the same procedure as done in the proof of [40, Theorem 1.10]. If \(RF\cup SF=\emptyset \), then X is homeomorphic to a 4manifold and the torus action in question has no fixed points. In this case, the result is exactly Case 1 of [40, Theorem 1.10].
Therefore, we assume that \(RF\cup SF\ne \emptyset \). The strategy to deal with this case follows along the lines of Case 2 of [40, Theorem 1.10]. Let \(RF^*\cup SF^*=\{ x_1^*,\ldots , x_r^*\}\). We split \(X^*\) as \(Y^*\cup D_1^*\cup \cdots \cup D_{r}^*\), where \(Y^*\) is a closed 2disk contained in the interior of \(X^*\), and each \(D_{i}^*\) is homeomorphic to \([1,1]\times I\), and satisfies: \(D_i^*\cap \left( RF^*\cap SF^*\right) =\{x_i^*\}\), and \(D_i^*\cap Y^{*}\) corresponds to \([1,1]\times \{1\}\). The set \(D_i^*\cap D_{i+1}^*\) corresponds to \(\{1\}\times I\) and the distribution of isotropies on \(D_i^*\) is exactly as in Lemma 5.7 (identifying \(D_i^*\cap \partial X^*\) with \([1,1]\times \{0\}\) and \(x_i^*\) with \(\{0\}\times \{0\}\)). Since \(Y^*\) does not contain orbits with nontrivial isotropy, the restriction of the quotient map \(\pi :\pi ^{1}(Y^*) \rightarrow Y^*\) is a principal \(T^2\)bundle. Furthermore, since \(Y^*\) is contractible, this principal bundle is trivial. Thus, there exists a crosssection \(\eta :Y^*\rightarrow \pi ^{1}(Y^*)\).
By considering the restriction of \(\eta \) to \(D_i^*\cap Y^{*}\) we fall into the hypotheses of Lemma 5.7. Hence, by this result, we can extend \(\eta \) to \(Y^*\cup D_1^*\). Slightly abusing the notation we do not rename the crosssection. Similarly, by Lemma 5.7 and Remark 1.9 in [40], the restriction of \(\eta \) to \(D_2^*\cap (Y^*\cup D_1^*)\) can be extended to a crosssection on \(Y^*\cup D_1^*\cup D_2^*\). Proceeding inductively, it is clear that there exists an extension of \(\eta \) to \(X^*\), thus proving the result. \(\square \)
5.2 Equivariant Classification
Theorem 5.11
Let \(X_1\) and \(X_2\) be two closed, oriented Alexandrov 4spaces each with an effective, isometric \({T^{2}}\)action. Then \(X_1\) and \(X_2\) are equivariantly homeomorphic if and only if \(X_1^*\) and \(X_2^*\) are isomorphic.
Proof
The “only if” part is clear. Therefore, we focus on showing that if \(X_1^*\) is isomorphic to \(X_2^*\), then \(X_1\) is equivariantly homeomorphic to \(X_2\). To this end, we begin by observing that if \(SF_i=\emptyset \), then \(X_i\) are homeomorphic to 4manifolds. In this case the proof is as in [40, 41]. Hence, in the following we assume \(SF_i\ne \emptyset \). Suppose there is an isomorphism \(h^*:X_1^*\rightarrow X_2^*\) of orbit spaces. We also consider the projection maps \(\pi _1:X_1\rightarrow X_1^*\) and \(\pi _2:X_2\rightarrow X_2^*\). Assume there are exceptional orbits \(x_1^*,\ldots ,x_k^*\) in the interior of \(X_1^*\). Since all the topologically singular points lie in the boundary of \(X_1^*\), this exceptional orbits are topologically regular points, and thus via the slice theorem we can find small tubular neighborhoods \(D_s\subset X\) such that \(D_s^*\smallsetminus \{x_s^*\}\) and \(h^*(D_s^*)\smallsetminus \{h^*(x^*)\}\) contain only principal orbits. Since \(H^2(\partial D_s^*,{\mathbb {Z}}\oplus {\mathbb {Z}})=0\) and \(H^2(\partial h^*(D_s^*),{\mathbb {Z}}\oplus {\mathbb {Z}})=0\), we have crosssection on the boundaries of \(D_s^*\) and \(h^*(D^*_s)\). Furthermore, assume the boundary of \(X_1^*\) consist of m connected components. Set \(U_j^*\) to be a closed annular neighborhood of the jth component.
Then for \(X_{1,1}=X_1\setminus (D_1^*\cup \cdots \cup D_k^*\cup U_1^*\cup \cdots \cup U_m^*)\), and \(X_{1,2} = X_2\setminus (h^*(D_1^*)\cup \cdots \cup h^*(D_k^*)\cup h^*(U_1^*)\cup \cdots \cup h^*(D_m^*))\) from the proof of Theorem 1.10 in [40], the sections on each \(\partial D_s^*\) can be extended to a section \(\sigma _1:X_{1,1}^*\rightarrow X_{1,1} = \pi _1^{1} (X^*_{1,1})\). The same is true for the sections over the boundaries of \(h^*(D_s^*)\); we can extend them to a crosssection \(\sigma _2:X_{1,2}^*\rightarrow X_{1,2} = \pi _2^{1}( X_{1,2}^*)\). Since the action of \(T^2\) over \(X_{1,1}\) and \(X_{1,2}\) is free, we conclude that \(X_{1,1}\) is equivariantly homeomorphic to \(X_{1,1}^*\times T^2\), and \(X_{1,2}\) is equivariantly homeomorphic to \(X_{1,2}^*\times T^2\). Thus we have an equivariant homeomorphism \(h:X_{1,1}^*\times T^2 \rightarrow X_{1,2}^*\times T^2\) given by \(h(x^*,\xi ,\zeta ) = (h^*(x^*), \xi ,\zeta )\).
The sections \(\sigma _1\) and \(\sigma _2\) can be extended over each annulus \(U_j^*\), and \(h^*(U_j^*)\) by Theorem 1.10 in [40] and Proposition 5.10. Consider \(x^*\in U_j^*\). Observe that for any \(y\in \pi _1^{1}(x^*)\), there exists and element \((\xi , \zeta )\in T^2\), such that \(y = (\xi , \zeta )\sigma _1(x^*)\); the same is true for \(h(x^*)\in h^*(U_j^*)\). Set \(X^*_{2,1} = X^*\smallsetminus (D^*_1\cup \cdots \cup D^*_k)\), \(X_{2,1} = \pi _1^{1}(X_{2,1}^*)\),and \(X_{2,2} = X_2\smallsetminus (D_1^*\cup \cdots \cup D_k^*)\), \(X_{2,2} = \pi _2^{1}(X_{2,2}^*)\). We can extend the equivariant homeomorphism h, to an equivariant homeomorphism \(h:X_{2,1}\rightarrow X_{2,2}\) as follows: for \((\xi ,\zeta )\sigma _{1}(x^*)\in U_j= \pi _1^{1}(U_j^*)\) we set \(h((\xi ,\zeta )\sigma _{1}(x^*)) = (\xi ,\zeta )\sigma _2(h^*(x^*))\).
To finish the proof, we have to extend the equivariant homeomorphism h to each \(D_s\). We recall that the actions on each \(D_s\) are completely understood (see [41, Sect 1.1]). In this sense, there is a unique equivariant way to attach each of them to \(X_{2,1}\) and \(X_{2,2}\). Therefore, we have an equivariant homeomorphism \(h:X_1\rightarrow X_2\) lifting the given isomorphism \(h^*\). \(\square \)
6 Construction of an Alexandrov Metric
We have so far showed that two closed, oriented Alexandrov 4spaces with an effective and isometric \(T^2\)action which have the same set of invariants must be equivariantly homeomorphic. To complete the equivariant classification we show that given an arbitrary set of invariants as in (5.2), there exists a closed orientable Alexandrov 4space X and an effective and isometric \(T^2\)action on X which has precisely this set of invariants. More precisely, we assume that we are given a compact 2dimensional topological manifold endowed with a set of invariants as in Sect. 5. Then using a result of Haefliger and Salem we construct a smooth orbifold with a smooth \({T^{2}}\)action giving rise to the given weighted orbit space. From this we construct an invariant Riemannian orbifold metric with curvature bounded below.
Since it is a central tool in this section, for the sake of completeness, we recall the result of Haefliger and Salem.
Proposition 6.1
[26, Proposition 4.5] Let W be a paracompact space and \(\{V_i\}_{i\in I}\) an open covering of W such that for each \(i\in I\), there exists a \(T^{n}\)orbifold \(X_i\) with orbit space equal to \(V_i\). Assume the following compatibility condition: Let \(\pi _i:X_i\rightarrow V_i\) be the projection map of each \(T^{n}\)action. For each \(w\in V_i\cap V_j\), there is an open neighborhood V of w in \(V_i\cap V_j\) and a \(T^n\)equivariant diffeomorphism of \(\pi _i^{1}(V)\) onto \(\pi _j^{1}(V)\) over V. Then to these compatible local data there is an associated cohomology class in \(H^{3}(W,{\mathbb {Z}}^n)\) whose vanishing is equivalent to the existence of a \(T^n\)orbifold X with orbit space W, such that \(\pi ^{1}(V_i)\) is locally equivalent to \(X_i\), where \(\pi :X\rightarrow W\) is the projection map.
In the following definition we make explicit the restrictions that the a given set of weights needs to satisfy in order to be considered the set of invariants of an \({T^{2}}\)action on a 4dimensional orbifold.
Definition 6.2
Consider \(X^*\), a compact topological 2manifold (possibly with boundary) which has associated to it the following set of weights:
We say that \(X^*\) is legally weighted if the determinant of two adjacent weights, (a, b) and (c, d), in \(X^*\) is nonzero, i.e., \(adbc\ne 0\).
Remark 6.3
The condition of being legally weighted is required to get, at a fixed point, as space of directions a suspension of a spherical space. In other words to exclude \(S^2\times S^1\) as space of directions (see [38]).
Remark 6.4
Let \(X^*\) be homeomorphic to a 2disk with two fixed points: one, topologically regular and the other, topologically singular. In other words we have the following weight information:
where \(\delta \ne \pm 1\). By Sect. 5, Item (iv), we exclude such weighted orbit space.
Remark 6.5
Observe that if X is a closed orientable 4dimensional Alexandrov space with an effective \({T^{2}}\)action by isometries, then the orbit space \(X^*\) with the set of invariants described in Sect. 3, is a legally weighted space.
The following theorem shows that legal sets of weights are realized by orbifolds.
Theorem 6.6
Let \(X^*\) be the legally weighted topological 2manifold with weights as in (6.1). Then there exists an orbifold \({\mathcal {O}}_X\) with a \({T^{2}}\)action, such that \({\mathcal {O}}_X / {T^{2}}\) is isomorphic to \(X^*\).
Proof
We begin by pointing out that if \(SF^*=\emptyset \) then the result follows directly from [40, 41], and in this case \({\mathcal {O}}_X\) is in fact a smooth 4manifold. Thus, in the following we assume that \(SF\ne \emptyset \). To prove the result in this case we construct an open cover of \(X^*\) that satisfies the hypothesis of [26, Proposition 4.5]: we give an open cover \(\{V^*_i\}\) of \(X^*\), such that for each open set \(V^*_i\), there exists an orbifold \(V_i\) with a smooth \({T^{2}}\)action, projection map \(\pi _i:V_i\rightarrow V_i/{T^{2}}\), such that \(V_i^*\) is the orbit space of the action, and for \(w^*\in V_i^*\cap V_{j}^*\), there exists an open neighborhood \(V^*\subset V_i^*\cap V_j^*\) of \(w^*\), and an equivariant diffeomorphism between \(\pi _{i}^{1}(V^*)\) and \(\pi _{j}^{1}(V^*)\).
Recall that \(t+s\) is the total number of boundary components of \(X^*\). Let us write \(\partial X^*= (\sqcup _{i=1}^q \partial X^*_i)\sqcup (\sqcup _{j=q+1}^{t+s}\partial X^*_j)\) for some \(q\ge 0\) in such a way that for \(q+1 \leqslant j \leqslant t+s\), the boundary component \(\partial X^*_j\) contains at least one SForbit, and for \(1 \leqslant i \leqslant q\), the boundary component \(\partial X^*_i\) contains no SForbit (i.e., it contains only C and RForbits). Here, we adopt the convention that if \(C^*,RF^*=\emptyset \) then \(q=0\) and \((\sqcup _{i=1}^q \partial X^*_i)=\emptyset \).
Consider the following open cover for \(X^*\). Assume that \(W_i^*\) and \(U_{i}^*\), \(q+1\le i\le t+s\), are annular neighborhoods around each boundary component with \(SF^*\)points, such that \({\bar{U}}_{i}^*\) is properly contained in \(W_i^*\), and \(W_i^*\cap W_j^*=\emptyset \), for all \(i\ne j\). Let us denote \(Y^*=X^*\setminus {\bigcup _{i=q+1}^{t+s} {\bar{U}}_i^*} \). Note that \(Y^*\) is a 2surface of genus g with boundary. We now divide each open neighborhood \(W_i^*\) into smaller neighborhoods \(Z^*_{ij}\), where each \(Z^*_{ij}\) is required to contain only a single fixed point, and \(\bigcup _j Z_{ij}^*=W_i^*\) (see Fig. 4).
Now we construct smooth 4orbifolds equipped with effective \({T^{2}}\)actions over each of the pieces of the open cover \(\{Y^*\}\cup (\cup _{ij}\{Z_{ij}^*\}\), and show that on the nontrivial intersections they are compatible in the sense of Proposition 6.1.
6.1 Orbifold Corresponding to \({\mathbf {Y}}^{*}\)
We point out that the number of boundary components in \(X^*\) containing only RForbits is given by \(qs\) in the notation above. With this observation we point out that by [40, 41], the set of invariants
determines a unique (up to equivariant homeomorphism) closed, smooth 4manifold \({\hat{Y}}\) with an effective \({T^{2}}\)action having these invariants. Note that \({\hat{Y}}^*\) has exactly \(q\ge 0\), boundary components: s boundary components containing only circular isotropy; and \(qs\) boundary components containing at least one regular fixed point. Recall that t is the number of boundary components of \(X^*\) containing fixed points. The number of boundary components containing SForbits is given by \(t+sq\). Let \({\hat{Y}}_1^*\) be the subset of \({\hat{Y}}^*\) obtained by taking out sufficiently small, disjoint, closed neighborhoods of \(t+sq\) (arbitrary) points with trivial isotropy. Then \({\hat{Y}}_1^*\) is isomorphic to \(Y^*\). Set Y to be the inverse image of \({\hat{Y}}_1^*\) under the projection map \({\hat{Y}}\rightarrow {\hat{Y}}^*\), associated to the \({T^{2}}\)action on \({\hat{Y}}\). Notice that Y is an open submanifold of \({\hat{Y}}\) which is invariant under the action of \({T^{2}}\).
6.2 Orbifolds Corresponding to \(\mathbf {Z_{ij}^*}\)
Let \(x_{ij}^*\in Z_{ij}^*\) be the fixed point in this neighborhood with isotropies \(G(a_{ij},b_{ij})\) and \(G(a_{i(j+1)},b_{i(j+1)})\), with \(a_{ij}b_{i(j+1)}a_{i(j+1)}b_{ij}=f_{ij}\). From Sects. 4.1 and 4.2, we know that the open cone over a lens space has an \({T^{2}}\)action such that the orbit space is isomorphic to \(Z_{ij}^*\). We point out that the cone over a lens space admits an orbifold structure. We denote such a cone by \(Z_{ij}\).
6.3 Compatibility of the Open Cover
We now show that over the overlaps in \(X^*\), the smooth 4orbifolds we have produced are equivariantly diffeomorphic. That is, we need to examine the intersection of \(Z_{ik}^*\) with \( Y^*\), and the intersection of \(Z_{ij}^*\) with \(Z_{i(j+1)}^*\), for \(j =1,\ldots , r_i\) with the identification \(r_i+1 = 1\).
In the first case, the intersection \(Z_{ik}^*\cap Y^*\) is the product of two open intervals \(I_1\times I_2\) (see Fig. 4). Set \(A = \pi ^{1}_{Y}(I_1\times I_2)\subset Y\). Observe that A is contained in a smooth manifold, and the action of \({T^{2}}\) on A is free. Since \(A/{T^{2}}= I_1\times I_2\) is contractible, we conclude that A is equivariantly diffeomorphic to \({T^{2}}\times I_1\times I_2\) via a diffeomorphism \(\psi _A:{T^{2}}\times I_1\times I_2\rightarrow A\). Analogously we point out, that \(B = \pi ^{1}_{Z_{ik}}(I_1\times I_2)\subset Z_{ik}\) is a manifold, and the action of \({T^{2}}\) on B is free. Since \(B/{T^{2}}= I_1\times I_2\) is contractible, we conclude that there exists an equivariant diffeomorphism \(\psi _{B}:B \rightarrow {T^{2}}\times I_1\times I_2\). Therefore, there exists an equivariant diffeomorphism \(\Psi :B\rightarrow A\) defined as \(\Psi = \psi ^{1}_A\circ \psi _B\).
We now focus ourselves on \(Z_{ij}^*\cap Z_{i(j+1)}^*\). We can consider the closure of \(Z_{ij}^*\cap Z_{i(j+1)}^*\) in \(X^*\), and since \(Z_{ij}\) and \(Z_{i(j+1)}\) are open cones over lens spaces, we can then assume \(Z_{ij}\) and \(Z_{i(j+1)}\) to be the closed cones. The intersection of the closures is homeomorphic to \([0,1]\times [0,1]\). Moreover, the edge \([0,1]\times \{0\}\) consists of orbits with isotropy \(G(a_{i(j+1)},b_{i(j+1)})\), for both actions of \({T^{2}}\) on \(Z_{ij}\) and \(Z_{i(j+1)}\) (see Fig. 4). Then by the proof of Lemma 5.5, the subset \(A =\pi ^{1}_{Z_{ij}}([0,1]\times [0,1])\subset Z_{ij}\) is equivariantly diffeomorphic to \([0,1]\times K(S^1)\times S^1\), where \(K(S^1)\) is the closed cone over \(S^1\). This same statement holds for \(B =\pi ^{1}_{Z_{i(j+1)}}([0,1]\times [0,1])\subset Z_{i(j+1)}\): it is equivariantly diffeomorphic to \([0,1]\times K(S^1)\times S^1\). Thus we conclude that A is equivariantly diffeomorphic to B.
Thus the conditions of compatibility in Proposition 6.1 are fulfilled and this procedure gives rise to a cohomology class in \(H^3(X^*, {\mathbb {Z}}^2)\). Since \(X^*\) is a 2dimensional manifold, the group \(H^3(X^*, {\mathbb {Z}}^2)\) vanishes, and the conclusion of Proposition 6.1 gives us a compact, smooth 4orbifold X (possibly with boundary) carrying an effective \({T^{2}}\)action which has exactly the collection
as its set of invariants. That is, \(X/{T^{2}}\) is isomorphic to \(X^*\). \(\square \)
Thus for a weighted surface \(X^*\), we have constructed a space X (the underlying space of a smooth 4orbifold) which admits a \({T^{2}}\)action, and realizes \(X^*\) as its orbit space. Now, we point out that this space X admits an Alexandrov metric.
Theorem 6.7
For each legally weighted surface \(X^*\), there exists a closed, orientable Alexandrov 4space admitting an effective and isometric \({T^{2}}\)action realizing \(X^*\) as its orbit space.
Proof
Let X be the underlying space of the orientable orbifold \({\mathcal {O}}_X\) obtained by Theorem 6.6. As in the case of Riemannian manifolds, for the \({T^{2}}\)action on \({\mathcal {O}}_X\) we can construct an invariant orbifold metric via [1, Theorem 3.65]. This induces the desired invariant Alexandrov metric. \(\square \)
Combining Theorems 5.11, 6.6, and 6.7 we conclude the following
Corollary 6.8
Let X be a closed orientable 4dimensional Alexandrov space with a \({T^{2}}\)action by isometries. Then X is equivariantly homeomorphic to an orbifold.
7 Basic Topological Recognition
In this section we prove a basic topological recognition result: we show that given a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action, it is always possible to decompose it as an “equivariant connected sum” of a closed smooth 4manifold with an effective \(T^2\)action, with a finite number of “simple” closed, orientable Alexandrov 4spaces with standard \({T^{2}}\)actions; the connected sum is done along the boundaries of tubular neighborhoods of some circle orbits. To make this statement precise we define these simple closed Alexandrov 4spaces with an isometric \({T^{2}}\)action.
Definition 7.1
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action. We say that X is simple if \(X^*\) is homeomorphic to a 2disk and \(E=\emptyset \), \(C=\emptyset \) and \(RF\cup SF\ne \emptyset \).
We observe that the set of invariants (5.2) associated to a simple Alexandrov 4space is of the form
Before proving the decomposition result of closed orientable 4dimensional Alexandrov spaces with a \({T^{2}}\)action by isometries, we present some examples of simple Alexandrov spaces. At the present moment we are not aware of a complete list of simple spaces.
Example 7.2
Let \({T^{2}}\) act on the lens space L(r, s) of cohomogeneity one with the group diagram \(({T^{2}}, 1, G(p,q), G(m,n))\), where \(pnmq=r\). Then \({{\,\mathrm{Susp}\,}}(L(r, s))\) with the suspension action of \({T^{2}}\) is a simple Alexandrov space. This action has two singular fixed points and its set of invariants is as follows:
Example 7.3
This example deals with a \({T^{2}}\)action on a weighted projective space, which is defined as follows: Consider the unit sphere
and define an action of the circle \(\mathrm {T^{1}}\) on \(\mathrm {S}^5\) as follows:
Here the \(r_i\)’s, are coprime integers. The quotient space of the sphere by this action
is called a weighted projective space. Note that \(W{\mathbb {P}}(r_1, r_2, r_3)\) is a compact Riemannian orbifold and in particular an Alexandrov space. We now define an effective \({T^{2}}\)action on \(W{\mathbb {P}}(r_1, r_2, r_3)\). We assume that each \(r_i\) is positive, and we choose three pairs of coprime integers \((m_1, n_1)\), \((m_2, n_2)\), and \((m_3, n_3)\), satisfying
We define an \({T^{2}}\)action on \(\mathrm {S}^5\) as
Since the actions of the circle \(\mathrm {T^{1}}\) commutes with the action of \({T^{2}}\) on \(\mathrm {S}^5\), the action of \({T^{2}}\) on \(\mathrm {S}^5\) descends to an action on \(W{\mathbb {P}}(r_1, r_2, r_3)\). This action has three fixed points and the set of invariants of the action is:
7.1 CEquivariant Connected Sums
We describe a similar construction to that of the usual equivariant connected sums which are performed at RForbits (see [40, 41]). The core difference is that it is performed at Corbits.
Let \(X_1\) and \(X_2\) be two closed, orientable Alexandrov 4spaces equipped with effective and isometric \({T^{2}}\)actions, and let \(\pi _i: X_i \rightarrow X_i^*\) denote the canonical projection maps for \(i=1,2\). We let \(C_i\), \(RF_i\), \(SF_i\) and \(E_i\) denote the subsets of \(X_i\) consisting of points on circular orbits, topologically regular fixed points, topologically singular fixed points and exceptional orbits, respectively, for \(i=1,2\). We assume that \(C_i\ne \emptyset \) for \(i=1,2\). Furthermore, we assume that there exist circular orbits \(x_i^*\in X_i^*\) having the same isotropy, that is, there exist \(x_{i}\in X_{i}\) with \(\pi _{i}(G(x_{i}))=x_{i}^{*}\) and \(G_{x_{i}}=G(m,n)\) for some m, n, with \(\mathrm {gcd}(m, n)=1\), for \(i=1,2\).
Now we consider invariant closed tubular neighborhoods \(\overline{B_i}\) of each orbit \(\pi _{i}^{1}(x_i)\), sufficiently small so that \(\overline{B_i}\cap SF_i= \overline{B_i}\cap RF_i=\overline{B_i}\cap E_i=\emptyset \) for \(i=1,2\). Since the actions are by isometries, \(\partial \overline{B_i}\) is an invariant subset of \(X_i\). It follows from the classification of cohomogeneity one 3manifolds [38] (see Sect. 2.5) that the restriction of the \({T^{2}}\)action on \(X_i\) to \(\partial \overline{B_i}\) is equivalent to the cohomogeneity one action with group diagram \(({T^{2}}, G(m,n), G(m,n), \{1\})\). Note that this in particular shows that \(\partial \overline{B_i}\) is equivariantly homeomorphic to \(S^2\times S^1\) for \(i=1,2\) (see Table 1). Hence, there exists an equivariant homeomorphism \(\Psi : \partial \overline{B_1}\rightarrow \partial \overline{B_2}\).
In a similar fashion to that of the RFequivariant connected sums we consider a space Z obtained by gluing \(X_1\setminus B_1\) with \(X_2\setminus B_2\) along their homeomorphic \(S^2\times S^1\) boundaries via \(\Psi \).
Definition 7.4
We say that the space Z is the Cequivariant connected sum of \(X_1\) and \(X_2\) with respect to the circular orbits \(x_1^*\) and \(x_2^*\) and denote it by \(X_1\#_C X_2\).
Remark 7.5
Since \(\partial \overline{B_{i}}\) is \({T^{2}}\)invariant and \(\Psi \) is an equivariant homeomorphism, one can define a homeomorphism \(\Psi ^{*}:(\partial \overline{B_{1}})^{*}\rightarrow (\partial \overline{B_{2}})^{*}\) such that the following diagram commutes:
Recall from Sect. 4.3 and the Slice Theorem that \(\overline{B_{i}}\) is equivariantly homeomorphic to \(({T^{2}}\times C(\mathrm {S}^{2}))/ G(m,n)\), where \(C(\mathrm {S}^{2})\) is a closed cone over \(\mathrm {S}^{2}\), and its boundary is equivariantly homeomorphic to \(({T^{2}}\times \mathrm {S}^{2}\times 1)/G(m,n)\). Note that \({B_i}\) is mapped to the shaded subset of \(X_{i}^{*}\) as in Fig. 5 and its boundary is mapped to the arc \(\widehat{abc}\).
Hence, the image of \(X_{i}\setminus B_{i}\) is merely \(X_{i}^{*}\) with the shaded area removed (see Fig. 6). It is clear from the gluing procedure that \(Z^{*}\) is just the gluing of \((X_1\setminus B_1)^{*}\) and \((X_2\setminus B_2)^{*}\) along \((\partial B_{1})^{*}\) and \((\partial B_{2})^{*}\) via the homeomorphism \(\Psi ^{*}\) as illustrated in Fig. 6, for \(X_1\) and \(X_2\) simply connected.
Therefore, we have shown the following lemma.
Lemma 7.6
Let \(X_{i}\), \(i=1, 2\) and \(X_1\#_C X_2\) be as above. Then we have
where \(\#_{C}\) on the right hand side means the gluing along \((\partial \overline{B_{i}})^{*}\) via the map \(\Psi ^{*}\) defined in Remark 7.5. In particular, if the boundary component of \(X_{1}^{*}\) containing \(x_{1}^{*}\) has only Corbits with isotropy G(m, n), then the boundary component of Z obtained as a result of Cequivariant connected sum and the boundary component of \(X_{2}\) containing \(x_{2}\) have the same weights.
Let us make some remarks. The equivariant homeomorphism type of \(X_1\#_C X_2\) depends on the choices of the circular orbits used in the construction. For the sake of simplicity, the notation we use does not explicitly display this fact. The space \(X_1\#_CX_2\) is naturally equipped with an effective \({T^{2}}\)action. Moreover, as the orbit space of such action is legally weighted, it follows from Theorem 6.6 that \(X_1\#_CX_2\) is equivariantly homeomorphic to the closed, orientable Alexandrov 4space having an effective and isometric \({T^{2}}\)action (uniquely determined up to equivariant homeomorphism) having \((X_1\#_CX_2)^*\) as orbit space.
With these definitions in hand we state the main result of the section.
Theorem 7.7
Let X be a closed, orientable Alexandrov 4space with an effective, isometric \({T^{2}}\)action. Then X is equivariantly homeomorphic to Cequivariant connected sums of a single closed orientable 4manifold with an effective \({T^{2}}\)action and a collection of simple Alexandrov 4spaces.
Proof
We begin by noting that if \(SF=\emptyset \) then the result is immediate as X is equivariantly homeomorphic to a 4manifold and there is no need to take Cequivariant connected sums. Therefore, we assume that \(SF\ne \emptyset \).
Let X be determined by the following set of invariants
By our assumption on SF we have that \(t>0\). Then we can assume without loss of generality, by relabeling the boundary components if necessary, that there exists a maximal index \(0\le q< t\) such that for each \(1\le \ell \le q\) we have \(f_{\ell ,w}=\pm 1\) for all \(1\le \ell \le q\); i.e., up to the index q all the boundary components containing fixed points do not contain SFpoints.
We now consider a collection of simple Alexandrov 4spaces \(X_j\) with \(j=1,\ldots , tq\), given by
satisfying that \((a^j_{1,w},b^j_{1,w})=(a_{j+q,w}, b_{j+q,w})\) for all \(w=1,\ldots , r_{j+q}\). We also consider a closed, orientable 4manifold M with an effective \({T^{2}}\)action determined by the invariants
where we recall that \(\big (({\overline{a}}_{\ell },{\overline{b}}_{\ell }), {\overline{f}}_{\ell }\big )\) denotes \(\big (({\overline{a}}_{\ell ,1},{\overline{b}}_{\ell ,1}), {\overline{f}}_{\ell ,1},\ldots ,({\overline{a}}_{\ell ,{\overline{r}}_{\ell }}, {\overline{b}}_{\ell ,{\overline{r}}_{\ell }}),{\overline{f}}_{\ell ,{\overline{r}}_{\ell }} \big )\), satisfying the following conditions:

(i)
\({\overline{s}}= s+tq\) and \({\overline{t}}= q\),

(ii)
\(({\overline{a}}_{\ell ,w},{\overline{b}}_{\ell ,w})= (a_{\ell ,w},b_{\ell ,w})\) (and therefore \({\overline{f}}_{\ell ,w}=f_{\ell ,w}=\pm 1\)) for all \(w=1,\ldots ,{\bar{r}}_{\ell }\), and \(\ell =1,\ldots , {\overline{t}}\),

(iii)
\(\left\langle p_i,q_i \right\rangle = \left\langle a_{i+q,1}, b_{i+q,1} \right\rangle \) for all \(i=1,\ldots , tq\).
By the third point in the conditions defining M, for each \(j=1,\ldots , tq\) there exist a circular orbit \(x^*\in M^*\) contained in the jth connected component of \(\partial M^*\) containing only circular orbits and a circular orbit \(x_j^*\in X_j^*\) having the same weight, namely \((a_{j+q,1}, b_{j+q,1})\). Therefore the Cequivariant connected sum of M and \(X_j\) can be performed with respect to \(x^*\) and \(x_j^*\) for all \(j=1,\ldots , tq\) yielding a space \(M\#_CX_1\#_C\ldots \#_C X_{tq}\) with an effective \({T^{2}}\)action whose orbit space is isomorphic to \(X^*\) by Lemma 7.6. Hence by Theorem 5.11 we have that X is equivariantly homeomorphic to \(M\#_CX_1\#_C\ldots \#_C X_{tq}\). \(\square \)
Remark 7.8
To achieve a full homeomorphism classification of closed, orientable Alexandrov 4spaces with effective and isometric \({T^{2}}\)actions it is sufficient to have a classification of the homeomorphism type of simple Alexandrov 4spaces and combine this with the previous homeomorphism classification of OrlikRaymond [40, 41] and Pao [42, 43].
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Diego Corro was supported by CONACyTDAAD Doctoral Scholarship No. 409912, DFG RTG2229 “Asymptotic Invariants and Limits of Groups and Spaces”, DGAPAUNAM Postdoctoral Fellowship of the Institute of Mathematics, DFGEigenestelle Fellowship CO 2359/11. Jesús NúñezZimbrón was supported by a MIUR SIRGrant ‘Nonsmooth Differential Geometry’ (RBSI147UG4) and a DGAPAUNAM Postdoctoral Fellowship of the Center of Mathematical Sciences. Masoumeh Zarei was Supported by DFG project AM 342/41 and DFG Grant GA 2050/21 within the Priority Program SPP2026 “Geometry at Infinity”. The authors would like to thank the IV Encuentro Conjunto RSMESMM where initial talks leading to this work took place. In particular, we thank F. GalazGarcía and L. Guijarro, organizers of the Riemannian/Metric geometry special session for very useful conversations. The authors also thank the anonymous referee for her or his helpful comments.
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Corro, D., NúñezZimbrón, J. & Zarei, M. Torus Actions on Alexandrov 4Spaces. J Geom Anal 32, 214 (2022). https://doi.org/10.1007/s1222002200935x
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DOI: https://doi.org/10.1007/s1222002200935x
Keywords
 4Manifold
 Torus action
 Alexandrov space
Mathematics Subject Classification
 53C23
 57S15
 57S25