2.3.1 Fiber types

We define the fiber types of a local circle action by the isotropy of the local action, as well as the isotropy representation and if any point on the orbit is a regular or singular point (see [Reference Galaz-García and Núñez-Zimbrón13, Section 3]). The possible types are as follows:

• • F-fibers are topologically regular fixed-point fibers.

• • SF-fibers are topologically singular fixed-point fibers.

• • E-fibers are those that correspond to $\mathbb {Z}_k$ isotropy, acting in such a way that local orientation is preserved.

• • SE-fibers correspond to $\mathbb {Z}_2$ isotropy, reversing the local orientation.

• • R-fibers are fibers that are not F-, $SF$ -, E-, or $SE$ -fibers.

The sets of points in F-, $SF$ -, E-, $SE$ -, and R-fibers are denoted by F, $SF$ , E, $SE,$ and R, respectively. The quotient space $X^{\ast }$ of the local circle action, with the quotient topology, admits a $2$ -orbifold structure (see, e.g., [Reference Galaz-García and Núñez-Zimbrón13]). Its boundary consists of the images of F-, $SF$ -, and $SE$ -fibers under the natural projection map, while the interior consists of R-fibers and a finite number of E-fibers.

2.3.2 Building blocks

A closed Alexandrov $3$ -space X with a local isometric circle action can be decomposed into building blocks of types F, $SF$ , $E,$ and $SE$ , defined by considering small invariant tubular neighborhoods of connected components of the strata in X consisting of fibers of the corresponding type. A building block is simple if its boundary is homeomorphic to a torus, and twisted if its boundary is homeomorphic to a Klein bottle. Note that the stratum of R-fibers is an $S^1$ -fiber bundle with structure group $\mathrm {O}(2)$ .

Theorem 2.2 (cf. [Reference Galaz-García and Núñez-Zimbrón13, Theorem B])

Let X be a closed, connected Alexandrov $3$ -space with $2r\geq 0$ topologically singular points. Then, the set of isometric local circle actions (up to equivariant equivalence) is in one-to-one correspondence with the set of unordered tuples

$$\begin{align*}\left\{b; \varepsilon, g, (f,k_1), (t,k_2), (s,k_3); \{ (\alpha_i, \beta_i) \}_{i=1}^n; (r_1,r_2, \ldots, r_{s-k_{3}}); (q_1, q_2, \ldots, q_{k_3})\right\}.\end{align*}$$

The definition of the symbols appearing in this result is as follows:

• • The numbers k, $k_1$ , $k_2$ , and $k_3$ satisfy $k_1 + k_2 + k_3 = k$ .

• • $(\varepsilon ,k)$ is the pair that classifies the $S^1$ -bundle of R-fibers according to [Reference Galaz-García and Núñez-Zimbrón13, Theorem 3.2].

• • $g\geq 0$ is the genus of $X^{*}$ .

• • The symbols $f, t, k_1, k_2$ are nonnegative integers such that $k_1 \leq f$ and $k_2\leq t$ , where $k_1$ is the number of twisted F-blocks and $k_2$ is the number of twisted $SE$ -blocks (therefore $f-k_1$ is the number of simple F-blocks and $t-k_2$ is the number of simple $SE$ -blocks).

• • n is the number of E-fibers and we let $\{ (\alpha _i, \beta _i)\}_{i=1}^n$ be the corresponding Seifert invariants (see [Reference Orlik and Raymond32] for the definition).

• • We let $s, k_3$ be nonnegative integers, where $k_3\leq s$ is the number of twisted $SF$ -blocks (thus $s-k_3$ is the number of simple $SF$ -blocks).

• • $(r_1, r_2, \ldots , r_{s-k_{3}})$ and $(q_1, q_2, \ldots , q_{k_3} )$ are $(s-k_{3})$ - and $k_3$ -tuples of nonnegative even integers corresponding to the number of topologically singular points in each simple and twisted $SF$ -block, respectively.

• • The invariant b is an integer (or integer mod $2$ ) representing an obstruction class in the following way: Let $X_0$ denote the subset of X obtained by removing from X sufficiently small (open) tubular neighborhoods of the F-, $SE$ -, E-, and $SF$ -fibers which are disjoint. Then $X_0$ consists of R-fibers. Observe that the Seifert invariants determine cross-sections on the boundaries of $X_0$ which correspond to tubular neighborhoods of E-fibers and that any section that could already be defined on $X_0$ can be extended to the tubular neighborhoods of F-, $SF$ -, and $SE$ - fibers (see [Reference Galaz-García and Núñez-Zimbrón13, Proof of Theorem B]). Thus, we define b as the obstruction class in

  $$\begin{align*}H^2(X^{\ast} \setminus \mathrm{int}(V_1^{\ast}\cup\dots\cup V_n^{\ast}), \partial (V_1^{\ast}\cup\dots\cup V_n^{\ast}) ;\mathbb{Z}), \end{align*}$$
  where the $V_i$ are the tubular neighborhoods of the E-orbits, to extending a cross-section defined on $\partial (V_1^{\ast }\cup \dots \cup V_n^{\ast })$ to all $X_0$ . It follows from [Reference Galaz-García and Núñez-Zimbrón13] and the manifold case in [Reference Fintushel10, Reference Orlik and Raymond32] that b is an integer (or an integer mod $2$ depending on the orientability of $X^{\ast }$ ), subject to the following restrictions: $b=0$ if $f+t>0$ or if $\varepsilon \in \{o_2,n_1,n_3,n_4\}$ and some $\alpha _i=2$ (see [Reference Galaz-García and Núñez-Zimbrón13, Theorem 3.2] for the precise definitions of the $o_i$ , $n_j$ , and $\alpha _l$ ); $b\in \{0,1\}$ if $f+t=0$ and $\varepsilon \in \{ o_2,n_1,n_3,n_4\}$ and all $\alpha _i\neq 2$ . In the remaining cases, b is an arbitrary integer. It is worth pointing out that b plays a role, for example, when $SF\cup F\cup SE \cup E=\emptyset $ and that the structure group reduces to $\mathrm {SO}(2)$ (see, e.g., the descriptions of b in Lemmas 2 and 3 in [Reference Orlik31, Section 1.9]).

3.2.1 Proof of Theorem A

We first prove the uniqueness statement. Let $X_1=\operatorname {\mathrm {Seif}}(B_1)$ and $X_2=\operatorname {\mathrm {Seif}}(B_2)$ be closed and connected generalized Seifert spaces and let $p_j\colon X_j\to B_j$ , $j=1,2$ denote the generalized Seifert fibrations. It is clear that if $X_1$ is fiberwise homeomorphic to $X_2$ , then $B_1$ is isomorphic to $B_2$ . We now prove that if there is an isomorphism of fiber spaces $\varphi \colon B_1 \to B_2$ , then $X_1$ is equivalent to $X_2$ .

Let $I^{\ast }_{j}=\{x^{\ast }_{j1}, x^{\ast }_{j2}, \ldots , x^{\ast }_{j\iota }\}\subseteq B_j$ for $j=1,2$ be the collection of points of I-fiber type on $B_j$ . We now choose disjoint closed disk neighborhoods $D_{1i}$ of each $x^{\ast }_{1i}$ such that $p_1^{-1}(D_{1i})\cong B\!\left (\mathrm {pt}\right )$ . Without loss of generality, as $\varphi $ is fiber-type preserving, we can assume $\varphi (x^{\ast }_{1i})=x^{\ast }_{2i}$ and denote $D_{2i}= \varphi (D_{1i})$ , $i=1,\ldots , \iota $ . Furthermore, we can assume that $D_{1i}$ is small enough so that $p_2^{-1}(D_{2i})\cong B\!\left (\mathrm {pt}\right )$ for all $i=1,\ldots ,\iota $ .

Now, observe that $\varphi $ restricts to a fiber-type preserving homeomorphism ${B_1 \setminus \bigcup _{i=1}^{\iota } \mathrm {int}(D_{1i}) \to B_2 \setminus \bigcup _{i=1}^{\iota } \mathrm {int}(D_{2i})}$ and consider for $j=1,2$

$$\begin{align*}Z_j:= X_j\setminus \bigcup_{i=1}^{\iota} p_j^{-1}\left( \mathrm{int}(D_{ji}) \right). \end{align*}$$

Then $\partial Z_j$ is a disjoint union of $\iota $ copies of Klein bottles $K^2$ . Observe moreover that $Z_j$ is a Seifert manifold with boundary.

We let $V_{ji}$ be twisted F-blocks for $j=1,2$ , $i=1,\ldots , \iota $ and consider the spaces $Y_j$ obtained by gluing each $V_{ji}$ to each boundary component of $Z_j$ via fiber-preserving homeomorphisms. To this end, recall from [Reference Galaz-García and Núñez-Zimbrón13] (see the subsection “Block Types”) that the fibrations restricted to $\partial V_{ji}$ are unique up to fiber-preserving homeomorphism; note that this is also the case for the fibration on the boundary of $B\!\left (\mathrm {pt}\right )$ . Further, note that the spaces $Y_j$ satisfy that $\operatorname {\mathrm {sing}}(Y_j)=\emptyset $ as all topologically singular points of $X_j$ must be on I-fibers and not on C-fibers. Thus, each $Y_j$ is a closed topological $3$ -manifold and carries a local circle action. Therefore, by the work of Orlik and Raymond [Reference Orlik and Raymond32] and Fintushel [Reference Fintushel10], the $Y_j$ are determined up to fiber-preserving homeomorphism by the same set of invariants, namely,

$$\begin{align*}\left\{b,\varepsilon,g, (f=\iota,k_1=\iota), (t=0,k_2=0), \left\{(\alpha_j,\beta_j) \right\}_{j=1}^{n} \right\}, \end{align*}$$

and there is a fiber-preserving homeomorphism $\Psi \colon Y_1 \to Y_2$ which is determined by cross-sections to the local actions away from the exceptional orbits.

Observe now that $\Psi (V_{1i})\subset Y_2$ is a twisted F-block so that, without loss of generality, we can assume $\Psi (V_{1i})=V_{2i}$ , for each $i=1,\ldots ,\iota $ . Thus, by restricting $\Psi ,$ we obtain a fiber-preserving homeomorphism $\Psi |_{Z_1}\colon Z_1 \to Z_2$ determined by cross-sections $\rho _j\colon \partial ( B_j \setminus \bigcup _{i=1}^{\iota } \mathrm {int}(D_{ji}))\to \partial Z_j$ , that is, such that

$$\begin{align*}\Psi\left( \rho_1\left( \partial ( B_1 \setminus \bigcup_{i=1}^{\iota} \mathrm{int}(D_{1i}))\right)\right) = \rho_2\left( \partial ( B_2 \setminus \bigcup_{i=1}^{\iota} \mathrm{int}(D_{2i}))\right). \end{align*}$$

Furthermore, as the local circle action restricted to $\partial Z_j$ is equivalent to the “free” local circle action, we can parametrize each component of $\partial Z_j$ , homeomorphic to the Klein bottle, seen as the nontrivial bundle $S^1\widetilde {\times }S^1$ as an equivalence class (with respect to the involution $\alpha $ as in (2.1)) of points $[e^{i\theta }, e^{i\eta }]$ , where $e^{i\theta }\in \rho _1\left ( \partial ( B_1 \setminus \bigcup _{i=1}^{\iota } \mathrm {int}(D_{1i}))\right ) \cong S^1$ and $e^{i\eta }$ parameterizes the fiber direction. Hence, $\Psi |_{\partial Z_1}$ is of the form

$$\begin{align*}\Psi|_{\partial Z_1}([e^{i\theta}, e^{i\eta}]) = [\Psi_1(e^{i\theta}), \Psi_{2,\theta}(e^{i\eta})], \end{align*}$$

where $\Psi _1$ and $\Psi _{2,\theta }$ are self-homeomorphisms of $S^1$ for each $\theta \in [0,2\pi )$ .

We now conclude by observing that $\Psi $ extends to a fiber-preserving homeomorphism $X_1\to X_2$ by parameterizing each $B\!\left (\mathrm {pt}\right )$ block by classes of points $[e^{i\theta }, re^{i\eta }],$ where $r\in [0,1]$ , and defining

$$\begin{align*}\Psi|_{B\!\left(\mathrm{pt}\right)}([e^{i\theta}, re^{i\eta}]):= [\Psi_1(e^{i\theta}), r\Psi_{2,\theta}(e^{i\eta})]. \end{align*}$$

We now prove the existence part of the theorem. By the classification of spaces with local circle actions [Reference Galaz-García and Núñez-Zimbrón13], there exists a space Y with a local circle action with the invariants

$$\begin{align*}\left\{b,\varepsilon,g, (f=\iota,k_1=\iota), (t=0,k_2=0), \left\{(\alpha_j,\beta_j) \right\}_{j=1}^{n} \right\}, \end{align*}$$

and an orbifold Riemannian metric of curvature bounded below. In a similar fashion to the uniqueness part of the proof, we take out all $\iota $ twisted F-blocks from Y, obtaining a Riemannian orbifold with boundary Z, curvature bounded from below, and a local circle action. We now take $\iota $ copies $V_i$ of $B\!\left (\mathrm {pt}\right )$ with the orbifold metric induced by the flat metric on $S^1\times D^2$ .

By the smooth Collar Neighborhood Theorem [Reference Lee24, Theorem 9.25], and since the boundary components of Z and each $V_i$ are Klein bottles contained in their smooth loci, $\partial Z$ and $\partial V_i$ admit smooth product collars, and we may assume the metrics to be product near the boundary and that their restrictions to corresponding components of $\partial Z$ and $\partial V_i$ are isometric. Therefore, we can glue the $V_i$ to Z fiber-preservingly by $\iota $ copies of the cylinder $I\times K^2$ in such a way that the metric on $\{0\}\times K^2$ coincides with the metric on the i-th boundary component of Z, and the metric on $\{1\}\times K^2$ coincides with the metric on the boundary of $V_i$ . Moreover, this can be done so that the glued space X is a smooth Riemannian orbifold which, by compactness, has curvature bounded below, and in particular, X is an Alexandrov space admitting a generalized Seifert fiber space structure with the invariants

$$\begin{align*}\left\{ b,(\varepsilon,k),g, \iota, \left\{ (\alpha_i, \beta_i) \right\}_{i=1}^n \right\}. \end{align*}$$

As we pointed out, $k=\iota $ so that we remove the superfluous invariant k and uniquely associate (up to fiber-preserving homeomorphism) the list

(3.2) $$ \begin{align} \left\{ b,\varepsilon,g, \iota, \left\{ (\alpha_i, \beta_i) \right\}_{i=1}^n \right\} \end{align} $$

to each generalized Seifert fiber space, as stated in Theorem A.

4.1.1 The lifted Seifert fibration

The existence and properties of the lifted Seifert fibration are given in the following proposition.

Proposition 4.2. Let $p\colon X \to B$ be a generalized Seifert fibration with X closed, connected, and not a manifold. Let $\operatorname {\mathrm {sing}}(X) \subset X$ denote the set of topologically singular points of X and let

$$\begin{align*}\pi\colon \widetilde X \longrightarrow X \end{align*}$$

be the canonical orientable double branched cover, branched precisely along $\operatorname {\mathrm {sing}}(X)$ . Then there exists a compact $2$ -orbifold $\widetilde B$ , a double branched cover

$$\begin{align*}\widetilde\pi\colon \widetilde B \longrightarrow B, \end{align*}$$

and a Seifert fibration

$$\begin{align*}\widetilde p\colon \widetilde X \longrightarrow \widetilde B, \end{align*}$$

such that:

1. (1) $\widetilde \pi $ is branched precisely over $p(\operatorname {\mathrm {sing}}(X)) \subset B$ .

2. (2) The following diagram commutes:

   (4.1)

   

3. (3) Let $\mathcal F$ denote the partition of $\widetilde X$ whose elements are the connected components of $\pi ^{-1}\!\big (p^{-1}(b)\big )$ for $b\in B$ . Then $\mathcal F$ is the fiber decomposition of $\widetilde p$ , and in particular every element of $\mathcal F$ is a circle.

4. (4) Let $\tau \colon \widetilde X\to \widetilde X$ be the canonical involution of the double branched cover $\pi $ . Then $\tau $ preserves the partition $\mathcal F$ and hence descends to an involution $\widetilde \tau \colon \widetilde B\to \widetilde B$ satisfying $\widetilde \pi \circ \widetilde \tau =\widetilde \pi $ . Moreover, the fibers of $\widetilde \pi $ are exactly the $\widetilde \tau $ -orbits. In particular, $\widetilde \tau $ fixes precisely the points of $\widetilde B$ lying over the branching set $p(\operatorname {\mathrm {sing}}(X))$ .

4.1.2 The lifted partition

Let $\mathcal F$ be as in the statement of the proposition. We call elements of $\mathcal {F}$ fibers.

4.1.3 The elements of $\mathcal {F}$ are circles

Let $b\in B$ . By the definition of a generalized Seifert fibration, every fiber $p^{-1}(b)$ is either a circle (a C-fiber) or a closed interval (an I-fiber), and I-fibers occur precisely over $p(\operatorname {\mathrm {sing}}(X))$ .

Suppose that $b\notin p(\operatorname {\mathrm {sing}}(X))$ . Then $p^{-1}(b)$ is a circle contained in $X\setminus \operatorname {\mathrm {sing}}(X)$ . Choose a sufficiently small orbifold neighborhood $U\subset B$ of b disjoint from $p(\operatorname {\mathrm {sing}}(X))$ . By the local description of generalized Seifert fibrations, the restriction $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ is fiber-preserving homeomorphic to a (classical) Seifert fibration of a solid torus (see Section 2.2). Since $p^{-1}(U)\subset X\setminus \operatorname {\mathrm {sing}}(X)$ , the restriction $\pi |_{\pi ^{-1}(p^{-1}(U))}\colon \pi ^{-1}(p^{-1}(U))\to p^{-1}(U)$ is the orientation double cover of $p^{-1}(U)$ . Thus, since $p^{-1}(U)$ is orientable, $\pi |_{\pi ^{-1}(p^{-1}(U))}$ is the trivial double cover, so $\pi ^{-1}(p^{-1}(U))$ consists of two connected components, each mapped homeomorphically onto $p^{-1}(U)$ by $\pi $ . In particular, $\pi ^{-1}(p^{-1}(b))$ consists of exactly two connected components, each mapped homeomorphically onto $p^{-1}(b)$ by $\pi $ . Since $p^{-1}(b)$ is a circle, each component is a circle. Therefore, every connected component of $\pi ^{-1}(p^{-1}(b))$ , that is, every element of $\mathcal F$ lying over b, is a circle.

Suppose now that $b\in p(\operatorname {\mathrm {sing}}(X))$ . Then $p^{-1}(b)$ is an I-fiber, and there exists an orbifold neighborhood $U\subset B$ of b such that $p^{-1}(U)$ is fiber-preserving homeomorphic to $B\!\left (\mathrm {pt}\right )$ with respect to the standard fibration $q\colon B\!\left (\mathrm {pt}\right ) \to K_1(S^1(1/2))$ sending the I-fiber $q^{-1}(o)$ to the vertex o. By the definition of the canonical orientable double branched cover, $\pi \colon \widetilde {X}\to X$ is a twofold branched cover with branching set $\operatorname {\mathrm {sing}}(X)$ . Under the fiber-preserving homeomorphism $p^{-1}(U)\cong B\!\left (\mathrm {pt}\right )$ , which identifies $\operatorname {\mathrm {sing}}(X)\cap p^{-1}(U)$ with $\operatorname {\mathrm {sing}}(B\!\left (\mathrm {pt}\right ))$ , the restriction $\pi |_{\pi ^{-1}(p^{-1}(U))}\colon \pi ^{-1}(p^{-1}(U))\to p^{-1}(U) $ is isomorphic to the model double branched cover $S^1\times D^2\to B\!\left (\mathrm {pt}\right )$ described in Lemma 4.1 (via a $\mathbb {Z}_2$ -equivariant homeomorphism of total spaces which, after identifying $p^{-1}(U)$ with $B\!\left (\mathrm {pt}\right )$ , commutes with the covering maps). Under this identification, the fiber $p^{-1}(b)$ corresponds to the I-fiber $q^{-1}(o)$ . In the model double branched cover $S^1\times D^2\to B\!\left (\mathrm {pt}\right )$ from Lemma 4.1, where $q\colon B\!\left (\mathrm {pt}\right )\to K_1(S^1(1/2))$ is induced by $(e^{i\theta },x)\mapsto x$ , the preimage of the I-fiber $q^{-1}(o)$ is the core circle $S^1\times \{0\}$ in $S^1\times D^2$ . Hence, $\pi ^{-1}(p^{-1}(b))$ is a circle. Combined with the previous case, we conclude that for all $b\in B$ , every element of $\mathcal F$ is a circle.

4.1.4 Local Seifert charts for $\mathcal {F}$

Let $L\in \mathcal F$ be a fiber and pick $\widetilde x\in L$ . Set $x:=\pi (\widetilde x)$ and $b:=p(x)$ . Choose an orbifold neighborhood $U\subset B$ of b so that the restriction $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ is fiber-preserving homeomorphic to one of the standard local models in the definition, either a (classical) Seifert fibration of a solid torus or the standard model $q\colon B\!\left (\mathrm {pt}\right )\to K_1(S^1(1/2))$ . We consider two cases, corresponding to whether $b\in p(\operatorname {\mathrm {sing}}(X))$ .

Suppose first that $b\notin p(\operatorname {\mathrm {sing}}(X))$ . Then $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ is fiber-preserving homeomorphic to a (classical) Seifert fibration of a solid torus. Moreover, the restriction $\pi |_{\pi ^{-1}(p^{-1}(U))}\colon \pi ^{-1}(p^{-1}(U))\to p^{-1}(U)$ is a regular double cover, since $U\cap p(\operatorname {\mathrm {sing}}(X))=\varnothing $ , and hence $p^{-1}(U)\subset X\setminus \operatorname {\mathrm {sing}}(X)$ . As noted above, since $p^{-1}(U)$ is orientable, $\pi |_{\pi ^{-1}(p^{-1}(U))}$ is the trivial double cover. Hence, $\pi ^{-1}(p^{-1}(U))$ has two connected components swapped by the deck involution $\tau \colon \widetilde {X}\to \widetilde {X}$ , the canonical involution of the double branched cover $\pi $ . In particular, $\pi ^{-1}(p^{-1}(b))$ has two connected components: two circles exchanged by $\tau $ , each mapping homeomorphically onto $p^{-1}(b)$ under $\pi $ . Therefore, $\pi ^{-1}(p^{-1}(U))$ is a $3$ -manifold covered by (classical) Seifert charts, and the fibers of $\mathcal F$ restricted to $\pi ^{-1}(p^{-1}(U))$ are exactly the fibers of the lifted Seifert charts.

Suppose now that $b\in p(\operatorname {\mathrm {sing}}(X))$ . Then $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ is fiber-preserving homeomorphic to the standard generalized Seifert fibration $q\colon B\!\left (\mathrm {pt}\right ) \to K_1(S^1(1/2))$ sending the I-fiber $q^{-1}(o)$ to the vertex o. Note that this homeomorphism identifies $\operatorname {\mathrm {sing}}(X)\cap p^{-1}(U)$ with $\operatorname {\mathrm {sing}}(B\!\left (\mathrm {pt}\right ))$ . By the definition of the canonical orientable double branched cover $\pi $ , the restriction $\pi |_{\pi ^{-1}(p^{-1}(U))}\colon \pi ^{-1}(p^{-1}(U))\to p^{-1}(U)$ is a double branched cover of $p^{-1}(U)$ with branching set $\operatorname {\mathrm {sing}}(X)\cap p^{-1}(U)$ . After identifying $p^{-1}(U)$ with $B\!\left (\mathrm {pt}\right )$ as above, Lemma 4.1 implies that the lifted partition by elements of $\mathcal F$ is locally given by a (classical) Seifert chart on a solid torus. Under this identification, the elements of $\mathcal {F}$ in $\pi ^{-1}(p^{-1}(U))$ coincide with the circle fibers of the standard Seifert fibration of the solid torus $S^1\times D^2$ . In particular, the quotient $\pi ^{-1}(p^{-1}(U))/\mathcal {F}$ is a disk and the map induced by $\pi $ between the local bases (quotients by circle fibers) $\pi ^{-1}(p^{-1}(U))/\mathcal {F}\to U$ is a twofold branched cover of $U\cong K_1(S^1(1/2))$ , branched over $b\in U$ (corresponding to the vertex of $K_1(S^1(1/2))$ ).

In both cases, we obtain a neighborhood $V\subset \widetilde X$ of $\widetilde x$ saturated by fibers of $\mathcal F$ , together with a homeomorphism from V to a (classical) Seifert chart modeled on $(S^1\times D^2)/\mathbb Z_k$ for some $k\ge 1$ , sending fibers of $\mathcal F$ to circle fibers. Thus, $\mathcal F$ defines a Seifert fibration on the $3$ -manifold $\widetilde X$ .

4.1.5 Definition of $\widetilde B$ and $\widetilde p$

Define the base space $\widetilde B := \widetilde X/\mathcal F$ , endowed with the quotient topology, and let ${\widetilde p\colon \widetilde X \longrightarrow \widetilde B}$ be the quotient map. By our previous analysis, around every point of $\widetilde X$ there are Seifert charts in which $\widetilde p$ corresponds to the standard projection $(S^1\times D^2)/\mathbb {Z}_k\to D^2/\mathbb {Z}_k$ . It follows that $\widetilde B$ admits a $2$ -orbifold structure whose local charts are of the form $D^2/\mathbb {Z}_k$ (with $k\geq 1$ depending on the chart), and with respect to this structure $\widetilde p$ is a Seifert fibration. The compactness of $\widetilde B$ follows from the compactness of $\widetilde X$ .

4.1.6 Definition of $\widetilde \pi $ and commutativity of diagram (4.1)

Let us first verify that the continuous map $p\circ \pi \colon \widetilde X\to B$ is constant on the elements of $\mathcal F$ . By construction, each fiber $L\in \mathcal {F}$ is contained in $\pi ^{-1}(p^{-1}(b))$ for some $b\in B$ . Hence, $p(\pi (\widetilde x))=b$ for all $\widetilde x\in L$ . Define $\widetilde \pi \colon \widetilde B \longrightarrow B$ by

(4.2) $$ \begin{align} \widetilde\pi\big(\widetilde p(\widetilde x)\big) := p(\pi(\widetilde x)). \end{align} $$

Since $p\circ \pi $ is constant on elements of $\mathcal {F}$ (equivalently, on the fibers of $\widetilde {p}$ ), the map $\widetilde {\pi }$ is well defined. Indeed, if $\widetilde {p}(\widetilde {x}) = \widetilde {p}(\widetilde {y})$ , then $\widetilde {x}$ and $\widetilde {y}$ lie in the same fiber $L\in \mathcal {F}$ . Hence, $p(\pi (\widetilde {x}))=p(\pi (\widetilde {y}))$ . This gives the commutative diagram in the statement.

4.1.7 Induced involution on the base and fibers of $\widetilde \pi $

Let $\tau \colon \widetilde X\to \widetilde X$ denote the canonical involution of the double branched cover $\pi $ . Hence, $\pi \circ \tau =\pi $ and $\tau ^2=\mathrm {id}_{\widetilde X}$ , and the fixed point set of $\tau $ is the branching set of $\pi $ (see the discussion of the canonical double branched cover in Section 2.1). Since $\pi \circ \tau =\pi $ , for every $b\in B$ , we have

$$\begin{align*}\tau\big(\pi^{-1}(p^{-1}(b))\big)=\pi^{-1}(p^{-1}(b)). \end{align*}$$

Since $\tau $ is a homeomorphism, it permutes the connected components of $\pi ^{-1}(p^{-1}(b))$ . Equivalently, $\tau $ preserves the partition $\mathcal {F}$ . Therefore, $\tau $ induces a continuous map $\widetilde \tau \colon \widetilde B\to \widetilde B$ given by

$$\begin{align*}\widetilde\tau\big(\widetilde p(\widetilde x)\big):=\widetilde p\big(\tau(\widetilde x)\big), \qquad \widetilde x\in \widetilde X. \end{align*}$$

Note that $\widetilde {\tau }$ is well defined because $\tau $ sends $\mathcal {F}$ -fibers to $\mathcal {F}$ -fibers. Moreover, $\widetilde \tau $ is an involution since $\widetilde \tau ^2(\widetilde p(\widetilde x))=\widetilde p(\tau ^2(\widetilde x))=\widetilde p(\widetilde x)$ . In particular, $\widetilde {\tau }$ is a homeomorphism. Finally, using the definition of $\widetilde \pi $ in (4.2) and $\pi \circ \tau =\pi $ , we obtain

$$\begin{align*}\widetilde\pi\big(\widetilde\tau(\widetilde p(\widetilde x))\big) =\widetilde\pi\big(\widetilde p(\tau(\widetilde x))\big) =p\big(\pi(\tau(\widetilde x))\big) =p\big(\pi(\widetilde x)\big) =\widetilde\pi\big(\widetilde p(\widetilde x)\big). \end{align*}$$

Hence, $\widetilde \pi \circ \widetilde \tau =\widetilde \pi $ .

We now identify the fibers of $\widetilde \pi $ with $\widetilde \tau $ -orbits. Fix $b\in B$ and consider the set $\widetilde \pi ^{-1}(b)\subset \widetilde B$ . By definition,

$$\begin{align*}\widetilde\pi^{-1}(b) =\Big\{\widetilde p(L)\ :\ L \text{ is a connected component of } \pi^{-1}(p^{-1}(b))\Big\}. \end{align*}$$

If $b\notin p(\operatorname {\mathrm {sing}}(X))$ , then $p^{-1}(b)$ is a C-fiber contained in $X\setminus \operatorname {\mathrm {sing}}(X)$ . Hence, $\pi ^{-1}(p^{-1}(b))$ is the disjoint union of two circles, and the involution $\tau $ exchanges these two components. Therefore, $\widetilde \pi ^{-1}(b)$ consists of two points $\{u,\widetilde \tau (u)\}$ with $\widetilde \tau (u)\neq u$ . If instead $b\in p(\operatorname {\mathrm {sing}}(X))$ , then $p^{-1}(b)$ is an I-fiber and, by Lemma 4.1, $\pi ^{-1}(p^{-1}(b))$ is connected. Hence, $\widetilde \pi ^{-1}(b)$ consists of a single point u, which is fixed by $\widetilde \tau $ . In particular, for all $u,v\in \widetilde B$ , we have $\widetilde \pi (u)=\widetilde \pi (v)$ if and only if $v\in \{u,\widetilde \tau (u)\}$ , that is, the fibers of $\widetilde \pi $ are exactly the $\widetilde \tau $ -orbits, with fixed points precisely over the branch points of $\widetilde {\pi }$ , that is, over $p(\operatorname {\mathrm {sing}}(X))$ .

4.1.8 The map $\widetilde \pi \colon \widetilde {B}\to B$ is a twofold branched cover with branching set $p(\operatorname {\mathrm {sing}}(X))$

Let $b\in B$ . Suppose first that $b\notin p(\operatorname {\mathrm {sing}}(X))$ . Choose a sufficiently small orbifold neighborhood $U\subset B$ of b disjoint from $p(\operatorname {\mathrm {sing}}(X))$ , so $p^{-1}(U)\subset X\setminus \operatorname {\mathrm {sing}}(X)$ . Then the restriction $\pi |_{\pi ^{-1}(p^{-1}(U))}\colon \pi ^{-1}(p^{-1}(U))\to p^{-1}(U)$ is a regular twofold cover. By our previous analysis, $\pi \colon \widetilde {X}\to X$ maps each $\mathcal {F}$ -fiber in $\pi ^{-1}(p^{-1}(U))$ onto a p-fiber in $p^{-1}(U)$ . Passing to the quotients by the circle fibers of the Seifert fibrations $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ and $\widetilde {p}|_{\widetilde {p}^{-1}(\widetilde {\pi }^{-1}(U))}\colon \widetilde {p}^{-1}(\widetilde {\pi }^{-1}(U))\to \widetilde {p}^{-1}(\widetilde {\pi }^{-1}(U))/\mathcal {F} = \widetilde {\pi }^{-1}(U)$ yields that $\widetilde \pi |_{\widetilde \pi ^{-1}(U)}\colon \widetilde \pi ^{-1}(U)\to U$ is an unbranched twofold covering of orbifolds.

Suppose now that $b\in p(\operatorname {\mathrm {sing}}(X))$ . Choose a sufficiently small neighborhood U of b so that the restriction $p|_{p^{-1}(U)}\colon p^{-1}(U)\to U$ is fiber-preserving homeomorphic to the standard generalized Seifert fibration $q\colon B\!\left (\mathrm {pt}\right ) \to K_1(S^1(1/2))$ sending the I-fiber $q^{-1}(o)$ to the vertex o. By our previous analysis, the induced map on the local bases is a twofold branched cover $D^2\to K_1(S^1(1/2))$ , branched at its vertex. Hence, $\widetilde \pi $ is branched over b.

By the first case, $\widetilde {\pi }\colon \widetilde {B}\to B$ is unbranched over $B\setminus p(\operatorname {\mathrm {sing}}(X))$ . The second case shows that $\widetilde {\pi }$ is branched over each $b\in p(\operatorname {\mathrm {sing}}(X))$ . Hence, the branching set of $\widetilde {\pi }$ is $p(\operatorname {\mathrm {sing}}(X))$ .

Thus, we have shown that $\widetilde p$ is a Seifert fibration $\widetilde X\to \widetilde B$ , that $\widetilde \pi \colon \widetilde B\to B$ is a twofold branched cover with branching set $p(\operatorname {\mathrm {sing}}(X))$ , and that $\mathcal F$ is precisely the fiber decomposition of $\widetilde p$ . This verifies item (3) by the construction of $\widetilde B$ and $\widetilde p$ , item (2) by (4.2), item (4) by the induced involution argument, and item (1) by the final local analysis.

4.1.9 The lifted Seifert invariants

We now determine the Seifert invariants of the Seifert fibration $\tilde {p}\colon \widetilde {X}\to \widetilde {B}$ . Let us denote the symbolic invariants associated with $X=\operatorname {\mathrm {Seif}}(B)$ as in (3.1) and those of $\tilde {X}=\operatorname {\mathrm {Seif}}(\tilde {B})$ by

(4.3) $$ \begin{align} \left\{ \tilde{b},\tilde{\varepsilon},\tilde{g}, \tilde{\iota}, \left\{ (\tilde{\alpha}_i, \tilde{\beta}_i) \right\}_{i=1}^{\tilde{n}} \right\}. \end{align} $$

It is then immediately clear that, since $\operatorname {\mathrm {sing}}(\tilde {X})=\emptyset $ , then $\tilde {\iota }=0$ . Near the exceptional C-orbits, the canonical involution of $\tilde {X}$ acts freely and, therefore, the preimage in $\tilde {X}$ of each exceptional orbit on X consists of two copies of itself. Thus, $\tilde {n}=2n$ and we can enumerate the Seifert invariants of $\tilde {X}$ , for example, as $(\tilde {\alpha }_{2k},\tilde {\beta }_{2k})=(\alpha _k, \beta _k)$ and $(\tilde {\alpha }_{2k-1},\tilde {\beta }_{2k-1})=(\alpha _k, \beta _k)$ for each $k=1,2,\ldots ,n$ . Because the local actions of $\mathbb {Z}_2$ on $\tilde {B}$ that give rise to B are orientation preserving, it follows that $\tilde {B}$ is orientable if and only if B is orientable. Therefore, $\tilde {\varepsilon } = \varepsilon $ .

We now compute $\chi (\tilde B)$ . Let $S\subset B$ be a disjoint union of $\iota $ small open $2$ -disks, one around each point of I-fiber type, and let $B^\circ =B\setminus S$ . The restriction of $\tilde \pi $ to $\tilde {B}^\circ =\tilde \pi ^{-1}(B^\circ )$ is a regular double covering

$$\begin{align*}\tilde\pi\colon \tilde{B}^\circ \longrightarrow\ B^\circ. \end{align*}$$

Hence,

$$\begin{align*}\chi(\tilde{B}^\circ)=2\chi(B^\circ)=2\big(\chi(B)-\iota\big). \end{align*}$$

To recover $\chi (\tilde {B})$ from $\chi (\tilde {B}^\circ ),$ we glue back the $\iota $ disks around the branch points. Each removed base disk has a single preimage disk in $\tilde B$ . Therefore,

$$\begin{align*}\chi(\tilde{B}) = \chi(\tilde{B}^\circ)+\iota=2\chi(B)-\iota. \end{align*}$$

When B is closed and orientable, $\chi (B)=2-2g$ . By the previous paragraph, $\tilde {B}$ is also orientable. Hence, $\chi (\tilde {B})=2-2\tilde g$ . Therefore, $\tilde {g}=2g+\frac {\iota }{2}-1$ and, in particular, $\iota $ must be even.

To compute $\tilde b$ , we proceed as follows. Let $B_0\subset B$ be given by removing small disjoint open disks around the $\iota>0$ points of I-fiber type. Set

$$\begin{align*}X_0=p^{-1}(B_0) = X\setminus\bigcup_{j=1}^\iota \mathrm{int}B(\mathrm{pt})_j. \end{align*}$$

Let Y be the closed Seifert $3$ -manifold obtained from $X_0$ by gluing, along each boundary Klein bottle, a twisted F-block (i.e., a solid Klein bottle) fiber-preservingly. Thus, we obtain Y from X by replacing each $B(\mathrm {pt})$ with a twisted F–block.

Consider the canonical double branched cover $\pi \colon \tilde {X}\to X$ . Over $X_0$ , the cover restricts to a regular double cover $\pi \colon \tilde X_0\to X_0$ . Each boundary Klein bottle $K^2\subset \partial X_0$ lifts to a boundary torus $T^2\subset \partial \tilde {X}_0$ . The boundary coverings $T^2\to K^2$ are the same regardless of whether we attach a $B\!\left (\mathrm {pt}\right )$ or a twisted F-block: in both cases, the boundary involution corresponds to the canonical orientable double covering $T^2\to K^2$ .

Fix boundary identifications compatible with the circle foliation on each $K^2 \subset \partial X_0$ . The lifts of the boundary gluing maps used to build X (attaching copies of $B\!\left (\mathrm {pt}\right )$ ) and Y (attaching twisted F-blocks) coincide on $\partial \tilde {X}_0$ , since in both cases, the boundary covering is the canonical double cover $T^2\to K^2$ . Consequently, $\tilde X$ and the orientable double cover $\tilde Y$ are obtained from the $\tilde X_0$ by attaching the same collection of solid tori along the same boundary maps. Hence, $\tilde X$ and $\tilde Y$ are fiber-preservingly homeomorphic. In particular, they determine the same Seifert fibration and invariants.

Now we compute Euler numbers. We follow Scott’s conventions (see [Reference Scott37, Section 3]). Since $\tilde {X}$ is a Seifert bundle over a closed orbifold with oriented total space, its (rational) Euler number is given by

(4.4) $$ \begin{align} e(\tilde{X})=-\left(\tilde{b}+\sum_{i=1}^{\tilde{n}}\frac{\tilde{\beta}_i}{\tilde{\alpha}_i}\right) = -\left(\tilde{b}+2\sum_{i=1}^n\frac{\beta_i}{\alpha_i}\right). \end{align} $$

The Euler number is defined to be zero if the total space of the Seifert bundle is non-orientable. Since Y contains a twisted F-block, which is a solid Klein bottle and hence non-orientable, Y is non-orientable. Thus, $e(Y)=0$ . For a finite covering $\hat {M}\to M$ of Seifert bundles of degree d, let l be the degree of the induced orbifold covering of bases and m the degree with which a regular fiber of $\hat {M}$ covers a regular fiber of M, so $d=lm$ . Then

$$\begin{align*}e(\hat{M}) = \frac{l}{m} e(M) \end{align*}$$

(see [Reference Scott37, Theorem 3.6] and compare with [Reference Jankins and Neumann21, Theorem 3.3]). Hence, since $\tilde Y$ is the orientable double cover of Y, we have

$$\begin{align*}e(\tilde Y) = 2e(Y) = 0. \end{align*}$$

Since $\tilde X\cong \tilde Y$ as Seifert bundles, we get $e(\tilde X) = 0$ , and it follows from (4.4) that

$$\begin{align*}\tilde{b} = -2\sum_{i=1}^n\frac{\beta_i}{\alpha_i}. \end{align*}$$

We now present two examples of non-equivalent generalized Seifert fibrations on the non-manifold Alexandrov space $\operatorname {\mathrm {Susp}}(P^2)\#\operatorname {\mathrm {Susp}}(P^2)$ , the connected sum of two copies of the suspension of $P^2$ . Here, the connected sum is defined by taking the connected sum in the manifold part of $\operatorname {\mathrm {Susp}}(P^2)$ (cf. [Reference Reyna, Galaz-García, Gómez-Larrañaga, Guijarro and Heil36]).

Example 4.3. (The double of $B(\mathrm {pt})$ )

Let

$$\begin{align*}X = B(\mathrm{pt})\ \cup_{K^2}\ B(\mathrm{pt}), \end{align*}$$

obtained by gluing two copies of $B(\mathrm {pt})$ along their common boundary $K^2$ via the identity map, which is fiber-preserving. Then X is a closed generalized Seifert fiber space with base $B\cong S^2$ with exactly two points of I-fiber type and no exceptional C-fibers. In the notation of Theorem A, the invariants of X are

$$\begin{align*}\{\, b=0,\ \varepsilon = \text{(orientable)},\ g=0,\ \iota=2,\ \{(\alpha_i,\beta_i)\}_{i=1}^{0}=\emptyset \,\}. \end{align*}$$

By Lemma 4.1, the canonical double branched cover of each half is the projection $S^1\times D^2 \to B(\mathrm {pt})$ , and the boundary covering is $T^2\to K^2$ . Gluing the two lifted boundaries via the identity map of $T^2$ (the fiber-preserving lift of the identity map on $K^2\cong \partial B\!\left (\mathrm {pt}\right )$ ) yields

$$\begin{align*}\tilde{X} \;=\; (S^1\times D^2)\ \cup_{T^2}\ (S^1\times D^2)\cong S^2\times S^1, \end{align*}$$

with its standard Seifert fibration $\tilde p:S^2\times S^1\to S^2$ . This illustrates Theorem B; the Seifert fibration of $\tilde {X}$ commutes with the branched cover, and the invariants transform as stated:

$$\begin{align*}\tilde\iota=0,\qquad \qquad \tilde\varepsilon=\varepsilon,\qquad \chi(\tilde B)=2\,\chi(B)-\iota=2\ \ (\text{hence }\tilde g=0). \end{align*}$$

For the underlying topology, X, being the double of $B\!\left (\mathrm {pt}\right )$ , is homeomorphic to $\operatorname {\mathrm {Susp}}(P^2)\#\operatorname {\mathrm {Susp}}(P^2)$ (see, e.g., [Reference Reyna, Galaz-García, Gómez-Larrañaga, Guijarro and Heil36, Lemma 5.1]).

Example 4.4 (Two I-fibers and one C-fiber $(2,1)$ )

Let $P\subset S^2$ be the $2$ -sphere with three open disks removed and let $Z\to P$ be the $\mathrm {O}(2)$ -bundle of R-fibers whose boundary has two twisted components (total-space boundary $K^2\sqcup K^2$ ) and one untwisted component (total-space boundary $T^2$ ). Define

$$\begin{align*}X = \big(Z\ \cup_{\,K^2\sqcup K^2}\ \big(B(\mathrm{pt})\sqcup B(\mathrm{pt})\big)\big)\ \cup_{\,T^2}\ V_{(2,1)}, \end{align*}$$

where $V_{(2,1)}$ denotes a Seifert solid torus of type $(\alpha ,\beta )=(2,1)$ . We glue the $K^2$ boundary components of Z and $B\!\left (\mathrm {pt}\right )$ via the identity map, which is fiber-preserving with respect to the chosen fiber-preserving identifications of each boundary component. We attach $V_{(2,1)}$ by a fiber-preserving homeomorphism $\varphi \colon \partial V_{(2,1)} \to \partial Z$ such that $\varphi _{*}(\mu )=2q_Z+h_Z$ in $H_1(\partial Z;\mathbb {Z})$ , where $\mu $ is the class of the meridian of $\partial V_{(2,1)}$ in $H_1(\partial V_{(2,1)};\mathbb {Z})$ , $h_Z$ is the class of a regular fiber on $\partial Z$ , and $q_Z$ is the class of a simple closed curve in $\partial Z$ transverse to the fibers such that $(q_Z,h_Z)$ is a $\mathbb {Z}$ -basis of $H_1(\partial Z;\mathbb {Z})$ (see, e.g., [Reference Seifert39, Lemma 6]) or [Reference Orlik31, Theorem 1.10]). Then X is a closed generalized Seifert fiber space with base $B\cong S^2$ having two points of I-fiber type and one exceptional C-fiber of type $(2,1)$ . In the notation of Theorem A, the invariants of X are

$$\begin{align*}\{\, b=0,\ \varepsilon=\text{(orientable)},\ g=0,\ \iota=2,\ \{(2,1)\}\,\}. \end{align*}$$

By Lemma 4.1, the canonical double branched cover of each $B(\mathrm {pt})$ is $S^1\times D^2$ , with boundary covering $T^2\to K^2$ . The Seifert solid torus $V_{(2,1)}$ does not intersect the branching set, so its preimage is the disjoint union of two Seifert solid tori of the same type $(2,1)$ . Gluing the lifted pieces, we obtain a Seifert fibration of the canonical double branched cover $\tilde X$ with base $\tilde {B}\cong S^2$ with two exceptional fibers $(2,1)$ and $\tilde b=-1$ . Hence, $e(\tilde X)=-(\tilde b+\tfrac 12+\tfrac 12)=0$ . In particular,

$$\begin{align*}\tilde X \cong M\big(S^2;\ \tilde b=-1;\ (2,1),(2,1)\big)\cong S^2\times S^1, \end{align*}$$

as recorded in [Reference Scott37, Section 4]. This illustrates Theorem B: in $\tilde {X}$ , the list of exceptional pairs is doubled and the Seifert fibration commutes with the branched covering. For the underlying topology, X is the quotient of $S^2\times S^1$ by an orientation-reversing involution with exactly four isolated fixed points. This involution is unique up to conjugacy (see [Reference Tollefson40, Theorem A]) and its quotient is homeomorphic to $\operatorname {\mathrm {Susp}}(P^2)\# \operatorname {\mathrm {Susp}}(P^2)$ (see also [Reference Galaz-Garcia and Guijarro11] and [Reference Reyna, Galaz-García, Gómez-Larrañaga, Guijarro and Heil36, Lemma 5.1]). The double branched cover $\pi \colon \tilde {X}\to X$ has exactly four branching points, corresponding to the four fixed points of the involution on $X\cong S^2\times S^1$ , and each one of the four branching points corresponds to an endpoint of the two I-fibers in $X\cong \operatorname {\mathrm {Susp}}(P^2)\#\operatorname {\mathrm {Susp}}(P^2)$ .