Strong Haken via Sphere Complexes
SStrong Haken via Sphere Complexes
Sebastian Hensel and Jennifer SchultensFebruary 22, 2021
Abstract
We give a short proof of Scharlemann’s Strong Haken Theorem for closed 3-manifolds(and manifolds with spherical boundary). As an application, we also show that givena decomposing sphere R for a 3-manifold M that splits off an S × S summand, anyHeegaard splitting of M restricts to the standard Heegaard splitting on the summand. Any (closed, oriented, connected) three-dimensional manifold M admits a Heegaard split-ting , that is, it can be decomposed into two three-dimensional handlebodies
V, V (cid:48) of thesame genus g along an embedded surface S ⊂ M : M = V ∪ S V (cid:48) . In theory, all information about the three-manifold is encoded in the identification of thetwo handlebodies. However, in practice, interpreting topological properties of M using aHeegaard splitting is often nontrivial.A basic example of this occurs when studying spheres in M . If α ⊂ S is a curve whichbounds disks D, D (cid:48) in both V and V (cid:48) , then gluing these disks yields a 2–sphere D ∪ D (cid:48) ⊂ M which intersects S in the single curve α . When essential, such a sphere is called a Hakensphere – but a priori it is completely unclear what kind of spheres in M are of this form.A classical theorem of Haken shows that if M admits any essential sphere σ , then it alsoadmits a Haken sphere σ (cid:48) . In fact, Scharlemann recently proved a Strong Haken Theorem ,showing that σ (cid:48) can in fact be chosen to be isotopic to σ : Theorem 1.1. (Strong Haken Theorem) Let M = V ∪ S V (cid:48) be a Heegaard splitting. Everyessential -sphere in M is isotopic to a Haken sphere for M = V ∪ S V (cid:48) . The purpose of this article is to give an independent, short proof of Theorem 1.1 forany M which is closed or has spherical boundary. We want to mention that Scharlemann’sversion of the strong Haken theorem is in fact more general, allowing for manifolds witharbitrary boundary, and also showing that any properly embedded disk is isotopic to aHaken disk. This more general case could also be obtained from our methods; for claritywe focus on the closed case throughout the article.To prove Theorem 1.1, we make crucial use of the (surviving) sphere complex , which is acombinatorial complex encoding the intersection pattern of essential (surviving) spheres in M . Such complexes have already been used successfully in the study of outer automorphism1 a r X i v : . [ m a t h . G T ] F e b roups of free groups (via mapping class groups of connected sums of S × S ). Here, weshow that this perspective can also be useful in streamlining arguments in low-dimensionaltopology. The other crucial ingredient is the classical Waldhausen theorem on Heegaardsplittings of the three-sphere. Together, these allow an inductive approach to Theorem 1.1.Our methods and results also allow to control Heegaard splittings of reducible manifolds.As a motivating example, we prove: Proposition 1.2.
Every Heegaard splitting of W n = n ( S × S ) is isotopic to a stabilizationof the standard Heegaard splitting. Combined the uniqueness for W with the Strong Haken Theorem, we obtain the fol-lowing structural result on Heegaard splittings of arbitrary reducible three-manifolds. Corollary 1.3.
Given a reducible -manifold M with a Heegaard splitting M = V ∪ S V (cid:48) ,any decomposing sphere that splits off a S × S summand can be isotoped so that S isstandard in this summand. Acknowledgements
We would like to thank Martin Scharlemann for finding a mistakein an earlier draft, and many helpful comments. –Manifolds In this section we recall some preliminaries on closed three-manifolds, their Heegaard split-tings, and spheres in such manifolds. The results presented here are classical.
Definition 2.1. (Heegaard splitting) A handlebody is a -manifold that is homeomorphicto a regular neighborhood of a graph in S . A Heegaard splitting of a -manifold M is adecomposition M = V ∪ S V (cid:48) , where V, V (cid:48) are handlebodies and S = ∂V = ∂V (cid:48) = V ∩ V (cid:48) .The surface S is called the splitting surface or Heegaard surface . Heegaard splittingsare considered equivalent if their splitting surfaces are isotopic.
Remark 2.2.
The Heegaard splitting M = V ∪ S V (cid:48) is completely specified by the pair ( M, S ) , so we will sometimes write ( M, S ) instead of M = V ∪ S V (cid:48) . Remark 2.3.
The connected sum of two -manifolds M , M with Heegaard splittings ( M , S ) , ( M , S ) inherits a Heegaard splitting ( M M , S S ) . This Heegaard split-ting is unique in the sense that it is completely determined by the construction. Later, wewill briefly consider a refined notion of equivalence for Heegaard splittings where we distin-guish between the two sides of the splitting surface. With respect to this refined notion ofequivalence, the Heegaard splitting of a connected sum is then not (a priori) unique, as theconstruction allows two different choices, namely which side of S is identified to which sideof S . Definition 2.4.
Given a Heegaard splitting ( M, S ) , the Heegaard splitting obtained fromthe pairwise connected sum ( M, S ) S , T ) , where T is the standard unknotted torus in S ,is called a stabilization of ( M, S ) . A Heegaard splitting is stabilized if it is the stabilizationof another Heegaard splitting and unstabilized otherwise. sphere that separates ( M, S ) S , T ) , i.e. a sphere that splits off a punctured -ballcontaining an unknotted punctured torus, is called a stabilizing sphere.A stabilizing pair of disks is a pair ( D, D (cid:48) ) of disks such that D is properly embeddedin V , D (cid:48) is properly embedded in V (cid:48) and ∂D ∩ ∂D (cid:48) is exactly one point. Remark 2.5.
A Heegaard splitting is stabilized if and only if it admits a stabilizing pairof disks. Indeed, consider the standard unknotted torus in the -sphere and observe that itseparates S into two solid tori. The boundaries of the meridian disks of these solid toriintersect in exactly one point. A crucial theorem of Waldhausen’s characterises all Heegaard splittings of the three-sphere. See [19].
Theorem 2.6 (Waldhausen’s Theorem) . Every Heegaard splitting of the three-sphere is astabilization of the unique standard genus Heegaard splitting.
A core tool in our argument is the following simplicial complex, which encodes the inter-section patterns of spheres in M . Definition 2.7 (Sphere complex) . A sphere S in a -manifold is compressible if it boundsa -ball. Otherwise, it is incompressible . We say that a sphere is peripheral if it is isotopicinto the boundary of the manifold.The sphere complex of a -manifold M is the simplicial complex S ( M ) determined bythe following three conditions:1. Vertices of S ( M ) correspond to isotopy classes of incompressible, non-peripheral em-bedded -spheres;2. Edges of S ( M ) correspond to pairs of vertices with disjoint representatives;3. The complex S ( M ) is flag. It is not hard to see that a simplex in the sphere complex corresponds to a collection ofnonisotopic spheres that can be realised disjointly.Furthermore, a standard argument involving surgery at innermost intersection circlesshows that the sphere complex of any closed 3–manifold is connected (if it is nonempty).See e.g. [6] for a proof in the case of doubled handlebodies, which also works in general.
Our central aim will be to understand how essential spheres in M interact with Heegaardsplittings of M . The following notion is crucial. Definition 2.8.
Let M = V ∪ S V (cid:48) be a Heegaard splitting. An essential sphere in M thatmeets the Heegaard surface S in a single simple closed curve is called a Haken sphere . A(not necessarily essential) sphere that intersects S in a single simple closed curve essentialin S is called a reducing sphere . A -simplex in the sphere complex of W , the double of a genus handlebody(alternatively, the connected sum of copies of S × S ). The following theorem was originally proved by Haken in [5]. Proofs can be found inthe standard references on 3-manifolds, see [7], [8], [18].
Theorem 2.9 (Haken’s Lemma) . If a closed three-manifold M contains an essential sphereand M = V ∪ S V (cid:48) is a Heegaard splitting, then M admits a Haken sphere. In general, the Haken sphere is obtained by modifying the given essential sphere bysurgery, and so cannot be guaranteed to be related to the sphere given at the outset.
In this section, we present versions of the results and notions of the previous section for3-manifolds with spherical boundary. These appear naturally in our inductive proof of theStrong Haken Theorem (even if one is just interested in proving it in the closed case). Forease of notation, if M is a 3-manifold with spherical boundary, then we call each componentof ∂M a puncture . Similarly, we call such a manifold a punctured manifold . To define Heegaard splittings of punctured manifolds, we use spotted handlebodies.
Definition 3.1. A spotted handlebody is a handlebody with a specified set of disks D (cid:116)· · ·(cid:116) D k in its boundary. Each disk is called a spot . A Heegaard splitting of a -manifold M withspherical boundary is a decomposition M = V ∪ S V (cid:48) where V, V (cid:48) are spotted handlebodieswith spots D (cid:116) · · · (cid:116) D k and D (cid:48) (cid:116) · · · (cid:116) D (cid:48) k , respectively, and S = ∂V − ( D (cid:116) · · · (cid:116) D k ) = ∂V (cid:48) − ( D (cid:48) (cid:116) · · · (cid:116) D (cid:48) k ) . Remark 3.2.
In a Heegaard splitting of a -manifold with spherical boundary each puncturemeets the splitting surface in a single simple closed curve. This simple closed curve is theboundary of a spot on each of the handlebodies.Suppose M , M are two punctured manifolds, with boundary components ∂ i ⊂ M i , and M = M ∪ ∂ = ∂ M is the manifold obtained by gluing the boundary components. Given Hee-gaard splittings of M , M , the manifold M inherits a Heegaard splitting which is obtainedby gluing the handlebodies at the corresponding spots. he glued boundary components yield an essential –sphere σ in M , which intersects theinduced Heegaard splitting in a single circle (i.e. it becomes a Haken sphere). Conversely,given a Haken sphere σ for any manifold M , one can cut the manifold and the splitting at σ . We need a version of Waldhausen’s Theorem in the context of punctured 3-spheres(which is a fairly straightforward consequence of Waldhausen’s theorem for S ). Theorem 3.3 (Waldhausen’s Theorem for punctured 3-spheres) . Every Heegaard splittingof a punctured -sphere is a stabilization of a unique standard genus Heegaard splitting.Proof.
Let M be a punctured 3-sphere and M = V ∪ S V (cid:48) a Heegaard splitting. Constructˆ M = S from M by attaching a 3-ball to each puncture. By Alexander’s Theorem, theresult does not depend on the attaching map. Moreover, the attaching maps can be chosenso that a meridional disk of each 3-ball caps off a component of ∂S . We thus obtain a closedsurface ˆ S that defines a Heegaard splitting S = ˆ V ∪ ˆ S ˆ V (cid:48) .Each 3-ball that has been attached to a puncture is a regular neighborhood of a pointand, as such, arbitrarily small. By Waldhausen’s Theorem, S = ˆ V ∪ ˆ S ˆ V (cid:48) is a stabilizationof the standard genus 0 Heegaard splitting of S . The stabilizing pairs of disks can bechosen to be disjoint from the attached 3-balls. Thus, after destabilizing, if necessary, wemay assume that S is genus 0.Hence, to show the theorem, it suffices to show that any genus 0 splitting of a punctured S is standard. To this end, observe that the spotted genus 0 handlebody V ⊂ S can beisotoped to be a regular neighbourhood of a graph Γ ⊂ V . The graph Γ can be chosen tohave the following form: it has one vertex v in the interior of M , and one vertex on eachboundary component. Each vertex on a boundary component is joined to v by an edge.Now, observe that any two such graphs are isotopic, as any two arcs from v to a boundarysphere are isotopic, by the Lightbulb Trick. This shows that any two genus 0 splittings ofa punctured S are isotopic.At this point, we briefly want to address the ambiguity appearing in the previous proofwhen filling the boundary and drilling it out again – namely, one can isotope a pair of sta-bilizing disks across a puncture. This leads to a homeomorphism of the manifold preservingthe Heegaard surface. Given a Heegaard splitting of a 3-manifold, the Goeritz group ofthe splitting is the group of isotopy classes of orientation preserving diffeomorphisms of themanifold that preserve the splitting. Loosely speaking, the Goeritz group will be small ifthe surface automorphism that defines the Heegaard splitting is complicated relative to thehandlebodies. Conversely, the Goeritz group will be as large as possible in the case of W n ,the manifold for which this surface automorphism is the identity, and the Goeritz group isequal to the handlebody group. Scharlemann finds a system of 4 g + 1 generators for theGoeritz group of a handlebody (see [15]). On the other hand, the Goeritz group of thethree-sphere is still largely mysterious. We refer the interested reader to the recent [17] andthere references therein. We now want to define a useful sphere complex for punctured manifolds. One obviouschange is that for the vertices one should also exclude peripheral spheres, i.e. spheres5hich are homotopic into the boundary (otherwise, such spheres are adjacent to any othervertex, rendering the resulting complex useless). However, even with this modification, theresulting sphere complex will be somewhat problematic for our purposes, as it may oftenbe disconnected. Namely, suppose that M is an aspherical three-manifold with infinitefundamental group. Let M be the manifold obtained from M by removing two open balls.The manifold M admits many essential non peripheral spheres obtained by joining the twopunctures by a nontrivial tube. In fact, by asphericity of M , any essential non peripheralsphere in M is of this form. In particular, no two such are disjoint.To sidestep this issue, we use the following variant of the sphere complex. Definition 3.4 (Surviving Sphere complex) . A sphere S in a punctured -manifold is almost peripheral if it bounds a punctured S in M . If S is not almost peripheral, then itis surviving .The surviving sphere complex of a -manifold M is the simplicial complex S s ( M ) de-termined by the following three conditions:1. Vertices of S s ( M ) correspond to isotopy classes of incompressible, embedded, surviving -spheres;2. Edges of S s ( M ) correspond to pairs of vertices with disjoint representatives;3. The complex S s ( M ) is flag. The terminology stems from the fact that the spheres “survive filling in the punctures”and is in analogy to the surviving curve complex used in the study of mapping class groupsof surfaces, see e.g. , [1, 4]. It turns out that these complexes are much better behaved inour setting.
Lemma 3.5.
Let M be a –manifold. Then the surviving sphere complex S s ( M ) is con-nected (if it is nonempty).Proof. Let σ, σ (cid:48) be two incompressible, embedded, surviving 2-spheres in M . Up to isotopy,we may assume that σ, σ (cid:48) intersect transversely. Further, we may assume that up to isotopy,the number of intersection components σ ∩ σ (cid:48) is minimal.Let C ⊂ σ ∩ σ (cid:48) be an innermost intersection circle, i.e. suppose that it bounds a disk D ⊂ σ with D ∩ σ (cid:48) = ∂D = C . Denote by S + , S − ⊂ σ (cid:48) the two disks bounded by C , anddenote by σ ± = S ± ∪ D the two 2–spheres obtained by disk-swapping. Observe that up toisotopy, both of these are disjoint from σ (cid:48) , and intersect σ in at least one fewer circle than σ (cid:48) . If either σ + , σ − were compressible, then we could reduce the number of components in σ ∩ σ (cid:48) by sliding σ over the ball bounded by the compressible sphere, which is impossibleby our choice.Assume that σ − is almost peripheral. Then, after filling in the punctures of M , thespheres σ and σ + are isotopic (by sliding D over the now unpunctured ball bounded by σ − ). In particular, σ + is surviving, as the same is true for σ .Hence, at least one of σ ± is surviving, and we are done. Indeed, repeating this processproduces a sequence of spheres corresponding to vertices in a path, in S s ( M ), between [ σ ]and [ σ (cid:48) ]. 6igure 2: In the proof of Lemma 3.5: the innermost intersection circle of σ, σ (cid:48) cuts σ (cid:48) intotwo disks S + , S − . If σ − is almost peripheral, then σ + is isotopic to σ after filling thepunctures. . Just as in the closed case, we call an essential sphere which intersects a Heegaard splittingof a punctured manifold in a single curve a
Haken sphere . For punctured manifolds, almostperipheral and surviving Haken spheres behave slightly differently.On the one hand, using the same strategy as in the proof of Theorem 3.3, we obtainthe following corollary of Theorem 2.9.
Theorem 3.6 (Surviving Haken’s Lemma) . If M contains an surviving sphere and M = V ∪ S V (cid:48) is a Heegaard splitting, then there is a surviving Haken sphere.Proof. Denote by σ an essential surviving sphere in M . Let M (cid:48) be the three-manifoldobtained from M by gluing a ball to each boundary component. Denote by B ⊂ M (cid:48) thedisjoint union of the resulting balls. By definition of almost peripheral, the image of σ in M (cid:48) is still essential. Thus, Haken’s lemma (Theorem 2.9) applies, and yields a Haken sphere σ (cid:48) ⊂ M (cid:48) . By an isotopy preserving the Heegaard surface, we may assume that σ (cid:48) is disjointfrom B . We can thus interpret σ (cid:48) as a sphere in M ⊂ M (cid:48) , where it is the desired Hakensphere.On the other hand, almost peripheral spheres are also Haken spheres: Lemma 3.7 (Almost Peripheral Strong Haken Theorem) . Let M be a three-manifold withat least two punctures, and M = V ∪ S V (cid:48) be a Heegaard splitting. Then any almost peripheralsphere σ in M is isotopic to a Haken sphere.Proof. We begin with the case where σ cuts off exactly two punctures δ , δ . The almostperipheral sphere σ is then isotopic to the boundary of a regular neighbourhood of δ ∪ α ∪ δ ,where α ⊂ M is a properly embedded arc. We may homotope α to lie in S , as any arc in ahandlebody is homotopic into the boundary. However, the arc may now not be embeddedanymore. We can remove the self-intersections by “popping subarcs over δ ∩ S ”. To bemore precise, parametrise α : [0 , → S so that it starts on δ ∩ S , and homotope so that allself-intersections are transverse. Consider the first self-intersection point α ( t ) = α ( s ) , t < s .In particular, this implies that α | [0 ,t ] is an embedded arc.Now homotope a small subarc α | [ s − (cid:15),s + (cid:15) ] to instead be the arc obtained by following α | [0 ,t ] backwards to δ ∩ S , following around δ ∩ S , and returning along α | [0 ,t ] (compare7igure 3: Removing self-intersections of an arc joining two spots. .Figure 3). This homotopy is possible in V , and the resulting arc has at least one fewerself-intersection.After a finite number of modifications of this type, the boundary of a regular neigh-bourhood of δ ∪ α ∪ δ (which is homotopic to σ ) interects S in a single curve, and thus isa Haken sphere. By a theorem of Laudenbach, see [9], homotopy and isotopy are the samefor spheres in 3–manifolds, hence the claim follows .Now we suppose σ is a sphere cutting off k > σ (cid:48) ,disjoint from σ , which cuts off 2 punctures, and which is contained in the punctured S bounded by σ . By the initial case, σ (cid:48) may be assumed to be Haken. Let M (cid:48) be the manifoldobtained by cutting M at σ (cid:48) , with the induced Heegaard splitting; observe that σ ⊂ M (cid:48) isstill almost peripheral, but now cuts off at most k − σ is a Hakensphere.Since any essential, non peripheral sphere in a punctured S is almost peripheral, thisimplies the following: Corollary 3.8 (Strong Haken Theorem for punctured 3–spheres) . Any essential sphere ina punctured -spheres is isotopic to a Haken sphere. n ( S × S ) In this section, we study Heegaard splittings of a specific manifold, namely
Definition 4.1.
We denote the double of the genus n handlebody by W n . It is the connectedsum of n copies of S × S . A reader only interested in the strong Haken theorem may safely skip ahead to thenext section. Our goal here will be to prove that, similar to Waldhausen’s theorem for thethree-sphere, all Heegaard splittings of W n are “standard” in the following sense. Definition 4.2.
A Heegaard splitting of W n is standard if it is the double of a genus n handlebody. A standard Heegaard splitting of W n is a Heegaard splitting that is the connectedsum of n copies of W with the standard Heegaard splitting. One could also avoid citing this theorem by isotoping α into a regular neighbourhood of S and resolvingcrossings of the projection to S by isotopies which slide strands over the puncture similar to Figure 3. W n are standard(although it is not entirely clear up to which equivalence relation, and the proof sketch isincomplete). In the unpublished preprint [12], Oertel and Navarro Carvalho prove the result,using results on the homeomorphism groups of handlebodies and W n (in a very similar wayto the argument we will use below). In this section, we show that these techniques couldalso be used to prove a Strong Haken theorem (and obtain the uniqueness of splittings asa corollary). We want to emphasize that this of course follows from the general StrongHaken theorem (Theorem 1.1), but consider the argument using homeomorphisms of W n interesting enough to warrant this alternate proof. Proposition 4.3.
Every unstabilized Heegaard splitting of W is standard.Proof. Suppose that W = V ∪ S V (cid:48) is a Heegaard splitting. We wish to show that W = V ∪ S V (cid:48) is standard. Since W is reducible, Haken’s lemma tells us that there is a Hakensphere R for W = V ∪ S V (cid:48) . Denote V ∩ R by D and V (cid:48) ∩ R by D (cid:48) . Note that all essentialspheres in W , in particular R , are isotopic to S × ( point ).We may assume that S intersects a bicollar of R in an annulus ( S ∩ R ) × [ − , S × [ − , i.e., a twice punctured 3-sphere thatinherits a Heegaard splitting. By Theorem 3.3, this Heegaard splitting is either of genus 0or stabilized.Since W = V ∪ S V (cid:48) is unstabilized, the Heegaard splitting obtained on the complementof S × [ − ,
1] must be of genus 0. Specifically, the splitting surface is a twice punctured2-sphere, i.e., an annulus. Hence we can reconstruct W = V ∪ S V (cid:48) : Indeed, say, V , iscomposed of a 3-ball attached to the two copies D × {± } of D . It follows that V is asolid torus. The same is true of V (cid:48) and hence W = V ∪ S V (cid:48) is the standard Heegaardsplitting.First, we have the following classical result due to Griffiths [3]. Theorem 4.4.
The action of the mapping class group of a handlebody V n on its fundamentalgroup π ( V n ) = F n induces a surjection Mcg( V n ) → Out( F n ) → . We remark that the kernel of this map is quite complicated, and generated by twistsabout disk-bounding curves [11]. Next, we need a theorem of Fran¸cois Laudenbach [9] (seealso [10] and [2] for a modern proof):
Theorem 4.5.
The action of the mapping class group of a doubled handlebody W n on itsfundamental group π ( W n ) = F n induces a short exact sequence → K → Mcg( W n ) → Out( F n ) → . The kernel K is finite, generated by Dehn twists about non-separating spheres, and actstrivially on the isotopy class of every embedded sphere or loop. Corollary 4.6.
For the standard Heegaard splitting of W n , every essential sphere in W n isisotopic to a Haken sphere.Proof. First observe that any two nonseparating spheres in W n can be mapped to eachother by a homeomorphism. Namely, the complement of such a sphere is homeomorphic9o W n − with two punctures. Similarly, separating spheres can be mapped to each other ifand only if the fundamental groups of the complements are free groups of the same rank(as the complement is a disjoint union of once-punctured W k and W n − k ).Next, observe that there are Haken spheres of all such possible types, obtained bydoubling a suitable disk in the handlebody.Let i : V n → W n be the inclusion induced by doubling. Observe that on the onehand, the boundary of V n maps under i to the standard Heegaard splitting of W n , andon the other hand i induces an isomorphism i ∗ of fundamental groups. For any outerautomorphism ϕ ∈ Out( π ( V n )) of the fundamental group of V n , by Theorem 4.4, there isa homeomorphism f : V n → V n inducing it. Let F : W n → W n be the homeomorphism of W n obtained by doubling f . Observe that F preserves the standard Heegaard splitting of W n by construction, and F induces ϕ via the isomorphism i ∗ : π ( V n ) → π ( W n ). Since ϕ was arbitrary, this shows that any outer automorphism of π ( W n ) can in fact be realisedby a homeomorphism of W n preserving the standard Heegaard splitting.Together with Laudenbach’s Theorem 4.5 this shows that any sphere is isotopic tothe image of a Haken sphere under a homeomorphism preserving the standard Heegaardsplitting – hence, it is isotopic to a Haken sphere. Lemma 4.7.
There is a unique Heegaard splitting of W n of genus n .Proof. Connected sum decompositions of W n are not unique. However, let W n = V ∪ S V (cid:48) be the standard Heegaard splitting and let W n = X ∪ Y X (cid:48) be any Heegaard splitting ofgenus n . By repeated application of Theorem 2.9 there are Haken spheres R ∪ · · · ∪ R n − for W n that cut W n = X ∪ Y X (cid:48) into standard Heegaard splittings of W . By Corollary4.6, R , . . . , R n − are also Haken spheres for W n = V ∪ S V (cid:48) . By an Euler characteristicargument, these cut W n = V ∪ S V (cid:48) into genus 1 Heegaard splittings of the summands. ByProposition 4.3 these are standard. In particular, Y is isotopic to S . Proof of Proposition 1.2.
For an unstabilized Heegaard splitting, this is Lemma 4.7. Fur-thermore, if n = 1, then this follows from Proposition 4.3. So suppose that n > W n = V ∪ S V (cid:48) is stabilized. By Corollary 4.6, there is a Haken sphere R that decomposes W n into W W n − . Moreover, by [13], one of the Heegaard splittings inherited by thesummands is stabilized. By induction, W n = V ∪ S V (cid:48) is a stabilization of a connected sumof standard Heegaard splittings, i.e., a stabilization of the standard Heegaard splitting of W n .We finish this section with a quick discussion on “flippable” Heegaard splittings. Namely,so far we have considered Heegaard splittings up to isotopy of the splitting surface. Onecould define a more refined “oriented” version of equivalence by requiring that the handle-bodies are isotopic (as opposed to just their boundary surface).We want to remark that for the manifolds W n we have been considering, the two notionsof equivalence are in fact equal. Given the uniqueness shown above, to see this we just haveto observe that the standard splittings of W n are “flippable” – i.e. there is an isotopyexchanging the “inside” and “outside” handlebodies.For W this can be seen by hand (by rotating the S factor of S × S ). Furthermore, byconsidering a separating Haken sphere, we see that the connected sum of flippable Heegaardsplittings is flippable and, similarly, that stabilizations of flippable Heegaard splittings are10igure 4: A doubled handlebody can be flipped flippable. By Theorem 1.2, all Heegaard splittings of W n are stabilizations of the standardHeegaard splitting. Hence all Heegaard splittings of W n are flippable. Combining the uniqueness of Heegaard splittings for W n (Proposition 1.2) with Corollary 4.6yields a Strong Haken Theorem for W n : any sphere in W n is isotopic to a Haken sphere.This statement was recently proved by Scharlemann [16] for all three-manifolds. In thissection, we provide a short independent proof of this theorem for closed manifolds andmanifold with spherical boundary.The following proof proceeds by two nested inductions. It naturally involves 3-manifoldswith spherical boundary, even if we just want to prove the theorem in the closed case.Recall that for such 3-manifolds, we decree that each boundary sphere (puncture) meetsthe splitting surface in a single simple closed curve. Proof of Theorem 1.1.
We prove the theorem by considering all punctured 3-manifolds andall Heegaard splittings, ordered according to a suitable complexity. Namely, if M = V ∪ S V (cid:48) is a Heegaard splitting, we define the complexity as the pair ( g ( S ) , n ( S )) of genus andnumber of spots of the handlebodies (ordered lexicographically). We perform a nestedinduction on the genus g and the number of boundary components n . The argument forthe inductive step is in fact the same in both cases, and so we describe both inductionssimultaneously. Induction Start g = 0 , n ≥ Induction Steps
Now suppose that the Strong Haken Theorem is known for all manifoldsof complexity at most ( g, n ), and suppose that M is a manifold of complexity ( g, n + 1), orsuppose that the Strong Haken Theorem is known for all manifolds of complexity ( g, k ) , k ≥
0, and M is a manifold of complexity ( g + 1 , M is isotopic to aHaken sphere. We thus have to show that surviving spheres in M are also isotopic to Hakenspheres. We now make the following Claim 5.1.
Suppose that R is a surviving Haken sphere in M , and suppose that R (cid:48) is asurviving sphere disjoint from R . Then R (cid:48) is a Haken sphere. roof of Claim. Denote by M − R the punctured 3–manifold obtained by cutting at R . M − R has two punctures more than M , corresponding to the two sides of R . M − R hasone or two components, depending on whether R is separating or not.Let M (cid:48) be the component of M − R containing R (cid:48) . This manifold inherits a Heegaardsplitting from V ∪ S V (cid:48) with splitting surface a component of S (cid:48) = S − ( R ∩ S ). If R ∩ S isnonseparating, then g ( S (cid:48) ) < g ( S ). If R ∩ S is separating, then either the genus or the numberof boundary components is smaller for S (cid:48) . In either case, ( g ( S (cid:48) ) , n ( S (cid:48) )) < ( g ( S ) , n ( S ))lexicographically (crucial here is the fact that R (cid:48) is an essential sphere in M (cid:48) : it cannotbecome compressible, as that would violate incompressibility in M , therefore S − S (cid:48) is nota disk; furthermore, it cannot become peripheral as it would then be peripheral in M , orisotopic to R , so S − S (cid:48) is not an annulus.)Now, if R (cid:48) is almost peripheral in M (cid:48) , then by Theorem 3.6 it is isotopic to a Hakensphere in M (cid:48) . Otherwise, since the complexity of the splitting of M (cid:48) is smaller than theoriginal one, we can use the inductive hypothesis on M (cid:48) to conclude that R (cid:48) is isotopic to aHaken sphere in M (cid:48) . Interpreting M (cid:48) as a submanifold of M , and using that the Heegaardsplitting of M (cid:48) is inherited from M , this shows that R (cid:48) is isotopic to a Haken sphere in M as well. //Now, if M contains any surviving spheres, then the Surviving Haken Lemma (Theo-rem 3.6) implies that there is a surviving Haken sphere σ . Connectivity of the survivingsphere complex (Lemma 3.5), together with the Claim then inductively implies that anysurviving sphere is isotopic to a Haken sphere. This finishes the proof of the inductivesteps. References [1] Bowden, Jonathan and Hensel, Sebastian and Webb, Richard,
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