1. Introduction
Throughout, let W be a compact, connected, oriented 3manifold having no spherical boundary components. Let G be a finite group of orientationpreserving diffeomorphisms of W. A subset $X \subset W$ is equivariant if for all $g \in G$ , either $g(X) \cap X = \varnothing$ or $g(X) = X$ . It is invariant if $g(X) = X$ for all $g \in G$ . Famously, the Equivariant Sphere Theorem [ Reference W., Simon and Tung Yau16 ] says that if W is reducible, then there exists an equivariant essential sphere. Consequently, if W is reducible, then there exists an equivariant system of essential spheres $S \subset W$ giving a factorisation of W into irreducible 3manifolds. The (disconnected) manifold $W_S$ created by cutting open along S and gluing in 3balls is said to be obtained by equivariant surgery on S. We extend the action of G across the 3balls.
Closed 3manifolds also have equivariant Heegaard splittings [ Reference Zimmermann45 ]. (A Heegaard splitting for a disconnected manifold is the union of Heegaard splittings of its components.) The equivariant Heegaard genus $\mathfrak{g}(W;G)$ is the minimum genus of an equivariant Heegaard splitting for W. (This is the same as the minimal genus of an invariant Heegaard surface for W.) If G is trivial, then $\mathfrak{g}(W) = \mathfrak{g}(W;G)$ is the Heegaard genus of W. In general, $\mathfrak{g}(W;G) \geq \mathfrak{g}(W)$ . A reducing sphere for a Heegaard surface $H \subset W$ is a sphere intersecting W in a single essential simple closed curve. If W is reducible, Haken’s lemma states that every Heegaard surface for W admits a reducing sphere [ Reference Haken7 ]. In particular, if W is reducible, every invariant Heegaard surface for W admits a reducing sphere.
Question 1: (Equivariant Haken’s Lemma) If W is reducible and if H is an invariant Heegaard surface for W, must there exist an equivariant essential sphere in W that is a reducing sphere for H?
One important consequence of Haken’s lemma is that Heegaard genus of 3manifolds is additive under connected sum. That is, $\mathfrak{g}(W) = \mathfrak{g}(W_S)$ . We could also ask:
Question 2: (Additivity of Equivariant Heegaard Genus) If W is reducible and if $S \subset W$ is an invariant system of reducing spheres such that each component of $W_S$ is irreducible, is $\mathfrak{g}(W;G) = \mathfrak{g}(W_S; G)?$
A positive answer to Question 1 implies a positive answer to Question 2. Both questions have a positive answer in the case when G acts freely on W (Corollary 2·8 below). Another reason to think that the answer to both questions might be “yes,” is that recently Scharlemann [ 27 ] showed that given a collection of pairwise disjoint essential spheres $S \subset W$ , there is an isotopy of any Heegaard surface H for W so that each component of S is a reducing sphere for H. In particular, if S is as in Question 2 and if $H \subset W$ is an invariant Heegaard surface, then H can be isotoped so that each component of S is a reducing sphere for H. Of course, the isotopy may destroy the fact that H is invariant.
In fact, the answers to Questions 1 and 2 are negative. Throughout, $X$ is either the cardinality or the number of components of X.
Theorem. Equivariant Heegaard genus can be subadditive, additive, or superadditive. In particular:

(i) if $k \geq 2$ is large enough, then there exists a reducible W, an invariant essential sphere S for W dividing W into two irreducible factors, and a cyclic group G of diffeomorphisms of W having order k such that
\begin{align*}\mathfrak{g}(W;G) \lt \mathfrak{g}(W_S; G);\end{align*} 
(ii) for $k \geq 2$ , there exists a reducible W, an invariant essential sphere S for W dividing W into two irreducible factors, and a cyclic group G of diffeomorphisms of W having order k such that
\begin{align*}\mathfrak{g}(W;G) = \mathfrak{g}(W_S; G);\end{align*} 
(iii) for $k \geq 2$ , there exists a reducible W having a finite cyclic group of diffeomorphisms G of order k, and an invariant essential sphere S dividing W into two irreducible manifolds, such that
\begin{align*}\mathfrak{g}(W;G) = \mathfrak{g}(W_S; G) + k  1.\end{align*}
Furthermore, $\mathfrak{g}(W;G)$ can be arbitrarily high for fixed k.
Conclusions (i) and (ii) are the content of Theorems 3·1 and 3·3. Conclusion (iii) is the content of Theorem 6·4. Consequently, no invariant Heegaard surface H for the manifolds W in Conclusions (i) or (iii) admits an equivariant reducing sphere. Perhaps not surprisingly we use facts about the nonadditivity of tunnel number to prove those theorems. And so we are left with:
Question3: What can we say about the relationship between $\mathfrak{g}(W;G)$ and $\mathfrak{g}(W_S;G)$ ?
We establish an upper bound for $\mathfrak{g}(W;G)$ relative to a system of separating summing spheres. We also note that Rieck and Rubinstein [ Reference Rieck and Hyam Rubinstein25 ] have produced an upper bound on $\mathfrak{g}(W;G)$ in the case when $G = \{1, \tau\}$ , with $\tau$ an involution. However, their upper bound does not apply in the case when W is reducible. Theorem 6·4 shows that the following inequality for $c = 1$ is sharp; the other inequality is likely also sharp.
Theorem 6·10. For any invariant system of summing spheres $S \subset W$ such that each component of S is separating,
where $c = 1$ if every point of W has cyclic stabiliser and $c = 2$ otherwise.
Turning to lower bounds, we show:
Theorem 5·5. Suppose that W is closed and connected and that G does not act freely. If $S \subset W$ is an invariant system of summing spheres, then
where $\nu$ is the number of orbits of the components of $(W_S)$ that are not $S^3$ or lens spaces and $\mu$ is the number of orbits of components of $W_S$ that are homeomorphic to $S^3$ .
This bound is somewhat weak since $\mathfrak{g}(W;G) \geq \mathfrak{g}(W) = \mathfrak{g}(W_S) \geq 2\nu$ . However, it does give some indication of how the order of G affects the equivariant Heegaard genus. Additionally, Examples 5·6 and 5·7 suggest we are unlikely to be able to improve the lower bound without additional hypotheses. In a different direction, we show that equivariant Heegaard genus is additive or superadditive when the factors are equivariantly comparatively small. A 3manifold W is equivariantly comparatively small relative to the action of a finite group of diffeomorphisms G if every equivariant essential surface $F \subset W$ has $\mathfrak{g}(F) \gt \mathfrak{g}(W;G)$ .
Theorem 5·10. Suppose that when $S \subset W$ is an invariant system of summing spheres, then every component of $W_S$ is equivariantly comparatively small. Then:
As a byproduct of our methods, we produce a new invariant which we call equivariant net Heegaard characteristic $\operatorname{net}x_\omega(W;G)$ . It is similar to equivariant Heegaard genus, but its additivity properties are entirely understood, at least when every component of S is separating:
Theorem 4·10. There exists an invariant system of summing spheres $S \subset W$ such that
and $W_S$ is irreducible. If each component of S is separating, then equality holds.
Like equivariant Heegaard genus, $\operatorname{net}x_\omega(W;G)$ is defined by minimising a certain quantity over a surface. In this case, however, it is not an equivariant Heegaard splitting but an equivariant generalised Heegaard splitting. Theorem 4·10 arises from the fact that equivariant spheres arise as thin surfaces. We explain this in Section 4 using orbifolds. Part of the point of this paper is to advertise the very useful properties of the invariant $\operatorname{net}x_\omega$ , which is closely related to the invariant “net extent” of [ Reference Taylor and Tomova37, Reference Taylor and Tomova38 ].
Orbifolds and Outline
Threedimensional orbifolds are the natural quotient objects resulting from finite diffeomorphism group actions on 3manifolds. At least since Thurston’s work on geometrisation [ Reference Thurston39 , chapter 13], 3dimensional orbifolds have been the essential tool for understanding finite group actions on manifolds; they are also important and interesting in their own right. A closed orientable 3orbifold is locally modelled on the quotient of a 3ball by a finite group of orientation preserving diffeomorphisms. The singular set is the set of points which are the images of those points with nontrivial stabiliser group. A closed, orientable 3orbifold can thus be considered as a pair (M,T) where M is a closed orientable 3manifold and $T \subset M$ is a properly embedded trivalent spatial graph with integer edge weights $\omega \geq 2$ . If e is an edge of T with $\omega(e) = w$ , then we say that e has weight w. If $v \in T$ is a vertex with incident edges having weights a,b,c, we let $x_\omega(v) = 1  (1/a + 1/b + 1/c)$ .
When (M,T) is a closed, 3dimensional orbifold, for each vertex v we have $x_\omega(v) \lt 0$ . This occurs if and only if, up to permutation, (a,b,c) is one of (2,2,k), (2,3,3), (2,3,4) and (2,3,5) for some $k \geq 2$ . Conversely, every trivalent graph T properly embedded in a compact (possibly with boundary) 3manifold and having edge weights satisfying $x_\omega(v) \lt 0$ at each trivalent vertex v describes a compact, orientable 3orbifold. We work somewhat more generally by taking edge weights in the set $\mathbb{N}^\infty_2 = \{n \in\mathbb{N} : n \geq 2\} \cup \{\infty\}$ . We define $1/\infty = 0$ . This is similar to what is allowed by the software Orb [ Reference Heard and Culler11 ] for studying the geometry of 3orbifolds. We consider the edges of weight $\infty$ as consisting of points belonging to a knot, link, or spatial graph in M on which G acts freely. Allowing such weights does not create any additional difficulties and means that our methods may be useful for studying symmetries of knots, links, and spatial graphs. Additionally, some of our bounds are achieved only when there are edges of infinite weight; as these bounds are asymptotically achieved when edges have only finite weights, it is conceptually clearer to allow edges of infinite weight. Henceforth, we define “3orbifold” as follows.
Definition 1·1. A 3orbifold is a pair (M,T) where M is a compact, orientable 3manifold, possibly with boundary and $T \subset M$ is a properly embedded trivalent graph such that each edge e has a weight $\omega(e) \in \mathbb{N}^\infty_2$ . We also require that at every trivalent vertex v, $x_\omega(v) \lt 0$ .
When studying symmetries of knots or spatial graphs in a 3manifold, vertices of degree 4 or more may arise, as may vertices incident to edges of infinite weight. Our framework can handle such vertices by considering them as boundary components. Our approach should be contrasted with the usual methods of proving Haken’s lemma which involve intricate methods of controlling the intersections between a Heegaard surface and an essential sphere. We replace those arguments with appeals to machinery that guarantee that essential spheres show up as thin surfaces in a certain type of thin position. The basic structure of our arguments is as follows.
In Section 2, we define orbifold Heegaard splittings; show that they are the quotients of invariant Heegaard surfaces of W by G; and show that each orbifold Heegaard splitting of the quotient orbifold lifts to a Heegaard splitting of W. We also review the correspondence between equivariant essential spheres and orbifold reducing spheres. Section 3 uses the correspondence to prove that equivariant Heegaard genus can be subadditive or additive. Section 4 adapts Taylor–Tomova’s version of thin position to orbifolds. There we define the “net Heegaard characteristic” $\operatorname{net}x_\omega$ of an orbifold and establish its correspondence with equivariant net Heegaard characteristic. The equivariant net Heegaard characteristic of W is bounded above by $2\mathfrak{g}(W;G)  2$ . We also prove Theorem 4·10. The key idea is to use Taylor–Tomova’s partial order on the orbifold version of Scharlemann–Thompson’s generalised Heegaard splittings for 3manifolds. The minimal elements of this partial order are called “locally thin.” We show that $\operatorname{net}x_\omega$ is nonincreasing under the partial order and that orbifold reducing spheres show up as thin surfaces in the generalised Heegaard splittings. These two properties make $\operatorname{net}x_\omega$ additive under orbifoldsums. Understanding the net Heegaard characteristic of the result of decomposing the orbifold by essential spheres allows us to produce lower bounds on $\operatorname{net}x_\omega$ , and thus on $\mathfrak{g}(W;G)$ . Section 6 establishes upper bounds on $\mathfrak{g}(W;G)$ by adapting the amalgamation of generalised Heegaard splittings to orbifolds. Although, in principle, this should be straightforward to those who understand amalgamation of generalised Heegaard splittings, significant issues arise; issues which in some sense seem to characterize the nonadditivity of equivariant Heegaard genus.
2. General Notions, Orbifold Sums, and Orbifold Heegaard Surfaces
Throughout all manifolds and orbifolds are orientable. If X is a topological space, we let $Y \sqsubset X$ mean that Y is a path component of X. We let $X \setminus Y$ denote the complement of an open regular neighbourhood of Y in X.
We refer the reader to [ Reference Boileau, Leeb and Porti1, Reference Daryl Cooper2 ] for more on orbifolds. Suppose that (M,T) is an orbifold. A properly embedded orientable surface $S \subset M$ , transverse to T, naturally inherits the structure of a 2suborbifold; that is a surface locally modelled on the quotient of a disc or half disc by a finite group of orientationpreserving isometries. We call the points $S \cap T$ the punctures of S. If $p \in S \cap T$ is a puncture its weight $\omega(p)$ is the weight of the edge intersecting it. The orbifold characteristic of S is defined to be:
This is the negative of the orbifold Euler characteristic of S. Observe that if $\pi\mskip0.5mu\colon\thinspace R \to S$ is an orbifold covering map of finite degree d, then $x_\omega(R) = dx_\omega(S)$ . An orbifold is bad if it is not covered by a manifold and good if it is. The bad 2dimensional orbifolds are spheres that either have a single puncture of finite weight or have two punctures of different weight. 2orbifolds that are spheres with three punctures are turnovers. A good connected 2orbifold S is spherical if $x_\omega(S) \lt 0$ ; euclidean if $x_\omega(S) = 0$ ; and hyperbolic if $x_\omega(S) \gt 0$ . A compact, orientable 3orbifold is good if it does not contain a bad 2orbifold [ Reference Daryl Cooper2 , theorem 2·5].
Definition 2·1. A 3orbifold (M,T) is nice if:

(i) for each $S \sqsubset \partial M$ , $x_\omega(S) \geq 0$ and if S is a sphere then $S \cap T \geq 3$ ;

(ii) the pair (M,T) has no bad 2suborbifolds and no spheres that are oncepunctured with infinite weight puncture;
The reason for forbidding bad 2suborbifolds is that we cannot surger along them to create a valid 3orbifold. The requirement that $x_\omega(S) \geq 0$ for $S \sqsubset \partial M$ helps with some of our calculations. The other requirements help with our use of the material from [ Reference Taylor and Tomova36, Reference Taylor and Tomova37 ]. We note that $(W, \varnothing)$ is a nice orbifold. (Recall $\partial W$ has no spheres by our initial definition of W.)
2·1. Factorisations
Petronio [ Reference Petronio24 ] proved a unique factorisation theorem for closed good 3orbifolds (M,T) without nonseparating spherical 2suborbifolds. We are working in a slightly more general context (because our orbifolds may not be closed, we allow good nonseparating spherical 2suborbifolds, and we allow infinite weight edges). Nevertheless, we adopt Petronio’s terminology and his results carry over to our setting as we now describe.
Suppose that $(M_1, T_1)$ and $(M_2, T_2)$ are distinct nice orbifolds. Let $p_1 \in M_1$ and $p_2 \in M_2$ . We can perform a connected sum of $M_1$ and $M_2$ by removing a regular neighbourhood of $p_1$ and $p_2$ and gluing the resulting 3manifolds together along the newly created spherical boundary components; after gluing the corresponding sphere is a summing sphere. To extend the sum to the pairs $(M_1, T_1)$ and $(M_2, T_2)$ we place conditions on the points. We require that they are either in the interiors of edges of the same weight or that they are on vertices with incident edges having matching weights. The gluing map is then required to match punctures to punctures of the same weight.
When $p_1$ and $p_2$ are disjoint from $T_1$ and $T_2$ , the sum is a distant sum and we write $(M,T) = (M_1, T_1) \#_0 (M_2, T_2)$ . When each $p_i$ is in the interior of an edge of $T_i$ , it is a connected sum and we write $(M,T) = (M_1, T_1) \#_2 (M_2, T_2)$ . When each $p_i$ is a trivalent vertex, it is a trivalent vertex sum and we write $(M,T) = (M_1, T_1) \#_3 (M_2, T_2)$ . This sum is welldefined, for a particular choice of vertices $p_1, p_2$ and bijection from the ends of edges incident to $p_1$ to those incident to $p_2$ . See [ Reference Wolcott42 , section 4 ]. The pair $\mathbb{S}(0) = (S^3, \varnothing)$ is the identity for $\#_0$ . The pair $\mathbb{S}(2) = (S^3, T)$ where T is the unknot is the identity for $\#_2$ . The pair $\mathbb{S}(3) = (S^3, T)$ where T is a planar (i.e. trivial) $\theta$ graph, is the identity for $\#_3$ . In each case, we allow T to have whatever weights make sense in context. Note that when $k\lt i \leq j$ , both factors of $\mathbb{S}(i) \#_k \mathbb{S}(j)$ are trivial. This means that some care is needed when discussing prime factorisations.
Conversely, suppose that (M,T) is a nice orbifold. Given a connected spherical 2suborbifold $S \subset (M,T)$ we may split (M,T) open along S and glue in two 3balls each containing a graph that is the cone on the points $S \cap T$ . This operation is called surgery along S. Observe that the result is still an orbifold and that if M is connected, each component of $(M,T)_S$ is incident to one or more scars from the surgery (i.e. the boundaries of the 3balls we glued in). If S is such a sphere or the disjoint union of such spheres, we denote the result of surgery along (all components of) S by $(M,T)_S$ . Each component of S produces two scars in $(M,T)_S$ ; we say those scars are matching. If S separates M, then surgery is the inverse operation to distant sum, connected sum, or trivalent vertex sum.
Since we will be dealing with reducible orbifolds, we need to work with slight generalizations of compressing discs. Suppose that $S \subset (M,T)$ is a surface. An scdisc for S is a zero or oncepunctured disc D with interior disjoint from $T \cup S$ , with boundary in $S \setminus T$ , and which is not isotopic (by an isotopy everywhere transverse to T) into S. If $\partial D$ does not bound a zero or oncepunctured disc in S, then D is a compressing disc or cut disc corresponding to whether D is zero or oncepunctured. A cdisc is a compressing disc or cut disc. Otherwise, D is a semicompressing disc or semicut disc respectively. In other words, a semicompressing disc or semicut disc is a zero or oncepunctured disc with inessential boundary in the surface but which is not parallel into the surface. The weight $\omega(D)$ of an scdisc D is equal to the weight of the edge of T intersecting it and 1 otherwise. Compressing a surface using an scdisc D decreases $x_\omega$ by $2/\omega(D)$ . If $\partial M$ admits an scdisc D, then $(M,T)\setminus D$ is the result of $\partial$ reducing (M,T).
If S does not admit a cdisc, it is cincompressible. A cessential surface is a surface $S \subset (M,T)$ where each component is:

(1) a cincompressible surface

(2) not parallel in $M \setminus T$ into $\partial (M\setminus T)$ (i.e. not $\partial$ parallel); and

(3) not an unpunctured sphere bounding a 3ball in $M \setminus T$ .
Observe that for a surface, being parallel in (M,T) to a component of $\partial M$ is more restrictive than being parallel into $\partial (M\setminus T)$ .
Definition 2·2. An orbifold (M,T) is orbifoldreducible if there exists a cessential spherical 2suborbifold $S \subset (M,T)$ .
Definition 2·3 (c.f. [ Reference Petronio24 ]). Suppose that (M,T) is a nice orbifold and that $S \subset (M,T)$ is a closed spherical essential 2orbifold. Then S is an system of summing spheres if $(M,T)_S$ is orbifoldirreducible. A system of summing spheres S is efficient if, for $i \in \{0,2,3\}$ , whenever an $\mathbb{S}(i)$ component of $(M,T)_S$ contains an ipunctured scar, then it contains the matching scar.
For a system of summing spheres $S \subset (M,T)$ , we call $(M,T)_S$ , a factorisation of (M,T) using S. The main difference between a factorisation using an efficient system of summing spheres and a prime factorisation is that a factor in an efficient factorisation may contain a nonseparating sphere. By work of Petronio and HogAngeloni–Matveev, in the absence of nonseparating spherical 2orbifolds, up to orbifold homeomorphism for an efficient system of summing spheres S, both S and its factorisation $(M,T)_S$ are unique; however, these homeomorphisms are not necessarily realisable by an isotopy in (M,T). We note that neither the Petronio, HogAngeloni–Matveev results nor the following theorem (which is based on those) is trivial. For instance, as observed in [ Reference Petronio24 , example 1·3], the connected sum of any knot in $S^3$ with $(S^1 \times S^2, S^1 \times \text{(point)})$ is pairwise homeomorphic to $(S^1 \times S^2, S^1 \times \text{(point)})$ . Consequently, $(S^1 \times S^2, S^1 \times \text{(point)})$ does not have a prime factorization. The crux of the existence of an efficient splitting system is deferred to Corollary 4·8 below and is a consequence of Taylor–Tomova’s work on thin position. Based on the discussion in [ Reference Petronio24 , section 3], the ability of thin position to handle nonseparating spherical 2suborbifolds seems to be an instance where the thin position techniques have an advantage over normal surface techniques. We relegate the proof to the appendix, since it is a slightly more elaborate version of Petronio’s and HogAngeloni–Matveev’s proofs.
Theorem 2·4 (after Petronio, HogAngeloniMatveev). Suppose that (M,T) is a nice orbifold that is orbifoldreducible. Then there exists an efficient system of summing spheres $S \subset (M,T)$ ; indeed, any system of summing spheres contains an efficient subset. Furthermore, any two such systems S, S’ are orbifoldhomeomorphic, as are $(M,T)_S$ and $(M,T)_{S'}$ .
The Equivariant sphere theorem is an important tool for studying group actions on 3manifolds. The statement we use is inspired by [ Reference Boileau, Leeb and Porti1 , theorem 3·23] and the subsequent remark.
Theorem 2·5 (Equivariant sphere theorem [ Reference W., Simon and Tung Yau16 ] (c.f. [ Reference Dunwoody4 ])). Suppose that $\rho\mskip0.5mu\colon\thinspace (W, T') \to (M,T)$ is a regular orbifold covering, with (W, T’) and (M,T) nice. Then (W, T’) is orbifoldreducible if and only if (M,T) is orbifoldreducible.
2·2 Orbifold Heegaard Splittings
As is well known, a Heegaard splitting of a closed 3manifold is a decomposition of the 3manifold into the union of two handlebodies glued along their common boundary. Every closed 3manifold has such a decomposition. 3manifolds with boundary have a similar decomposition into two compressionbodies glued along their positive boundaries. Zimmermann (e.g. [ Reference Zimmermann44 – Reference Zimmermann46 ]) defined Heegaard splittings of closed 3orbifolds and used them to study equivariant Heegaard genus. In his decompositions, an orbifold handlebody is an orbifold that is the quotient of a handlebody under a finite group of diffeomorphisms. He gives an alternative description in terms of certain kinds of handle structures [ Reference Zimmermann45 , proposition 1]. We adapt this latter definition to define orbifold compressionbodies. Petronio [ Reference Petronio23 ] also defines handle structures for 3orbifolds; the definitions differ only on the definition of 2handles. Zimmermann’s definition has the advantage that handle structures can be turned upside down. Most of our definitions apply to graphs in 3manifolds more generally, so we state them in that generality if possible.
Definition 2·6 (Handle structures). A ball 0handle or ball 3handle is a pair $(W, T_W)$ where W is a 3ball and $T_W$ is the cone on a finite (possibly empty) set of points in $\partial W$ . We also call ball 0handles and 3handles, trivial ball compressionbodies. We set $\partial_+ W = \partial W$ and $\partial_ W = \varnothing$ . A product 0handle or product 3handle is a pair $(W, T_W)$ pairwise homeomorphic to $(F \times I, p \times I)$ where F is a closed orientable surface, $p \subset F$ is finitely (possibly zero) many points. We also call product 0handles and product 3handles, trivial product compressionbodies. We set $\partial_\pm W$ to be the preimage of $F \times \{\pm 1\}$ . The attaching region for a 0handle is the empty set and the attaching region for a 3handle $(W, T_W)$ is $(\partial_+ W, T_W \cap \partial_+ W)$ . A trivial compressionbody is either a trivial ball compressionbody or a trivial product compressionbody.
A 1handle or 2handle is a pair $(H, T_H)$ pairwise homeomorphic to $(D^2 \times I, p \times I)$ where p is either empty or is the center of $D^2$ . The attaching region of a 1handle is the preimage of $(D^2 \times \partial I, p \times \partial I)$ . The attaching region of a 2handle is $((\partial D^2) \times I, \varnothing)$ .
A vpcompressionbody $(C, T_C)$ is the union of finitely many 0handles and 1handles so that the following hold:

(1) the 0handles are pairwise disjoint, as are the 1handles;

(2) 1handles are glued along their attaching regions to the positive boundary of the 0handles, and are otherwise disjoint from the 0handles;

(3) if $(H, T_H)$ is a 1handle such that one component (D,p) of its attaching region is glued to a 0handle $(W, T_W)$ , and if $T_H \neq \varnothing$ , then $p \in T_H \cap \partial_+ W$ ;

(4) C is connected.
See Figure 1 for an example. The “vp” stands for “vertex punctured” and is used since drilling out vertices changes trivial ball compressionbodies with vertices into trivial product compressionbodies. If $(C, T_C)$ is a vpcompressionbody, we let $\partial_ C$ be the union of $\partial_ W$ over the 0handles $(W, T_W)$ and we let $\partial_+ C = \partial C \setminus \partial_ C$ . Edges of $T_C$ disjoint from $\partial_+ C$ are called ghost arcs. Closed loops disjoint from $\partial C$ are core loops. Edges with exactly one endpoint on $\partial_+ C$ are vertical arcs and edges with both endpoints on $\partial_+ C$ are bridge arcs. Dually, vpcompressionbodies may be defined as the union of 2handles and 3handles. Equivalently, if $(C, T_C)$ is a connected pair with one component of $\partial C$ designated as $\partial_+ C$ , then $(C, T_C)$ is a vpcompressionbody if and only if there is a collection of pairwise disjoint scdiscs $\Delta$ for $\partial_+ C$ such that the result of $\partial$ reducing $(C, T_C)$ along $\Delta$ is the union of trivial ball compressionbodies and trivial product compressionbodies. The collection $\Delta$ is a complete collection of scdiscs for $(C, T_C)$ if it is pairwise nonparallel. If $T_C$ has weights such that $(C, T_C)$ is both a vpcompressionbody and an orbifold, then we call $(C, T_C)$ an orbifold compressionbody.
For a nice orbifold (M,T), an orbifold Heegaard surface is a transversally oriented separating connected surface $H \subset (M,T)$ such that H cuts (M,T) into two distinct orbifold compressionbodies, glued along their positive boundaries. We define the Heegaard characteristic of (M,T) to be:
where the minimum is over all orbifold Heegaard surfaces H for (M,T). The invariant $x_\omega(M,T)$ is twice the negative of the “Heegaard number” defined by Mecchia–Zimmerman [ Reference Mecchia and Zimmermann15 ]. Dividing by 2 would also make the comparison with [ Reference Taylor and Tomova37, Reference Taylor and Tomova38 ] easier. However, since orbifold Euler characteristic need not be integral, making that normalisation would unpleasantly complicate some of the calculations in this paper.
The equivariant Heegaard characteristic of W is
where the minimum is taken over all invariant Heegaard surfaces for H. Zimmermann proved the following when W is closed; the proof extends to the case when W has boundary as we explain. See also [ Reference Futer5 ] for a similar result related to strong involutions on tunnel number 1 knots. The statement is deceptively simple as it implies (and its proof relies on) the Smith Conjecture as well as a thorough understanding of group actions on 3balls and products. See the remark on page 52 of [ Reference Boileau, Leeb and Porti1 ].
Lemma 2·7 (after Zimmermann [ Reference Zimmermann45 ]). Suppose that W has orbifold quotient (M,T). Every invariant Heegaard surface for W descends to an orbifold Heegaard surface for (M,T) and every orbifold Heegaard surface for (M,T) lifts to an invariant Heegaard surface for W. Consequently,
Proof. Suppose that Y is a compressionbody (i.e. orbifold compressionbody with empty graph) and that G is a finite group of orientationpreserving diffeomorphisms of Y. We show the quotient orbifold is an orbifold compressionbody. If $\partial_ Y = \varnothing$ (i.e. Y is a handlebody), then the quotient orbifold is an orbifold compressionbody by [ Reference Zimmermann45 ]. In particular, the quotient of a 3ball is an orbifold trivial ball compressionbody. Consider the case when $Y = F \times I$ for a closed, connected, oriented surface F. Since each element of G is orientationpreserving, no element interchanges the components of $\partial Y$ . If $Y = S^2 \times I$ , we cap off $\partial_ Y$ with a 3ball, extend the Gaction across the 3ball, and appeal to the 3ball case to observe that the quotient orbifold is a trivial product orbifold compressionbody. Suppose that $F \neq S^2$ . Let DY be its double and observe that we can extend the action of G to DY. The 3manifold DY is a Seifert fiber space with no exceptional fibers, namely $F \times S^1$ . Consider an embedded torus DQ that is the double of an essential spanning annulus Q in Y. By [ Reference Hass9 , theorem 1·5], DQ can be isotoped to DQ $^{\prime}$ so that for each $g \in G$ , gDQ $^{\prime}$ is vertical in DQ and is isotopic to gDQ. Since each $g \in G$ preserves each component of $\partial Y$ , there is an annulus $Q' \subset Y$ , vertical in Y, such that for each $g \in G$ , gQ $^{\prime}$ is isotopic to gQ by a proper isotopy in Y. Applying this observation to each annulus in a collection of spanning annuli in Y cutting Y into 3balls, we can then conclude that the quotient orbifold of Y by the action of G is a trivial product orbifold, as desired. Finally, suppose that $\partial_+ Y$ is compressible. By the Equivariant Disc Theorem [ Reference William and Yau17 ], Y admits an equivariant essential disc. Boundaryreducing along this disc and inducting on $x_\omega(\partial_+ Y)  x_\omega(\partial_ Y)$ shows that the quotient orbifold is again an orbifold compressionbody. Our lemma then follows from the definition of invariant Heegaard surfaces, orbifold Heegaard surfaces, and the multiplicativity of orbifold characteristic under finite covers.
Corollary 2·8. If G acts freely on W, then there exists an invariant system of summing spheres $S \subset W$ such that $W_S$ is irreducible and $\mathfrak{g}(W;G) = \mathfrak{g}(W_S;G)$ .
Proof. Let (M,T) be the quotient orbifold of the action of G on W. Assume the action is free so that $T =\varnothing$ . There exists an efficient system of summing spheres $\overline{S} \subset M$ . By Haken’s Lemma, $\mathfrak{g}(M) = \mathfrak{g}(M_{\overline{S}})$ . Let S be the preimage of $\overline{S}$ in W. By Lemma 2·7 (after converting from Euler characteristic to genus), $\mathfrak{g}(W;G) = \mathfrak{g}(W_S; G)$ .
3. Examples of Additivity and SubAdditivity
As an example of how to work with orbifold Heegaard surfaces, in this section we show that equivariant Heegaard genus can be both subadditive and additive. The examples of superadditivity require different techniques. Our examples arise from cyclic branched covers of knots in $S^3$ . We begin with a preliminary calculation.
Suppose that $K \subset S^3$ is a knot with weight $k \geq 2$ . When we ignore the weight k, an orbifold Heegaard surface H for $(S^3, K)$ of genus g is a (g,b)bridge surface where $b = H \cap K/2$ [ Reference Doll3 ]. Conversely, remembering the weight k, makes each (g,b)bridge surface H for K into an orbifold Heegaard surface for $(S^3, K)$ . If H is such a surface, observe that
If H is disjoint from K, then K lies as a core loop of the handlebody to one side of H. The tunnel number $\mathfrak{t}(K)$ of K is the minimum of $\mathfrak{g}(H)  1$ over all such H. If H is any (g,b)bridge surface for K, then by attaching tubes along the arcs of $K \setminus H$ , we can convert H into a $(g + b, 0)$ bridge surface for K. Consequently, $\mathfrak{t}(K) \leq g + b  1$ and $x_\omega(S^3, K) \leq 2\mathfrak{t}(K)$ . The same technique shows that if, for fixed g, H minimises b, then $b \leq \mathfrak{b}(K)$ , where, $\mathfrak{b}(K)$ is the bridge number of K. (Recall $\mathfrak{b}(K)$ is the minimal b $^{\prime}$ such that K admits a (0,b $^{\prime}$ )bridge surface.)
For $i = 1,2$ , let $K_i \subset S^3$ be a knot of weight k and let $H_i$ be a $(g_i, b_i)$ bridge surface for $K_i$ . Assume that $b_1, b_2 \gt 0$ . If we perform the connected sum of $K_1$ and $K_2$ using points of $K_1 \cap H_1$ and $K_2 \cap H_2$ , then $H = H_1 \# H_2$ is a $(g_1 + g_2, b_1 + b_2  1)$ bridge surface for $K = K_1 \# K_2$ . Note that if S is the summing sphere for $(S^3, K)$ arising from our choice of connected sum, then $x_\omega(S) =  2/k$ . Consequently,
Theorem 3·1. If $k \geq 2$ is large enough, then there exists a reducible W, an invariant essential sphere S for W dividing W into two irreducible factors, and a cyclic group G of diffeomorphisms of W having order k such that
Proof. We recall that there are many examples (e.g. [ Reference Morimoto21, 22, Reference Schirmer33 ]) of prime knots $K_1, K_2$ for which tunnel number is subadditive. That is, $\mathfrak{t}(K_1) + \mathfrak{t}(K_2)  \mathfrak{t}(K_1 \# K_2) \geq 1$ . Recall $K= K_1 \# K_2$ . Let $\overline{S}$ be the summing sphere. By [ Reference Schubert34 ], $\mathfrak{b}(K) = \mathfrak{b}(K_1) + \mathfrak{b}(K_2)  1$ . Suppose that $k \gt \mathfrak{b}(K)$ . Let $H_i$ be a $(g_i, b_i)$ bridge surface for $K_i$ such that $x_\omega(H_i) = x_\omega(S^3, K_i)$ . According to the preamble, $x_\omega(H_i) \geq 2\mathfrak{t}(K_i)  2b_i/k$ . Thus,
Passing to the kfold cyclic branched cover W over K, with G the deck group and S the preimage of $\overline{S}$ , produces the desired examples.
Remark 3·2. From Morimoto’s examples of tunnel number degeneration [ Reference Morimoto21 ] we can see that in the proof of Theorem 3·1, we can choose the desired $K_1$ and $K_2$ so that $\mathfrak{b}(K) \leq 6$ . Thus, $k \geq 7$ suffices in the statement of Theorem 3·1.
We now turn to examples of additivity.
Theorem 3·3. For $k \geq 2$ , there exists a reducible W, an equivariant essential sphere S for W dividing W into two irreducible factors, and a cyclic group G of diffeomorphisms of W having order k such that
Proof. Fix $k \geq 2$ . For $i = 1,2$ , let $K_i$ be a torus knot of type (p,q) with p, q relatively prime and $p \gt q \geq 5$ . By [ Reference Schubert34 ], each $\mathfrak{b}(K_i) = q$ . Also, each $K_i$ admits a (1,1)bridge surface $H_i$ . Thus, $x_\omega(S^3, K_i) = x_\omega(H_i) = 2(1  1/k)$ . As in the preamble, let $K = K_1 \# K_2$ and $H = H_1 \# H_2$ so that H is a (2,1)bridge surface for K. Let $\overline{S}$ be the summing sphere. By the calculations in the preamble,
Let H $^{\prime}$ be a (g,b)bridge surface for K such that $x_\omega(H') = x_\omega(S^3, K)$ . In particular, for that genus g, the number of punctures 2b is minimal. Also, since $x_\omega(H') \leq x_\omega(H)$ :
If $g = 0$ , then by Schubert’s result [ Reference Schubert34 ] that bridge number is $(\!1)$ additive, we have $b = 2q  1$ . In which case,
a contradiction to the fact that $q \geq 5$ and $k \geq 2$ .
If $g = 1$ , our inequality shows that $b \leq 3$ . Since K is not the unknot, $b \geq 1$ . Doll studied the situation when $g = 1$ and showed that (for our choice of $K_1$ and $K_2$ ), $b \geq q  1$ . (The proof for arbitrary $K_1$ and $K_2$ can be found in the solution [ Reference Doll3 , section 5] of his Conjecture (1.1’) for the case $g = 1$ .) In which case,
But this contradicts the assumption that $k \geq 2$ .
If $g = 2$ , and $b = 0$ , then the tunnel number of K would be 1, contradicting the fact that K is composite. Thus, $g = 2$ and $b \geq 1$ . Since $x_\omega(H') \leq x_\omega(H)$ , we have $b = 1$ and so, $x_\omega(H') = x_\omega(H)$ .
Thus, $x_\omega(S^3, K) = x_\omega(H') = x_\omega(H) = x_\omega(S^3, K_1) + x_\omega(S^3, K_2)  x_\omega(S)$ . Passing to the kfold cyclic branched cover W over K, with G the deck group and S the preimage of $\overline{S}$ , produces the desired examples.
4. Orbifold Thin Position
Building on a long line of work concerning thin position, beginning with Gabai [ Reference Gabai6 ] and particularly including Scharlemann–Thompson [ Reference Scharlemann and Thompson28, 29 ], Hayashi–Shimokawa [ Reference Hayashi and Shimokawa10 ] and Tomova [ Reference Tomova40 ], Taylor–Tomova created a thin position theory for spatial graphs in 3manifolds. In this section, we explain the minor adaptions needed to make it work for 3orbifolds.
Definition 4.1 (TaylorTomova [ Reference Taylor and Tomova36 ]). Let M be a compact, orientable 3manifold and $T \subset M$ a spatial graph. A properly embedded closed surface $\mathcal{H} \subset (M,T)$ is a multiple vpbridge surface if the following hold:

(i) $\mathcal{H}$ is the disjoint union of $\mathcal{H}^+$ and $\mathcal{H}^$ where each of $\mathcal{H}^\pm$ are the union of components of $\mathcal{H}$ ;

(ii) each component of $(M,T)\setminus \mathcal{H}$ is a vpcompressionbody;

(iii) $\mathcal{H}^+ = \bigcup \partial_+ C$ and $\mathcal{H}^ \cup \partial M = \bigcup \partial_ C$ where each union is over all components $(C, T_C) \sqsubset (M,T) \setminus \mathcal{H}$ .
When $\mathcal{H}$ has a transverse orientation, we can consider the dual digraph; this is the digraph with a vertex for each vpcompressionbody and an oriented edge corresponding to each component of $\mathcal{H}$ . We consider such $\mathcal{H}$ equipped with a transverse orientation such that the dual digraph is acyclic and each $(C,T_C) \sqsubset (M,T)\setminus \mathcal{H}$ is a cobordism from $\partial_ C$ to $\partial_+ C$ . Equipped in this way, $\mathcal{H}$ is an oriented multiple vpbridge surface. We let $\mathbb H(M,T)$ be the set of oriented multiple vpbridge surfaces up to isotopy transverse to T. When (M,T) is an orbifold, we call the elements of $\mathbb H(M,T)$ multiple orbifold Heegaard surfaces.
After assigning an orientation, every orbifold Heegaard surface for a nice orbifold (M,T) is an element of $\mathbb H(M,T)$ . Since every spatial graph in a 3manifold can be put into bridge position with respect to any Heegaard surface for the 3manifold, $\mathbb H(M,T) \neq \varnothing$ . For $\mathcal{H} \in \mathbb H(M,T)$ , observe that $\mathcal{H}^ = \varnothing$ if and only if $\mathcal{H} = \mathcal{H}^+$ is connected. If $T = \varnothing$ , multiple vpbridge surfaces induce the generalised Heegaard splittings of [ Reference Scharlemann, Schultens and Saito31 ]. The following lemma is a straightforward extension of Lemma 2·7.
Lemma 4·2. Suppose that $(M,T) \to (M', T')$ is a finitesheeted orbifold cover and that $\mathcal{H} \in \mathbb H(M',T')$ . Then the preimage of $\mathcal{H}$ is a multiple orbifold Heegaard surface for (M,T).
When (M,T) does not contain any oncepunctured spheres, Taylor and Tomova [ Reference Taylor and Tomova37 ] define an invariant called “net extent” on elements of $\mathbb H(M,T)$ . We now adapt that invariant to the orbifold context.
Definition 4·3. Suppose that (M,T) is a nice orbifold. For $\mathcal{H} \in \mathbb H(M,T)$ , the net Heegaard characteristic is
We define the net Heegaard characteristic of (M,T) to be:
We define $\operatorname{net}x_\omega(W;G)$ similarly, but minimise only over invariant $\mathcal{H} \in \mathbb H(W,\varnothing)$ .
Proposition 5·3 below ensures that $\operatorname{net}x_\omega(M,T)$ is welldefined and that there exists $\mathcal{H} \in \mathbb H(M,T)$ with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . The proof of the next lemma follows easily from Lemma 4·2.
Lemma 4·4. Suppose that W has quotient orbifold (M,T). Then
In [ Reference Taylor and Tomova36 ], Taylor and Tomova defined a set of operations on elements of $\mathbb H(M,T)$ and used them to define a partial order called thins to and denoted $\mathcal{J} \to \mathcal{H}$ . A minimal element in the partial order is said to be locally thin. The operations involved in the definition of the partial order are all versions of the traditional “destabilisation” and “weak reduction” of Heegaard splittings of 3manifolds and “unperturbing” of bridge surfaces for links. We do not need the precise definitions of the operations in this text, but we do need the following information. For our purposes, we group the operations into four categories (deferring to [ Reference Taylor and Tomova36 ] for precise definitions):

(I) destabilisation, meridional destabilisation, $\partial$ destabilisation, meridional $\partial$ destabilisation, ghost $\partial$ destabilisation, meridional ghost $\partial$ destabilisation;

(II) unperturbing, undoing a removable arc;

(III) consolidation;

(IV) untelescoping.
The next lemma summarises the key aspects of these operations. Pay particular attention to the disc D used in the operations (I). In Lemma 4·6, we will need to analyse the effect of compressing along this disc on net Heegaard characteristic.
Lemma 4·5 (Taylor–Tomova). The following hold:

(i) all of the operations listed in (I) involve replacing a thick surface $J \sqsubset \mathcal{J}^+$ with a new thick surface H such that H is obtained from J by compressing along a compressing disc or cut disc D and, if $\partial D$ separates J, discarding a component. The component that is discarded is parallel to a surface obtained by tubing together some components of $\partial M$ and vertices of T along edges of T disjoint from $\mathcal{H}$ ;

(ii) all of the operations in (II) remove two punctures from a thick surface $J \sqsubset \mathcal{J}^+$ ;

(iii) consolidation removes a thick surface and a thin surface from $\mathcal{J}$ that together bound a product vpcompressionbody with interior disjoint from $\mathcal{J}$ ;

(iv) untelescoping replaces a thick surface $J \sqsubset \mathcal{J}^+$ with two new thick surfaces $H_1$ and $H_2$ and creates additional thin surfaces F. These surfaces arise from a pair of disjoint scdiscs on opposite sides of J. $H_1$ and $H_2$ are each obtained (up to isotopy) by compressing along one of the discs and F is obtained by compressing along both.
Lemma 4·6. Suppose that (M,T) is an orbifold. As an invariant on $\mathbb H(M,T)$ , $\operatorname{net}x_\omega$ is nonincreasing under the partial order $\to$ .
Proof. This lemma follows fairly directly from Lemma 4·5. Suppose that one of the moves in a thinning sequence replaces a thick surface J with another thick surface H. Consider, first, the possibility that the move was of type (I). Let D be the disc we compress along, as in Lemma 4·5. If $\partial D$ is nonseparating on J, we have
Thus, in such a case, the move does not increase $\operatorname{net}x_\omega$ . If $\partial D$ separates J, recall that after compressing we discard one component J $^{\prime}$ of the resulting surface. That is, $H \cup J'$ is the result of the compression of J. We have
Suppose, in order to obtain a contradiction, that $x_\omega(H) \gt x_\omega(J)$ . Then
where the last inequality follows from the fact that J $^{\prime}$ contains a scar from the compression by D. Since J $^{\prime}$ is a closed surface, it must be a sphere. We recall from Lemma 4·5 that it is parallel to a certain surface S obtained by tubing together components of $\partial M$ and vertices of T along edges of T disjoint from $\mathcal{H}$ . Let $\Gamma$ be the graph with a vertex for each component of $\partial M$ and each vertex of T that goes into the creation of S and with edges the edges we tube along. Since J $^{\prime}$ is a sphere and is parallel to S, $\Gamma$ must be a tree and each component of $\partial M$ that is a vertex of T is a sphere. If $\Gamma$ has an edge, there are at least two leaves and, by our definition of orbifold, each must be incident to at least two vertical arcs, giving S at least 4 punctures. But in that case $x_\omega(J') \geq 0$ , a contradiction. Thus, $\Gamma$ is an isolated vertex; that is, J $^{\prime}$ is parallel to either a component of $\partial M$ or to a vertex of T. If it is a component of $\partial M$ , then $x_\omega(J') \geq 0$ , by hypothesis. Thus, J $^{\prime}$ is parallel to a vertex v of T. The vertex v is trivalent with incident edges having weights a,b,c and
If $\omega(D) = 1$ , then we have a contradiction. If $\omega(D) \neq 1$ , then D was a cut disc and so one of a,b,c is equal to $\omega(D)$ . Thus, in this case also, we have a contradiction. We conclude that $x_\omega( H) \leq x_\omega(J)$ and that none of the moves of type (I) increase $\operatorname{net}x_\omega$ .
The moves of type (II) remove two punctures from a thick surface and so cannot increase $\operatorname{net}x_\omega$ . If $H \sqsubset \mathcal{H}^+$ and $F \sqsubset \mathcal{H}^$ are parallel, then $x_\omega(H) = x_\omega(F)$ and so consolidation does not change $\operatorname{net}x_\omega$ . Finally, consider the operation of untelescoping. The three new surfaces $H_1$ , $H_2$ , and F (as in the statement of Lemma 4·5) are all obtained by compressions along scdiscs for J. An easy computation shows that $x_\omega(J) = x_\omega(H_1) + x_\omega(H_2)  x_\omega(F)$ and so untelescoping also leaves $\operatorname{net}x_\omega$ unchanged.
The next theorem is key to our endeavors. We have stated only what we need for this paper. We say that $\mathcal{H}^+$ is scstrongly irreducible if it is not possible to untelescope it (i.e. use move (IV) above).
Theorem 4·7 (TaylorTomova). Suppose that T is a spatial graph in an orientable 3manifold M such that no component of $\partial M$ is a sphere with two or fewer punctures and there is no oncepunctured sphere in (M,T). Then for every $\mathcal{J} \in \mathbb H(M,T)$ there exists a locally thin $\mathcal{H} \in \mathbb H(M,T)$ such that $\mathcal{J} \to \mathcal{H}$ . Furthermore, for any locally thin $\mathcal{H}$ the following hold:

(i) Each component of $\mathcal{H}^+$ is scstrongly irreducible in $(M,T)\setminus \mathcal{H}^$ ;

(ii) $\mathcal{H}^$ is cessential in (M,T);

(iii) If (M,T) contains a cessential sphere with 3 or fewer punctures, then so does $\mathcal{H}^$ ;

(iv) If (M,T) is irreducible and if a component of $\mathcal{H}^+$ is a sphere with three or fewer punctures then $\mathcal{H} = \mathcal{H}^+$ ;

(v) If $(C, T_C) \sqsubset (M,T)\setminus \mathcal{H}$ is a trivial product compressionbody, then $\partial_ C \subset \partial M$ .
Proof. We note that in [ Reference Taylor and Tomova36 ] saying that T is irreducible, by definition, means that (M,T) does not contain a oncepunctured sphere. The existence of $\mathcal{H}$ given $\mathcal{J}$ is [ Reference Taylor and Tomova36 , theorem 6·17]. Conclusions (i), (ii), and (v) can be found as conclusions (i), (iv) and (iii) of [ Reference Taylor and Tomova36 , theorem 7·6], respectively. Conclusion (iii) follows from [ Reference Taylor and Tomova36 , theorem 8·2]. Conclusion (iv) is similar to the proof of conclusion (v) of [ Reference Taylor and Tomova36 , theorem 7·6]. The details are similar to some of the arguments that follow, so we omit them here.
Examining the proof of conclusion (iii) in Theorem 4·7 provides us with more information about orbifolds. In particular, it produces another proof of the existence of systems of summing spheres. By Theorem 2·4, this also implies that efficient systems of summing spheres exist. The proof is nearly identical to that of [ Reference Taylor and Tomova36 , theorem 8·2] and [ Reference Taylor and Tomova37 , proposition 5·1]. Since the argument is completely topological, it is also the case that if $\mathcal{H}$ is locally thin, then $\mathcal{H}^$ contains a collection of turnovers cutting (M,T) into suborbifolds that contain no essential turnovers. For convenience, we provide the proof in the Appendix.
Corollary 4·8. Suppose that (M,T) is a nice orbifold and let $\mathcal{H} \in \mathbb H(M,T)$ be locally thin. Then $\mathcal{H}^$ contains a system of summing spheres for (M,T).
Theorem 4·9. Suppose that (M,T) is a nice orbifold. Let $S \subset (M,T)$ be an efficient system of summing spheres. Then
If each component of S is separating, then equality holds.
Proof. We first show that $\operatorname{net}x_\omega(M,T) \geq \operatorname{net}x_\omega((M,T)_S)  x_\omega(S)$ . Choose $\mathcal{J} \in \mathbb H(M,T)$ such that $\operatorname{net}x_\omega(\mathcal{J}) = \operatorname{net}x_\omega(M,T)$ . This is possible by Proposition 5·3 below. By Theorem 4·7 and Lemma 4·6, there exists a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\mathcal{J} \to \mathcal{H}$ and $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . By Corollary 4·8, there exists an efficient set of summing spheres $S \subset \mathcal{H}^$ . As we remarked, S is unique up to orbifold homeomorphism, as is $(M,T)_S$ . When we split (M,T) open along S, each component of S is converted to two boundary components of $(M,T) \setminus S$ . Boundary components are not included in the sum in the definition of $\operatorname{net}x_\omega$ and capping them off with trivial ball compressionbodies does not change that. The result follows.
When each component of S is separating, the proof that $\operatorname{net}x_\omega(M,T) \leq \operatorname{net}x_\omega((M,T)_S)  x_\omega(S)$ is nearly identical to that of [ Reference Taylor and Tomova37 , theorem 5·5]. In each component of $(M,T)_S$ mark the points where sums will be performed. Since each component of S is separating, the dual graph to S is a tree. In each component $(M_i, T_i)$ of $(M,T)_S$ choose $\mathcal{H}_i \in \mathbb H(M_i, T_i)$ such that $\operatorname{net}x_\omega(\mathcal{H}_i) = \operatorname{net}x_\omega(M_i, T_i)$ . Again this is possible by Proposition 5·3 below. By transversality we may also assume each $\mathcal{H}_i$ is disjoint from the marked points. As in the proof of [ Reference Taylor and Tomova37 , theorem 5·5], we may reverse orientations on the $\mathcal{H}_i$ as necessary to ensure that their union with S is an oriented multiple orbifold Heegaard surface for (M,T). The desired inequality follows.
Theorem 4·10. There exists an invariant system of summing spheres $S \subset W$ such that
and $W_S$ is irreducible. If each component of S is separating, then equality holds.
Proof. If W is not orbifoldreducible, then the theorem is vacuously true. Otherwise, it is orbifoldreducible. By the Equivariant Sphere Theorem, the quotient orbifold (M,T) is orbifoldreducible. We verify that (M,T) is nice. Note that as T is the singular set, no edge has infinite weight. Let $P \sqsubset \partial M$ be a 2sphere. If $x_\omega(P) \lt 0$ , then its preimage in $\partial W$ is the union of essential spheres, but there are none. Thus, $x_\omega(P) \geq 0$ . Since (M,T) is covered by a manifold, there are no bad 2suborbifolds. Consequently, (M,T) is nice.
By Lemma 4·4, $\operatorname{net}x_\omega(W;G) = G\operatorname{net}x_\omega(M,T)$ . By Theorem 4·9, there is an efficient system of summing spheres $\overline{S}$ for (M,T) such that $\operatorname{net}x_\omega(M,T) \geq \operatorname{net}x_\omega((M,T)_{\overline{S}})$ and equality holds if every component of $\overline{S}$ is separating. Let S be the lift of $\overline{S}$ to W. If a component of $\overline{S}$ is nonseparating, each component of its preimage in W would be nonseparating.
We have:
If each component of S is separating, then so is each component of $\overline{S}$ and equality holds. Let W $^{\prime}$ be a component of $W_S$ . Its image in (M,T) is a component of $(M,T)_S$ . If W $^{\prime}$ were orbifoldreducible, then its image would be also, by the Equivariant Sphere Theorem. But this contradicts the fact that $\overline{S}$ is an efficient system of summing spheres.
Suppose that some $S_0 \sqsubset S$ is inessential. Then it bounds a 3ball $B \subset W$ . Without loss of generality, we may assume that $S_0$ is innermost; i.e. the interior of B is disjoint from S. The image of B in (M,T) is then the quotient of B by its stabiliser. By [ Reference Zimmermann45 ], it is a trivial ball compressionbody and its boundary is inessential. This contradicts the fact that $\overline{S}$ is efficient. Thus, each component of S is essential.
Remark 4·11. The proof of Theorem 4·9 demonstrates the advantage that invariant multiple vpbridge surfaces have over invariant Heegaard surfaces. Although there is no guarantee that if W is reducible there is an equivariant sphere intersecting a minimal equivariant Heegaard splitting in a single closed loop, we can guarantee that there is an equivariant sphere showing up as a thin surface in an equivariant generalised Heegaard splitting of W.
5. Lower Bounds
The main purpose of this section is to find lower bounds on $\operatorname{net}x_\omega(M,T)$ for an orbifold (M,T) and use that to prove Theorems 5·5 and 5·10. Along the way, we prove Proposition 5·3 which guarantees that $\operatorname{net}x_\omega(M,T)$ is welldefined and that there exists $\mathcal{H} \in \mathbb H(M,T)$ with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ .
5·1. Analysing Orbifold Compressionbodies
Definition 5·1. A lens space is a closed 3manifold of Heegaard genus 1, other than $S^3$ or $S^1 \times S^2$ . A core loop in a solid torus $D^2 \times S^1$ is a curve isotopic to $\{\text{point}\} \times S^1$ . A core loop in a lens space or $S^1 \times S^2$ is a knot isotopic to the core loop of one half of a genus 1 Heegaard splitting. A Hopf link in $S^3$ , $S^1 \times S^2$ , or a lens space is a 2component link such that there is a Heegaard torus separating the components and so that each component is a core loop for the solid tori on opposite sides of a Heegaard torus. A pillow $(C, T_C)$ is a vpcompressionbody with boundary a 4punctured sphere that is the result either of joining two (3ball, arc) trivial ball compressionbodies by an unweighted 1handle or joining two (3ball, trivalent graph) compressionbodies by a weighted 1handle. (See Figure 2.) An orbifold that is a pillow or trivial ball compressionbody is Euclidean if the boundary surface is. A Euclidean double pillow is a pair $(S^3, T)$ with an orbifold bridge surface H such that each of $(S^3, T)\setminus H$ is a Euclidean pillow. (The terminology stems from the fact that Euclidean orbifolds admit a complete metric locally modeled on Euclidean 2 or 3space.)
If $(C, T_C)$ is the disjoint union of orbifold compressionbodies, let $N(C,T_C) = x_\omega(\partial_+ C)  x_\omega(\partial_ C)$ . Our key identity for an multiple orbifold Heegaard surface $\mathcal{H} \in \mathbb H(M,T)$ is:
where we sum over the vpcompressionbodies $(C, T_C) \sqsubset (M,T) \setminus \mathcal{H}$ . This follows immediately from the fact that each component of $\mathcal{H}^+$ and each component of $\mathcal{H}^$ appears exactly twice as a boundary component of $(M,T)\setminus \mathcal{H}$ .
Lemma 5·2. Suppose that $(C, T_C)$ is an orbifold compressionbody with no component of $\partial_ C$ a oncepunctured sphere. If $N(C, T_C) \lt 0$ , then $(C, T_C)$ is a trivial ball compressionbody. Also, if $N(C, T_C) = 0$ and $\partial_ C = \varnothing$ , then $(C, T_C)$ is one of:

(i) Euclidean trivial ball compressionbody;

(ii) Euclidean pillow;

(iii) (solid torus, $\varnothing$ );

(iv) (solid torus, core loop).
Proof. Let $\Delta$ be a complete collection of scdiscs for $(C, T_C)$ such that $\partial$ reducing $(C, T_C)$ along $\Delta$ results in trivial vpcompressionbodies $(C', T'_C)$ . Each disc of $\Delta$ , leaves 2 “scars” on $\partial_+ C'$ . If E is a scar, let $\omega(E) = 1$ if it is unpunctured and otherwise let $\omega(E)$ be the weight of the puncture. Let $(C_0, T_0) \sqsubset (C', T'_C)$ . Let $N'(C_0, T_0)$ be equal to the sum of $N(C_0, T_0)$ with ${1}/{\omega(E)}$ for all scars E on $\partial_+ C_0$ . Observe that
where the sum is taken over all $(C_0, T_0) \sqsubset (C', T')$ .
If $(C_0, T_0)$ is a product compressionbody then $N'(C_0, T_0) \geq N(C_0, T_0) = 0$ with equality if and only if every scar on $\partial_+ C_0$ has weight $\infty$ . Suppose that $(C_0, T_0)$ is a trivial ball compressionbody.
Case 1: $T_0 = \varnothing$ .
If $\Delta = \varnothing$ , then $(C, T_C) = (C_0, T_0)$ and $N(C, T_C) = 2$ . Otherwise, by the choice of $\Delta$ , $\partial_+ C_0$ contains at least 2 scars, each of weight 1. If it contains exactly 2, then $(C, T_C)$ is (solid torus, $\varnothing$ ). If it has at least 3 scars, then $N'(C_0, T_0) \geq 1$ .
Case 2: $T_0$ is an arc of weight k.
If $\Delta = \varnothing$ , then $(C, T_C) = (C_0, T_0)$ and $N(C, T_C) = {2}/{k} \geq 1$ . If $N(C, T_C) = 0$ , then $k = \infty$ . If $\Delta \neq \varnothing$ , then $\partial_+ C_0$ contains at least one scar. By our choice of $\Delta$ , either $(C, T_C)$ is (solid torus, core loop) or $\partial_+ C_0$ contains at least 1 scar of weight 1. In which case, $N'(C_0, T_0) \geq 0$ . Equality holds if and only if $k = 2$ , there is exactly one scar and it has weight 1.
Case 3: $T_0$ contains an interior vertex.
Note that $N(C_0, T_0) = x_\omega(\partial_+ C) = x_\omega(v) \lt 0$ where v is the internal vertex of $T_0$ . If $\Delta = \varnothing$ , then we have our result. If $\Delta \neq \varnothing$ , then $\partial_+ C_0$ contains at least one scar and it either has weight 1 or has weight equal to the weight of one of the punctures on $\partial_+ C$ . In which case, $N'(C_0, T_0) \geq 0$ . Equality holds only when there is exactly one scar, it contains a puncture, and the two punctures not contained in the scar both have weight 2.
This concludes our analysis of the individual cases and, in particular, we may assume that $\Delta \neq \varnothing$ and that $(C, T_C)$ is neither (solid torus, $\varnothing$ ) or (solid torus, core loop). By our analysis, each component $(C_0, T_0) \sqsubset (C', T'_C)$ has $N'(C_0, T_0) \geq 0$ . Thus, $N(C, T_C) \geq 0$ . Suppose that $N(C, T_C) = 0$ . Then $N'(C_0, T_0) = 0$ for each component of $(C', T'_C)$ . Consequently, each component is one of:

(i) a product compressionbody such that every scar has infinite weight;

(ii) a trivial ball compressionbody containing an arc and with a single scar of weight 1;

(iii) a trivial ball compressionbody containing a trivalent vertex and with edges of weight (2,2,k) with $k \geq 2$ . It has a single scar of weight k.
The compressionbody $(C, T_C)$ can be reconstructed by attaching possibly weighted 1handles to the scars on (C $^{\prime}$ , T $^{\prime}$ ). Thus our result holds if $N(C, T_C) \leq 0$ .
We next have two propositions whose proofs are closely related.
Proposition 5·3. Suppose that (M,T) is a nice orbifold and that $\mathcal{H} \in \mathbb H(M,T)$ . Then either $\operatorname{net}x_\omega(\mathcal{H}) \geq ({1}/{2})x_\omega(\partial M)$ or (M,T) is one of $\mathbb{S}(0)$ , $\mathbb{S}(2)$ or $\mathbb{S}(3)$ . Furthermore, there exists a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . If (M,T) is an $\mathbb{S}(i)$ , for $i \in \{0,2,3\}$ , then any locally thin $\mathcal{H}$ is an ipunctured sphere.
Proposition 5·4. Suppose that (M,T) is a nice, closed orbifold that is orbifoldirreducible and that $\mathcal{H} \in \mathbb H(M,T)$ is locally thin. Then either $\operatorname{net}x_\omega(\mathcal{H}) \geq 1/6$ or one of the following exceptional cases holds:

(i) $\mathbb{S}(0)$ , $\mathbb{S}(2)$ , or $\mathbb{S}(3)$ ;

(ii) $M = S^3$ or a lens space and T is a core loop, Hopf link and $\mathcal{H}$ is an unpunctured torus; or

(iii) (M,T) is a Euclidean double pillow and $\mathcal{H}$ is a fourpunctured sphere.
The remainder of the section is devoted to the proofs of these propositions. A key bookkeeping device for a vpcompressionbody $(D, T_D)$ is its ghost arc graph. This is the graph $\Gamma$ whose vertices are the components of $\partial_ D$ and the vertices of $T_D$ . The ghost arcs of $T_D$ are the edges. For example, if $(D, T_D)$ has a single ghost arc and it joins distinct components of $\partial_ D$ , then $\Gamma$ is a single edge. The key observation is that if $\partial_+ D$ is a sphere, then $\Gamma$ is acyclic and if $\partial_+ D$ is a torus, then $\Gamma$ contains at most one cycle. If it contains a cycle, then $\partial_ D$ is the union of spheres (that is, it does not contain a torus).
Begin by assuming only that (M,T) is a nice orbifold. Let $\mathcal{H} \in \mathbb H(M,T)$ be locally thin. Recall from Corollary 4·8 that $\mathcal{H}^$ contains an efficient system of summing spheres S. Assume, for the time being, that $S = \varnothing$ ; equivalently, that (M,T) is orbifoldirreducible. Since each component of $\mathcal{H}^$ is cessential in (M,T), this also implies that no $S_0 \sqsubset \mathcal{H}^$ is a sphere with $S_0 \cap T \leq 3$ and $x_\omega(S_0) \lt 0$ .
Case 1: Some $(C, T_C) \sqsubset (M,T) \setminus \mathcal{H}$ has $N(C, T_C) \lt 0$ .
By Lemma 5·2, $(C, T_C)$ is a trivial ball compressionbody. Observe that $\partial_+ C$ is a sphere with 0, 2 or 3 punctures. Let $(D, T_D) \sqsubset (M,T)\setminus \mathcal{H}$ be the other vpcompresionbody having $\partial_+ D = \partial_+ C$ . If $\partial_ D = \varnothing$ , then $M = C \cup D$ and $(D, T_D)$ is also a trivial ball compressionbody. In this case, (M,T) is either $\mathbb{S}(0)$ , $\mathbb{S}(2)$ or $\mathbb{S}(3)$ . Assume there exists $F \sqsubset \partial_ D$ .
Let $\Gamma$ be the ghost arc graph for $(D, T_D)$ . As $\partial_+ D$ is a sphere, $\Gamma$ is acyclic and the components of $\partial_ D$ are all spheres. Since (M,T) is nice, none of them are oncepunctured. Since $S = \varnothing$ , F is at least thricepunctured and has $x_\omega(F) \geq 0$ . If $\Gamma$ contains an isolated vertex, $(D, T_D)$ is a product. Since $\mathcal{H}$ is locally thin, $F = \partial_ D \subset \partial M$ . If $T_C$ contains an interior vertex v, we must have $0 \gt x_\omega(v) = x_\omega(F) \geq 0$ , a contradiction. If $T_C$ does not contain an interior vertex, then F is twicepunctured, contradicting our definition of nice 3orbifold. Thus, we may assume that $\Gamma$ does not have an isolated vertex. Since no component of $\partial_ D$ is a twicepunctured sphere, each leaf of $\Gamma$ is incident to at least two vertical arcs, so there is at most one leaf. Since $\Gamma$ is acyclic, this is a contradiction. Consequently, (M,T) is one of the exceptional cases in the statement of Proposition 5·3.
Case 2: Every $(C, T_C) \sqsubset (M,T) \setminus \mathcal{H}$ has $N(C, T_C) \geq 0$ .
By (i), we see that $\operatorname{net}x_\omega(\mathcal{H}) \geq x_\omega(\partial M)/2$ . Let L be the product of all the finite weights on T. Note that for any $\mathcal{J} \in \mathbb H(M,T)$ , the quantity $2L\operatorname{net}x_\omega(\mathcal{J})$ is an integer, as is $2Lx_\omega(\partial M)/2$ . By Theorem 4·7, for any $\mathcal{J} \in \mathbb H(M,T)$ , there exists a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\mathcal{J} \to \mathcal{H}$ . By Lemma 4·6, $\operatorname{net}x_\omega(\mathcal{J}) \geq \operatorname{net}x_\omega(\mathcal{H})$ . If (M,T) is one of the exceptional cases from Proposition 5·3, then by the analysis in Case 1, $\mathcal{H}$ is connected and so $L\operatorname{net}x_\omega(\mathcal{J})$ is bounded below by a constant depending only on (M,T). If (M,T) is not one of the exceptional cases from Proposition 5·3, then we see that $2L\operatorname{net}x_\omega(\mathcal{J}) \geq 2Lx_\omega(\partial M)/2 \geq 0$ . Thus, in either case, since the invariant $2L\operatorname{net}x_\omega$ defined on $\mathbb H(M,T)$ is integervalued and bounded below by a number depending only on it achieves its minimum on a locally thin element of $\mathbb H(M,T)$ . That element also minimizes $\operatorname{net}x_\omega$ . This concludes the analysis when $S = \varnothing$ for the proof of Proposition 5·3.
Now suppose that $S \neq \varnothing$ . Expand S to include all summing spheres in $\mathcal{H}^$ ; continue to call it S. As we have observed previously,
Let $(M_0, T_0) \sqsubset (M,T)_S$ and let $\mathcal{H}_0 = (\mathcal{H}\setminus S) \cap M_0$ . If $\operatorname{net}x_\omega(\mathcal{H}_0) \lt 0$ , then $(M_0, T_0)$ is one of the exceptional cases from Proposition 5·3. Since $\mathcal{H}$ is locally thin, each component of S is essential, so $(M_0, T_0) \neq \mathbb{S}(0)$ . If $(M_0, T_0) = \mathbb{S}(2)$ , then at least one of the components S $^{\prime}$ of S used to sum with $(M_0, T_0)$ must be unpunctured. Thus, if $T_0$ has weight k, we have:
Similarly, if $(M_0, T_0)$ is an $\mathbb{S}(3)$ , then at least one of the components $S' \sqsubset S$ used to sum with $(M_0, T_0)$ must be either unpunctured or twice punctured and with the weight of the punctures equal to the weight c of one of the edges of $T_0$ . In that case, letting a,b be the weights of the other punctures,
Consequently, $\operatorname{net}x_\omega(\mathcal{H}) \geq x_\omega(\partial M)/2$ , even in this situation. As before, the quantity $L\operatorname{net}x_\omega$ is an integervalued invariant on $\mathbb H(M,T)$ bounded below by a constant depending only on (M,T) and so, as before, there is a locally thin $\mathcal{H} \in \mathbb H(M,T)$ with $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . This concludes the proof of Proposition 5·3.
Henceforth, suppose that (M,T) is closed and orbifoldirreducible and not one of the exceptional cases from Proposition 5·3. By our previous remarks, this implies that $N(C, T_C) \geq 0$ for every $(C, T_C) \sqsubset (M,T)\setminus \mathcal{H}$ . The dual digraph to $\mathcal{H}$ is acyclic, so it has at least one source and one sink. The sources and sinks are exactly those $(C, T_C) \sqsubset (M,T)\setminus \mathcal{H}$ with $\partial_ C = \varnothing$ . Suppose that $(C, T_C)$ is one such. Note that $N(C, T_C) = x_\omega(\partial_+ C)$ . Let $(D, T_D) \sqsubset (M,T)\setminus \mathcal{H}$ be the other orbifold compressionbody with $\partial_+ D = \partial_+ C = H$ .
Observe that $x_\omega(H) \geq \chi(H) + H \cap T/2$ . Equality holds only if every puncture on H has weight 2. Consequently, if $1/6 \gt x_\omega(H)$ , then H is a sphere with $H \cap T \leq 4$ . If $H \cap T \leq 3$ , then by our analysis above (M,T) is one of the exceptional cases from Proposition 5·3. Consider, therefore, the case that $H \cap T = 4$ . If at least one puncture does not have weight 2, then $x_\omega(H) \geq 1/6$ , so assume that each puncture has weight 2. If $\partial_ D = \varnothing$ , then $\mathcal{H}$ divides (M,T) into two Euclidean pillows. Suppose $\partial_ D \neq \varnothing$ and let $\Gamma$ be the ghost arc graph for $(D, T_D)$ as above. It is acyclic. Each component of $\partial_ D$ is a sphere with at least three punctures, since (M,T) is orbifoldirreducible and nice. Also $\partial_ D \subset \partial M$ since M is closed. An isolated vertex of $\Gamma$ is a sphere incident to at least three vertical arcs and a leaf is a sphere incident to at least two vertical arcs. Since H has four punctures, if $\Gamma$ has an isolated vertex, that vertex is the entirety of $\Gamma$ and it is incident to four vertical arcs. This implies $(D, T_D)$ is a product and contradicts local thinness of $\mathcal{H}$ . Thus, $\Gamma$ has two leaves, each incident to two vertical arcs. At least one of those leaves F is a component of $\partial_ D$ (the other may be a vertex of T). Since (M,T) is orbifoldirreducible, $x_\omega(F) \geq 0$ . Consequently, at least two of the arcs incident to F have weight at least 3. At least one of those is a vertical arc, contradicting the fact that each puncture of H has weight 2. Consequently, $N(C, T_C) = x_\omega(H) \geq 1/6$ .
Since the dual digraph to $\mathcal{H}$ has at least one source and one sink either (M,T) is one of the exceptional cases of Proposition 5·3, or $\mathcal{H}$ is a four punctured sphere dividing (M,T) into two Euclidean pillows, or $2\operatorname{net}x_\omega(\mathcal{H}) \geq 2\cdot (1/6)$ . This concludes the proof of Proposition 5·4.
Figure 3 shows that our bound of 1/6 is asymptotically sharp.
5·2. Equivariant Heegaard Genus and the Order of the Group
Theorem 5·5. Suppose that W is closed and connected and that G does not act freely. If $S \subset W$ is an invariant system of summing spheres, then
where $\nu$ is the number of orbits of the components of $(W_S)$ that are not $S^3$ or lens spaces and $\mu$ is the number of orbits of components of $W_S$ that are homeomorphic to $S^3$ .
Proof. Let (M,T) be the quotient orbifold and note that no edge of T has infinite weight. Let $\overline{S}$ be the image of S and note that each component of $(M,T)_{\overline{S}}$ is orbifoldirreducible by the definition of “system of summing spheres” and Theorem 2·5. By Theorem 2·4, $\overline{S}$ contains an efficient subset. Let $S_0 \subset S$ be the preimage of a component of $\overline{S}$ that is not in our chosen efficient subset. Since W is closed, each component of $W_S$ that is not a component of $W_{S_0}$ is a homeomorphic to $S^3$ . Thus, passing from S to $S\setminus S_0$ increases the righthand side of our inequality by $G/2  1$ . Since G does not act freely, it is not the trivial group. We conclude that it is enough to prove our result when $\overline{S}$ is efficient. Furthermore, by Theorem 2·4, we may prove it when $\overline{S}$ is any efficient system of summing spheres for (M,T), not merely the given one.
Recall that $x_\omega(W;G) \geq G\operatorname{net}x_\omega(M,T)$ . By Theorem 4·7 and Proposition 5·3, there is a locally thin $\mathcal{H} \in \mathbb H(M,T)$ such that $\operatorname{net}x_\omega(\mathcal{H}) = \operatorname{net}x_\omega(M,T)$ . Furthermore, $\mathcal{H}^$ contains an efficient set of summing spheres (Corollary 4·8). Call them $\overline{S}$ . Let S be the preimage of $\overline{S}$ in W. By Theorem 4·9, we have
Suppose some $(M_i, T_i)\sqsubset (M,T)_{\overline{S}}$ is an $\mathbb{S}(0)$ . Since $\overline{S}$ is efficient, the matching scar to every scar in $(M_i, T_i)$ is also in $(M_i, T_i)$ . It follows that $(M,T)_{\overline{S}} = (M_i, T_i)$ . In which case, $T = \varnothing$ , and G acts freely, contrary to hypothesis. Henceforth, we may assume no $(M_i, T_i) \sqsubset (M,T)_S$ is an $\mathbb{S}(0)$ . Suppose that $(M_i, T_i)$ is an $\mathbb{S}(2)$ . Then $x_\omega(M_i, T_i) \geq 1$ and each component of the preimage of $(M_i, T_i)$ in $W_S$ , is a copy of $S^3$ . Similarly, if $(M_i, T_i)$ is an $\mathbb{S}(3)$ , then $x_\omega(M_i, T_i) \geq 1/2$ and each component of the preimage of