Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins
DDiffeomorphisms of the 4-sphere, Cerf theory andMontesinos twins D AVID
T G AY One way to better understand the smooth mapping class group of the 4–spherewould be to give a list of generators in the form of explicit diffeomorphismssupported in neighborhoods of submanifolds, in analogy with Dehn twists onsurfaces. As a step in this direction, we describe a surjective homomorphism froma group associated to loops of 2–spheres in S × S ’s onto this smooth mappingclass group, discuss two natural and in some sense complementary subgroups ofthe domain of this homomomorphism, show that one is in the kernel, and givegenerators as above for the image of the other. These generators are described astwists along Montesinos twins, i.e. pairs of embedded 2–spheres in S intersectingtransversely at two points. Given a smooth oriented manifold X , let Diff + ( X ) be the space of orientation preservingdiffeomorphisms of X (fixed on a collar neighborhood of ∂ X if ∂ X (cid:54) = ∅ ). Here, inspiredheavily by Watanabe’s work [10] on homotopy groups of Diff + ( B ) and Budney andGabai’s work [2] on knotted 3–balls in S , we initiate a study of π ( Diff + ( S )), i.e.the smooth mapping class group of the 4–sphere. We know very little about thisgroup except that it is abelian and that every orientation preserving diffeomorphismof S is pseudoisotopic to the identity; the group could very well be trivial, like thetopological mapping class group. Ideally we would like to find a generating set forthis mapping class group defined explicitly and geometrically, for example as explicitdiffeomorphisms supported in neighborhoods of explicit submanifolds of S , in analogywith Dehn twists as generators of the mapping class groups of surfaces. In this paperwe construct a surjective homomorphism from a limit of fundamental groups of certainembedding spaces of 2–spheres in 4–manifolds onto π ( Diff + ( S )), we describe onegeometrically natural subgroup of the domain of this homomorphism which we showto be in its kernel, and we describe a “complementary” geometrically natural subgroupand give an explicit list of generators as above for its image in π ( Diff + ( S )). a r X i v : . [ m a t h . G T ] F e b David T Gay
Given smooth manifolds X and Y , let Emb ( X , Y ) denote the space of embeddingsof X into Y . Let n ( S × S ) † refer to a punctured n ( S × S ). (The puncture isnot important until the next paragraph.) Let S n = Emb ( (cid:113) n S , n ( S × S ) † ). This S stands for “spheres”, as in “space of embeddings of collections of spheres”. Fix a point p ∈ S and let (cid:113) n ( S × { p } ) ⊂ n ( S × S ) † denote the union of one copy of S × { p } in each S × S summand of n ( S × S ) † . This will be our basepoint in S n , whichwe will often suppress from our notation, with the understanding that S n is a pointedspace. We will also be interested in two subspaces of S n : Let S n denote the subspaceof embeddings with the property that for each i and j the i ’th component of (cid:113) n S intersects the { p } × S in the j ’th summand of n ( S × S ) † transversely at δ ij points.Let (cid:98) S n denote the subspace of embeddings with the property that the image of (cid:113) n S isdisjoint from (cid:113) n ( S × { p (cid:48) } ) for some fixed p (cid:48) (cid:54) = p ∈ S . Note that our basepoint liesin both of these subspaces.There is a natural homomorphism H n : π ( S n ) → π ( Diff + ( S )) defined as follows,and with more care in the next section: Given a loop of embeddings α t : (cid:113) n S (cid:44) → n ( S × S ) † ⊂ n ( S × S ) representing some a ∈ π ( S n ), for each t build a 5–dimensional cobordism by attaching first n , × S with some standard fixed attaching data in { } × S , so that the top boundary iscanonically identified with n ( S × S ), and then attach n α t as the attaching map. Varying t , we get a bundle over S of 5–dimensionalcobordisms; since attaching 3–handles along the basepoint embedding (cid:113) n ( S × { p } )canonically cancels the 2–handles, the bundle of cobordisms is a cobordism from S × S to an interesting S –bundle over S . We define H n ( a ) to be the monodromyof this bundle.Thanks to the puncture, we have a natural basepoint-preserving inclusion : S n (cid:44) →S n + , respecting the inclusions of S n and (cid:98) S n , and thus inclusion-induced homomor-phisms ∗ : π ( S n ) → π ( S n + ) which commute with H n and H n + and with theinclusion-induced homomorphisms ı ∗ : π ( S n ) → π ( S n ) and ı ∗ : π ( (cid:98) S n ) → π ( S n ).As a consequence, we have limit groups which we will denote π ( S ∞ ), π ( S ∞ ), π ( (cid:98) S ∞ ) (it is not important for us to think about the limiting spaces, just the groups,but this notation is convenient), limiting inclusion-induced homomorphisms betweenthem, and a limit homomorphism H ∞ : π ( S ∞ ) → π ( Diff + ( S )). Our first result is: Theorem 1
The homomorphism H ∞ : π ( S ∞ ) → π ( Diff + ( S )) is surjective andthe kernel of H ∞ contains ı ∗ ( π ( S ∞ )) . Our second result characterizes H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) in terms of a countable list of explicit iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins generators which we will now describe. This could in principle be all of π ( Diff + ( S )),although we have no evidence for or against that possibility.A Montesinos twin in S is a pair W = ( R , S ) of embedded 2–spheres R , S ⊂ S intersecting transversely at two points. For us, the 2–spheres are both oriented. Mon-tesinos shows [8] that the boundary of a neighborhood ν ( W ) of R ∪ S is diffeomorphic to S × S × S and that in fact there is a canonical parametrization S l × S R × S S ∼ = ∂ν ( W ),canonical up to postcomposing with diffeomorphisms of ∂ν ( W ) which are isotopic tothe identity and precomposing with independent diffeomorphisms of S l , S R and S S .This parametrization is characterized by S l × { b } × { c } being homologically trivialin H ( S \ ( R ∪ S )), i.e. a “longitude”, { a } × S R × { c } being a meridian for R , and { a } × { b } × S S being a meridian for S . This then parametrizes a neighborhood of ∂ν ( W ) as [ − , × S l × S R × S S . We adopt the orientation conventions that S R and S S have the standard meridian orientations coming from the orientations of R and S ,that [ − ,
1] is oriented in the outward direction from ν ( W ), and that S l is oriented sothat the orientation of [ − , × S l × S R × S S agrees with the standard orientation of S . Definition 2
Given a Montesinos twin W in S , parametrize a neighborhood of ∂ν ( W )as [ − , × S l × S R × S S as above. Let τ l : [ − , × S l → [ − , × S l denote a right-handed Dehn twist. The twin twist along W , denoted τ W , is the diffeomorphism of S which is the identity outside this neighborhood of ∂ν ( W ) and is equal to τ l × id S R × id S S inside this neighborhood.By the canonicity of our parametrization, τ W is well-defined up to isotopy, i.e. [ τ W ]is a well-defined class in π ( Diff + ( S )). Incidentally, we have the following as aconsequence of our orientation conventions: Lemma 3 If W = ( R , S ) is a Montesinos twin, then [ τ W ] − = [ τ ( S , R ) ] = [ τ ( R , S ) ] = [ τ ( R , S ) ] . Proof
Either switching the spheres S and R , or reversing the orientation of one ofthem, reverses the orientation of S R × S S , which then forces the reversal of the orientationof [ − , × S l , changing a positive Dehn twist to a negative Dehn twist.We now describe a family of twins W ( i ) = ( R ( i ) , S ), for i ∈ N ∪ { } . Figure 1illustrates W (3); W ( i ) is the same but with i turns in the spiral rather than 3 turns.Figure 2 and 3 give two alternate descriptions of this twin. Orientations are not madeexplicit here since our main claim is simply that the twists invoved generate a certain David T GayR (3) S Figure 1: An illustration of W (3) = ( R (3) , S ). The picture mostly happens in the slice { t = } ⊂ R = { ( x , y , z , t ) } ⊂ R ∪ {∞} = S . The ring labelled S is a slice through S , which shrinks to a point as we move forwards and backwards in the “time” coordinate t .The “snake whose tail passes through his head” is R (3), which is projected onto { t = } ,intersecting itself along one circle in the middle of the red disk (the “snake’s left ear hole”) andalong another circle in the middle of the blue disk (the “right ear hole”). Blue and red indicatethat these disks are pushed slightly forwards (blue) and backwards (red) in time to resolve theseintersections; otherwise R (3) lies in the slice { t = } . The bright pink dot where the ringpierces the tail is the positive point of intersection of R (3) and S ; the faint green dot on theback side of the tail is the negative intersection point. group, and the inverse of a generator is still a generator. Note that both R ( i ) and S are individually unknotted 2–spheres, and that the twin W (0) is the trivial “unknottedtwin”.Our second result is: Theorem 4
The subgroup H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) of π ( Diff + ( S )) is generated by the twintwists { τ W ( i ) | i ∈ N } . (In fact these automorphisms τ W ( i ) are also examples of the “barbell maps” discussedin [2]; readers familiar with barbell maps should be able to use the description of W ( i )in Figure 3 to see the connection.) Question 5
Is [ τ W ] ∈ H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) for an arbitrary Montesinos twin? Moregenerally, given any embedding of S × Σ (cid:44) → S , for closed surface Σ , a tubular iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins Figure 2: Another illustration of W (3) = ( R (3) , S ). Here we have drawn an immersed pair ofdisks, one green and one pink, with mostly ribbon intersections except for one nonribbon arc.Pushing these two disks into R + or R − and resolving ribbon intersections in the usual waygives two embedded disks intersecting each other transversely once, and then taking one copyin R + and one in R − glued along their common boundary, i.e. doubling the ribbon disks,gives R (3) (green) and S (pink) in R ⊂ S .Figure 3: Yet another illustration of W (3). Here we have drawn two disjoint, embedded2–spheres in S (the two thick circles, becoming 2–spheres when shrunk to points forwardsand backwards in time) and an arc connecting them. Push a finger from one of the spheres outalong this arc and then do a finger move when you encounter the other sphere, creating a pairof transverse intersections, and the result is W (3). David T Gay neighborhood gives an embedding of [ − , × S × Σ (cid:44) → S which gives a diffeomor-phism τ × id Σ , where τ is the Dehn twist on [ − , × S . Are such diffeomorphismsin H ∞ ( ı ∗ ( π ( (cid:98) S ∞ )))?One could try to answer these questions either through Cerf theory, by explicitlyidentifying a pseudoisotopy from a given diffeomorphism of S to the identity, andthen extracting a loop of attaching spheres for 5–dimensional 2–handles, or one couldtry to work explicitly with the diffeomorphisms in S and try to find relationshipsamongst such twists, to relate them to twists along our standard Montesinos twins W ( i ).The bigger questions are the following, with affirmative answers to both showing thatthe smooth mapping class group of S is trivial: Question 6 Is H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) trivial?Theorem 4 could help prove this if one can exhibit explicit isotopies from τ W ( i ) to id S . Question 7 Is H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) = π ( Diff + ( S ))?Since ı ∗ ( π ( S ∞ )) is in the kernel of H ∞ , we know that H ∞ factors through the quotientmap π : π ( S ∞ ) → π ( S ∞ ) / (cid:104) ı ∗ ( π ( S ∞ )) (cid:105) , where (cid:104) H (cid:105) denotes the normal closure ofa subgroup H . Thus one way to show that H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) = π ( Diff + ( S )) wouldbe to show that π ◦ ı : π ( (cid:98) S ∞ ) → π ( S ∞ ) / (cid:104) ı ∗ ( π ( S ∞ )) (cid:105) is surjective. On the otherhand this does not need to be true for the answer to this question to be “yes”, since thekernel of H ∞ could presumably be much larger than (cid:104) ı ∗ ( π ( S ∞ )) (cid:105) .In the next section we elaborate on the connection between loops of embeddings ofcertain spheres and self-diffeomorphisms of other spheres, setting up the general theoryin various dimensions and codimensions. After that, we devote one section to the proofof Theorem 1, and we break the proof of Theorem 4 into the three remaining sections.The author would like to thank Bruce Bartlett, Sarah Blackwell, Mike Freedman,Robert Gompf, Jason Joseph, Danica Kosanovic, Peter Lambert-Cole, Gordana Matic,Benjamin Ruppik, Rob Schneiderman, Peter Teichner and Jeremy Van Horn-Morrisfor helpful conversations along the way, and most especially Hannah Schwartz forpointing out Montesinos’s work and Dave Gabai for initial inspiration and for pointingout the mistake in the first version that claimed far too much, and both Hannah andDave for many clarifying conversations about loops of spheres.Much of this work was carried out during the author’s year at the Max Planck Institutefor Mathematics in Bonn, and thus the author gratefully acknowledges the institute’s iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins support in the form of the 2019-2020 Hirzebruch Research Chair. This work was alsosupported by individual NSF grant DMS-2005554 and NSF Focused Research Groupgrant DMS-1664567. Here we describe the homomorphism H ∞ : π ( S ∞ ) → π ( Diff + ( S )) as a special caseof a more general family of homomorphisms turning loops of framed embedded spheresin various dimensions into bundles of cobordisms and hence into self-diffeomorphismsof smooth manifolds. This should mostly be standard “Cerf theory”. In the introductionabove, we had 2–spheres embedded in 4–manifolds, but we did not mention framingsof these 2–spheres. Below, we will work with framed spheres and then later when werelate this back to the terminology of the introduction, we will see where the framingscome from.Fix an m –manifold X , for some m ≥
2, fix integers 0 < k < m and n ≥
0, and let n ( S k × S m − k ) † denote a punctured n ( S k × S m − k ). As in the introduction, the punctureis needed so that we can view n ( S k × S m − k ) † as a subspace of n + ( S k × S m − k ) † . Nowconsider the following space of embeddings of collections of framed spheres: F S n ( X , k ) = Emb ( (cid:113) n ( S k × B m − k ) , X n ( S k × S m − k ) † ))Picking a fixed point p ∈ S m − k and a disk neighborhood U of p parametrized as B m − k ,we get a natural basepoint (cid:113) n ( S k × U ) ⊂ X n ( S k × S m − k ) † ) for F S n ( X , k ), whichwe will again generally suppress in our notation, understanding that F S n ( X , k ) is apointed space.Then we get a homomorphism F H n : π ( F S n ( X , k )) → π ( Diff + ( X ))defined as follows. Represent an element of π ( F S n ( X , k )) by a 1–parameter family ofembeddings β t : (cid:113) n ( S k × B m − k ) (cid:44) → X n ( S k × S m − k ) † ) with β = β = (cid:113) n ( S k × U ).We will use this to build a ( m + Z β which is a cobordism from S × X to some X bundle over S , and the monodromy of this bundle will be H n ([ β t ]). Tobuild Z β , first let Y be [0 , × X with n ( m + k –handles attachedalong n unlinked 0–framed unknotted S k − ’s in a ball in X . Thus Y is a cobordismfrom X to X n ( S k × S m − k )). Now consider S × Y , which is a cobordism from S × X to S × X n ( S k × S m − k )). Identifying S with [0 , / ∼ t as the S –coordinate, we can now use each β t as the attaching map for n ( m + k + { t } × X n ( S k × S m − k )) ⊂ { t } × Y ; the David T Gay result is our construction of Z β . Thus Z β is a bundle over S , each fiber of whichis a ( m + X to an m –manifold X t , built with n k –handles and n ( k + k + β = β is the canonical cancelling handle for the corresponding k –handle, and thus X iscanonically diffeomorphic to X . Therefore the full ( m + Z β is a cobordism from S × X to some X bundle over S with a monodromy which iswell-defined by the homotopy class of the loop of embeddings β t . This defines F H n and it is straightforward to see that F H n is a group homomorphism.Thanks to the punctures, we have basepoint preserving inclusions . . . ⊂ F S n ( X , k ) ⊂ F S n + ( X , k ) ⊂ . . . and thus induced maps on π and thus a direct limit . . . → π ( F S n ( X , k )) → π ( F S n + ( X , k )) → . . . → π ( F S ∞ ( X , k ))Again, we do not really care about the limiting spaces, just the groups. Thus oneshould think of an element of π ( F S ∞ ( X , k )) as an equivalence class of loops insome F S n ( X , k ), where two such loops are equivalent if they become homotopic afterincluding into some F S N ( X , k ) for some N ≥ n . It is not hard to see that these inducedmaps on π commute with the F H n homomorphisms, so that finally we get the limithomomorphism F H ∞ : π ( F S ∞ ( X , k )) → π ( Diff + ( X ))As in the unframed setting of the introduction, we have two natural subspaces of F S n ( X , k ): Let F S n ( X , k ) denote those embeddings of (cid:113) n ( S k × B m − k ) into X n ( S k × S m − k )) for which the S k × { } in the i ’th S k × B m − k transversely intersects the { p } × S m − k in the j ’th S k × S m − k summand transversely at δ ij points. Let (cid:100) F S n denotethe subspace of embeddings with the property that the image of (cid:113) n ( S k × B m − k ) isdisjoint from (cid:113) n ( S k × { p (cid:48) } ) for some fixed p (cid:48) ∈ S m − k \ U . Note that our basepointlies in both of these subspaces.It is standard that ı ∗ ( π ( F S n ( X , k ))) is in the kernel of F H n , because when the S k × { } ’s are dual to the { p } × S m − k ’s for all t in a loop of embeddings α t , then forall t the k –handles and ( k + l’unicit´e demort [3]). Thus the cobordism Z α becomes a bundle over S with each fiber supportinga Morse function without critical points, so that the monodromy at the top is id X .Now we discuss the relationship to spaces of spheres without framings. Let S n ( X , k ) = Emb ( (cid:113) n S k , X n ( S k × S m − k ) † )) iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins There is an obvious “framing forgetting” map of pointed spaces F : F S n ( X , k ) → S n ( X , k )given by restricting an embedding of S k × B m − k to S k = S k × { } . Palais [9] showsthat such maps are locally trivial and thus satisfy the homotopy lifting property. Thefiber of F is the space of framings of a fixed S k , i.e. (up to homotopy) maps from S k to SO ( m − k ). Note that the fiber over the basepoint is actually a subspace of F S n ( X , k )and thus π of the fiber lands in the kernel of F H n . As a consequence, even though F ∗ : π ( F S n ( X , k )) → π ( S n ( X , k ))may not be injective, if the fiber is not simply connected, we still have that F H n induces a well-defined homomorphism H n from the image F ∗ ( π ( F S n ( X , k ))) of F ∗ in π ( S n ( X , k )) to π ( Diff + ( S m ). All of this also commutes with the inclusion mapsfrom n to n + H ∞ .When m = k =
2, the fibers of F are connected, i.e. a sphere in a 4–manifoldwith self-intersection 0 has only one framing up to isotopy (because π ( SO (2)) = F ∗ : π ( F S n ( S , → π ( S n ( S , = S )is surjective and so F H n and F H ∞ induce homomorphisms H n and H ∞ : π ( S ∞ ) → π ( Diff + ( S m ) H ∞ In this section we will prove Theorem 1, i.e. that H ∞ : π ( S ∞ ) → π ( Diff + ( S ))is surjective. (We have already seen in the previous section that the kernel of H ∞ contains ı ∗ ( π ( S ∞ )).) Since H ∞ is a limit of maps H n , what we are really trying toprove is the following: Theorem 8
For any orientation preserving diffeomorphism φ : S → S , there issome n ∈ N and some loop α t in S n based at the basepoint (cid:113) n ( S × { p } ) , such that H n ([ α t ]) = [ φ ] ∈ π ( Diff + ( S ) . In other words, once we prove this theorem then we have:
Proof of Theorem 1
This is an immediate corollary to Theorem 8 and the discussionin the previous section. David T Gay
In fact, to make things more concrete, we will prove this:
Theorem 9
For any orientation preserving diffeomorphism φ : S → S , there is a –manifold Z with the following properties: (1) Z is a bundle over S and is a cobordism from S × S to S × φ S = ([0 , × S ) / (1 , p ) ∼ (0 , φ ( p )) , with the three fibrations over S , of Z , S × S and S × φ S , being compatible. (2) For each t ∈ S , the fiber Y t over t in Z , being a –dimensional cobordismfrom S to a –manifold X t which is diffeomorphic to S , comes with a handledecomposition with n –handles and n –handles. (3) The n –handles for each Y t are attached along a fixed standard collection of n standardly framed S ’s in S , so that the –manifold immediately above these –handles is canonically identified with n ( S × S )(4) The n –handles for each Y t are attached along a moving family of n framed S ’s in n ( S × S ) such that, ignoring the framings, these S ’s are given by aloop of embeddings of (cid:113) n S into n ( S × S ) all missing a single fixed pointand starting and ending at the standard embedding (cid:113) n ( S × { p } ) , i.e. a basedloop α t in S n . Then:
Proof of Theorem 8
This is an immediate corollary of Theorem 9, since the homo-morphism H n comes precisely from constructions of cobordisms as in Theorem 9.We will use Cerf theoretic techniques, beginning with a pseudoisotopy. Lemma 10
Every orientation preserving self-diffeomorphism of S is pseudoisotopicto the identity. Proof
Consider an orientation preserving diffeomorphism φ : S → S . Let f : S → R be projection onto the last coordinate in R , and for any interval I ⊂ R ,let S I = f − ( I ). Let V be a smooth vector field on S \ { (0 , , , , , ± } which isorthogonal to all level sets of f and scaled so that df ( V ) =
1. Let X = S − , ∪ φ S , ,where φ : ∂ S , = S → − S = ∂ S − , is now seen as an orientation reversinggluing diffeomorphism. Arrange the gluing (i.e. the smooth structure on X ) so thatthe vector field V on the two halves of X is still a smooth vector field on X , which wecall V X . Note that X also inherits the Morse function f , which we label f X : X → R , iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins and we can use the same notation X I = f − X ( I ) ⊂ X . The point is that if I ⊂ ( −∞ , I ⊂ [0 , ∞ ) then X I = S I , i.e. they are actually equal sets, not just diffeomorphicmanifolds.Now note that X is homotopy equivalent to S and therefore [6, 7] diffeomorphic to S . For some small (cid:15) > Φ : S → X isthe identity on S − , − + (cid:15) ] = X [ − , − + (cid:15) ] and on S − (cid:15), = X [1 − (cid:15), . Using flow along V and V X , respectively, and the standard identification of ∂ S − , − + (cid:15) ] with S , wecan parametrize both S − + (cid:15), − (cid:15) ] and X [ − + (cid:15), − (cid:15) ] as [ − + (cid:15), − (cid:15) ] × S and then Φ restricts to give a diffeomorphism from [ − + (cid:15), − (cid:15) ] × S to itself. Furthermore, thismap is the identity on {− + (cid:15) } × S and by continuity must equal φ on { − (cid:15) } × S .After reparametrizing [ − + (cid:15), − (cid:15) ] as [0 ,
1] we get the desired pseudoisotopy.Now we begin the Cerf theory.
Proof of Theorem 9
Given the orientation preserving diffeomorphism φ : S → S ,let Φ : [0 , × S → [0 , × S be a pseudoisotopy from the identity to S , i.e. Φ (0 , p ) = (0 , p ) and Φ (1 , p ) = (1 , φ ( p )). Let f : [0 , × S → [0 ,
1] be projectiononto the first factor, let V be the unit vector field on [0 , × S in the [0 ,
1] direction,let ( f , V ) = Φ ∗ ( f , V ) = ( f ◦ Φ , D Φ − ( V )), and let ( f t , V t ) be a generic homotopyof Morse functions with gradient-like vector fields from ( f , V ) to ( f , V ). Hatcherand Wagoner (Chapter VI, Proposition 3, page 214 of [5]) show that this family offunctions with gradient-like vector fields can be homotoped rel t ∈ { , } so as toarrange the following properties (see Figure 4):• The only Morse critical points of any f t are critical points of index 2 and 3.• None of the functions f t have two critical points mapping to the same criticalvalue.• All critical points of index 2 stay below all critical points of index 3. In otherwords, for every t , if p is a critical point of index 2 for f t and q is a critical pointof index 3 for f t then f t ( p ) < f t ( q ).• There are no handle slides. In other words, none of the vector fields V t haveflow lines between critical points of the same index.• All births of cancelling pairs of critical points happen before all deaths ofcancelling pairs, and no two births or deaths happen at the same time. David T Gay • At the moments of birth and death, the handle pair dying or being born does notrun over any other handles. In other words, there are no V t flow lines between anon-Morse birth/death critical point and any other critical point.Now consider the 6–manifold Z = [0 , × [0 , × S / (1 , s , p ) ∼ (0 , Φ ( s , p )). We willshow that this has the properties advertised in the statement of Theorem 9. Currentlywe have the first property, namely that Z is a bundle over S and is a cobordism from S × S to S × φ S = ([0 , × S ) / (1 , p ) ∼ (0 , φ ( p )), with the three fibrations over S ,of Z , S × S and S × φ S , being compatible. Each fiber Y t of Z → S gets a handledecomposition from ( f t , V t ) (allowing for birth and death handle decompositions) butit will take some work to arrange that these satisfy the remaining properties as stated inTheorem 9. First let us characterize each fiber Y t , with its given handle decomposition,as much as possible in terms of handle attaching data in { t } × S ⊂ S × S .Suppose that the births in our Morse functions f t occur at times 0 < t < t < . . . < t n < / / < t (cid:48) n < . . . < t (cid:48) < t (cid:48) <
1. We willshow how each cobordism Y t can be described as built from { t } × [0 , × S accordingto handle attaching data, and to do this we establish a few conventions. First, all ofour embeddings of spheres and disks of various dimensions are framed embeddings,but to keep the terminology minimal we will sometimes suppress mention of framings.Later we will make sure to carefully relate this back to the case of loops of unframedspheres. Second, given a framed embedding (cid:15) of a sphere in a manifold X , let X ( (cid:15) )denote the result of surgering X along (cid:15) . Third, a framed embedding δ of a disk D k into an n –manifold X gives us several auxiliary pieces of information: We get a framed iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins embedding α of S k − = ∂ D k into X . We get the surgered manifold X ( α ). And, in X ( α ), there is a natural framed embedding β of S k into X ( α ) which coincides with δ on the part of X ( α ) away from the surgery, and which coincides with the co-core of thesurgery inside the surgered region. This is the basic model for the result of attaching a( n + k –( k +
1) handle pair to [0 , × X , with the attachingdata for the pair being described entirely in X by the framed disk δ . Thus we alsosee that X ( α )( β ) is canonically identified with X . Here we will work with the case k = k + =
3, but later in the paper we will be interested in the case k = k + = δ ’s, α ’s or β ’s are embed-dings of disjoint unions of disks or spheres. Using this, we now describe the form ofthe explicit t –varying handle attaching data that gives the fiberwise construction of Z ,i.e. the data that shows how to construct each Y t starting with { t } × [0 , × S andattaching various 5–dimensional handles.• For 0 ≤ t < t , no handles are attached, i.e. Y t = { t } × [0 , × S .• For t = t , there is a framed embedding δ of a disk D into S such that Y t is built from { t } × [0 , × S by attaching a cancelling pair of a 5–dimensional 2–handle and 3–handle, with the 2–handle attached along theframed embedding α t = δ | S , and with the 3–handle attached along theresulting framed embedding β t of S into S ( α t ).• For t ≤ t ≤ t , we have a t –varying framed embedding α t = α t of S into S and a t –varying framed embedding β t = β t of S into S ( α t ). For t ≤ t < t , Y t is built from { t } × [0 , × S by attaching a 2–handle along α t and then a3–handle along β t . At t = t , the α t and β t agree with the α t and β t from theprevious point.• For t = t , there is a framed embedding δ of D into S disjoint from theimages of α t and β t , such that Y t is built from { t } × [0 , × S by attaching- first a 5–dimensional 2–handle along α t ,- then a cancelling 2–3–pair in which the 2–handle is attached along α t = δ | S and the 3–handle is attached along the resulting framed 2–sphere β t in S ( α t )( α t ), and- then a 3–handle attached along β t , which can be seen as a framed 2–sphere in S ( α t ), in S ( α t )( α t ) or in S ( α t )( α t )( β t ) ∼ = S ( α t ).• For t ≤ t ≤ t , we have a t –varying framed embedding α t = α t (cid:113) α t of S (cid:113) S into S and a t –varying framed embedding β t = β t (cid:113) β t of S (cid:113) S David T Gay into S ( α t ), agreeing with the α t , α t , β t and β t of the preceding point when t = t , so that Y t is built from { t } × [0 , × S by attaching 2–handles along α t and then 3–handles along β t .• This process continues with each birth at time t i governed by a new framed disk δ i , generating a new framed S α it i and a new framed S β it i , which then join theprevious framed spheres to create α t = α t (cid:113) . . . (cid:113) α it and β t = β t (cid:113) . . . (cid:113) β it ,which are the attaching spheres for 2– and 3–handles for t t ≤ t < t i + .• Reversing time we see the deaths governed by (most likely quite different) disks δ (cid:48) n , . . . , δ (cid:48) and the same pattern of framed S ’s and S ’s in between these times.• For t n ≤ t ≤ t (cid:48) n , there is a t –parametrized family α t of framed embeddings of (cid:113) n S into S and a t –parametrized family β t of framed embeddings of (cid:113) n S into S ( α t ) which constitute the attaching data for the n n Y t in this range.We will now improve the format of this data somewhat. First, we can arrange that forsome small (cid:15) >
0, on the time interval t ≤ t ≤ t + (cid:15) the embeddings α t and β t areindependent of t , i.e. the first framed circle and sphere do not move for a short timeafter their birth. Next, since a birth happens at a point, we can make the second birthhappen at an earlier time so that in fact t < t < t + (cid:15) . Repeating this, and doing thesame in reverse with the deaths, we can assume that on the whole interval t ≤ t ≤ t n and on the whole interval t (cid:48) n ≤ t ≤ t (cid:48) , the α t and β t are independent of t except forthe fact that at each t i a new α it and β it is added to the mix (and ditto for the deaths).Thus now the governing data for the constructions of each Y t can be more succinctlydescribed by the following data:• A framed embedding δ = δ (cid:113) . . . (cid:113) δ n of (cid:113) n D into S (for the births), definingframed embeddings α of (cid:113) n S into S and β of (cid:113) n S into S ( α ).• A framed embedding δ (cid:48) = δ (cid:48) (cid:113) . . . (cid:113) δ (cid:48) n of (cid:113) n D into S (for the deaths),defining framed embeddings α (cid:48) of (cid:113) n S into S and β (cid:48) of (cid:113) n S into S ( α (cid:48) )• A t –parameterized family, for t ∈ [1 / , / α t = α t (cid:113) . . . (cid:113) α nt of (cid:113) n S into S , with α / = α and α / = α (cid:48) .• A t –parameterized family, for t ∈ [1 / , / β t = β t (cid:113) . . . (cid:113) β nt of (cid:113) n S into S ( α t ), with β / = β and β / = β (cid:48) .We would now like to modify this data (by a homotopy of the family ( f t , V t )) so that δ = δ (cid:48) is a fixed standard embedding and so that the family α t is actually invariantwith respect to t , i.e. α t = α = α (cid:48) for all t . Since any collection of n disks is isotopicto any other, it is easy to arrange that δ = δ (cid:48) (we tack the necessary isotopies on at the iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins beginning and end of the t –parameterized family of handle attaching data). Thus wenow assume δ = δ (cid:48) is standard.Next focus on α t , which is a loop of embeddings of S starting and ending at a standardembedding. By an isotopy rel endpoints we can arrange that α t is independent of t at a fixed point on S (since π ( S ) = π ( S ) = S , i.e. isotopies of isotopies, and can be extended to all of S by theparametrized isotopy extension theorem, so that the other α it ’s move out of the waywhen we move α t . Also, when we move the α it ’s, we can realize this as a homotopy of( f t , V t ) by just modifying the gradient-like vector field V t in f − t [ a , b ] for some small0 < a < b with b below the lowest critical value of all the f t ’s. Thus we do not needto worry about how this affects the framed 2–spheres β t , they will still be a family asabove of framed embeddings of (cid:113) n S into S ( α t ). Now we use the fact that the spaceof embeddings of B in B with fixed endpoints on ∂ B is simply connected [1] tofinally arrange that α t is independent of t , i.e. that α t = α / = α / .We could proceed to do the same thing for α t if we knew that the parametrized isotopyextensions involved when fixing α t (and after that, α t up to α nt ) would not mess upthe work we have already done to fix α t . Recall that δ is a fixed disk bounded bythe (now constant in t ) circle α t ; if the other circles α t , . . . , α nt do not pass through δ , then we can arrange that the isotopy extensions do not need to move δ and hencedo not need to move α t . (This is because the isotopies of isotopies of circles involvedare ultimately 3–dimensional objects and can be arranged to miss points, and hencedisks. They cannot be assumed in general to miss circles.) To deal with this issue, let δ t be a family of framed embeddings of D into S , for t ∈ [1 / , / ∂δ t = α t and δ / = δ , such that δ t is disjoint from α t (cid:113) . . . (cid:113) α nt for all t ∈ [1 / , / δ out of the way as the S ’s α t (cid:113) . . . (cid:113) α nt move around and pass through δ . Note that there is no reasonto expect that δ / = δ . However, in the time interval [ t (cid:48) , t (cid:48) ], after the death of allof the cancelling handle pairs involving α t , . . . , α nt , we can reverse the isotopy of the δ t ’s and get back to δ t = δ . Finally, looking at the entire family of disk embeddings δ t from t = / t = t (cid:48) , this is homotopic rel endpoints to a constant family ofembeddings δ t = δ = δ (cid:48) , and this homotopy, i.e. isotopy of isotopies, extends to aparametrized ambient isotopy which again moves α t , . . . , α nt for t ∈ [1 / , / δ t , . . . , δ nt are not moved by this ambient isotopy againbecause they are disks and hence can be avoided by the isotopy of the family of disks δ t .Thus we get α t = α / = α / = ∂δ , and that α t (cid:113) . . . α nt is disjoint from δ David T Gay for all t . Now we can repeat this with α t up to α nt . This gets us to our desiredsituation where δ = δ (cid:48) , α t = α / = α / = ∂δ , and δ is a standard framedembedding of (cid:113) n D into S . With this setup, we can now move the birth of thefirst 2–3–pair closer to t = t = Z = [0 , × [0 , × S / (1 , s , p ) ∼ (0 , Φ ( s , p )), and repeat with the second pair and soon. After this, Y is built from [0 , × S by attaching n standard cancelling 2–3–pairsand Y t is built by attaching n n framed circles α and n β t of (cid:113) n S .This is precisely the conclusion of the theorem. – –handle pairs into – –handle pairs We now need to work toward the connection with Montesinos twins and the proof ofTheorem 4, that twists along Montesinos twins generate the subgroup of π ( Diff + ( S ))corresponding, via the 2–3–handle pair construction above, to loops of 2–sphereswhich remain disjoint from parallel copies of the basepoint 2–spheres. More precisely,recall that (cid:98) S n is the space of embeddings of (cid:113) n S into n ( S × S ) † which remaindisjoint from (cid:113) n ( S × { p (cid:48) } for some fixed p (cid:48) ∈ S , with basepoint being (cid:113) n ( S × { p } )for p (cid:54) = p (cid:48) . We want to study the subgroup H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))). To relate this toMontesinos twins we will first need to relate it to families of 1–2–handle pairs comingfrom loops of framed circles in n ( S × S ). In particular, using the notation fromSection 2, in this section we will prove: Theorem 11
For any n , H n ( ı ∗ ( π ( (cid:98) S n ))) = F H n ( π ( F S n ( S , . This means that any isotopy class of diffeomorphisms of S that can be realized by afamily of cobordisms built with n (cid:98) S n can alsobe realized by a family of cobordisms built with n F S n ( S , n circlesin n ( S × S ) † . Proof
Consider a cobordism Z from S × S to S × φ S built as before as a family Y t of cobordisms, such that each Y t is built by attaching n 2–handles to [0 , × S andthen n α t of framedembeddings of (cid:113) n S into S , except that in fact α t = α does not vary with t and is a iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins standard embedding, so that S ( α t ) is canonically identified with n ( S × S ). The 3–handles are attached along a family β t of framed embeddings of (cid:113) n S into n ( S × S ),with β = β = (cid:113) n ( S × { p } ), and with each β t disjoint from (cid:113) n ( S × { p (cid:48) } ).We will now introduce two 5–dimensional analogues of the “dotted circle” notationtraditionally used for 4–dimensional 1–handles. A 5–dimensional 1–handle attachedto the boundary of a 5–manifold in such a way that the 1–handle could be cancelled (which here means that its attaching S bounds a B and that the framing of the S comes from a framing of the B ) can be represented by a “dotted unknotted 2–sphere” together with a choice of –ball bounded by the –sphere in the 4–manifold,which means we push the interior of the 3–ball into the 5–manifold and carve out itsneighborhood. The B bounded by the dotted S intersects the B bounded by the 1–handle’s attaching S transversely once. Likewise, a 5–dimensional 2–handle attachedto the boundary of a 5–manifold in such a way that it could be cancelled (which heremeans that its attaching framed S bounds a framed B ) can be represented by a “dottedcircle” together with a choice of –disk bounded by the circle in the 4–manifold. Herethe B bounded by the dotted circle intersects the B bounded by the 2–handle’sattaching circle transversely once.We emphasize in both these 5–dimensional cases that, using “dotted” notation, inprinciple we need to know what the disks/balls are which we are going to push in andcarve out, with the 4–dimensional case being perhaps exceptional because an unknotin a S bounds a unique disk. Budney and Gabai have shown [2] that unknotted2–spheres in S can bound “knotted” 3–balls, and of course, although all S ’s in S are unknotted, 2–knots can be tied into any spanning disk for such an S . What reallymatters is whether these spanning disks and balls are isotopic in dimension 5, and weleave this as an interesting question; the Budney-Gabai examples are in fact isotopic in B , but there might in principle be more complicated examples that remain nonisotopiceven when pushed into B . However, here we will bypass this issue by working with“dotted disks” instead of “dotted circles” and “dotted balls” instead of “dotted spheres”.Now, given our handle attaching data α t and β t used to build Z , since the α t ’s areinvariant in t , and α t = α bounds a fixed collection of framed disks δ , we caninstead represent the 2–handles by a (for now, t –invariant) t –parametrized familyof n dotted disks α • t : (cid:113) n B (cid:44) → S . Note that α • t is not an extension of α t , butis rather an embedding of dual disks to the fixed disks bounded by α t . Also, weinsist on maintaining the subscript t even though these are t –invariant because wewill shortly modify the family so as to lose t –invariance. Now the instructions forbuilding the cobordism Z are to build each Y t from [0 , × S by pushing the interiorof each of these disks from { } × S into the interior of [0 , × S and removing their David T Gay neighborhoods, and then attaching 3–handles along β t . Note that, after carving out thedisks but before attaching the 3–handles, the upper boundary of this cobordism Y t isthe surgered 4–manifold S ( ∂α • t ). In other words, when looking at the 4–dimensionalboundary, we cannot tell whether we carved out the dotted disks or attached 2–handlesalong their boundaries, because the resulting surgeries are the same.The crucial point here is that, because each β t is disjoint from (cid:113) n ( S × { p (cid:48) } ), we canisotope the family β t so that it never goes over the surgered region of S ( ∂α • t ), andthus the entire handle attaching data now lives in S . Thus we can now describe each Y t , and thus Z , via data entirely lying in S , i.e. α • t (cid:113) β t : ( (cid:113) n B ) (cid:113) ( (cid:113) n S ) → S .This is not an embedding but is an embedding when restricted to (cid:113) n B and to (cid:113) n S ,and the only intersections occur between (cid:113) n S and the interiors of the disks (cid:113) n B . Attimes t = t =
1, each S intersects its corresponding B transversely once andis disjoint from all the other B ’s, i.e. the spheres and disks are in “cancelling position”Our goal is now to “switch the dots from the circles to the spheres”, i.e. to think of β t as being dotted spheres, thus corresponding to 5–dimensional 1–handles, and tothink of ∂α • t as attaching circles for 2–handles, rather than dotted circles describing2–handles. However, as discussed above, before we can “put dots on” β t we need toextend them to be embeddings of balls not just of spheres. In other words, we wantto extend β t : (cid:113) n S (cid:44) → S to β • t : (cid:113) n B (cid:44) → S . Unfortunately, if we could do thiswithout moving β t and also achieving the property that β • = β • , we would then haveshown that β t was trivial in π ( Emb ( (cid:113) n S , S )), and we do not (at least this authordoes not) know enough about π ( Emb ( (cid:113) n S , S )) to make such an assertion. So weproceed carefully, and in fact we will end up modifying both α • t and β t , abandoning α • = α • and β = β , but maintaining the property that α • and β are in cancellingposition.Using the isotopy extension theorem we can easily extend β t to β • t : (cid:113) n B (cid:44) → S (pick B ’s bounded by β and move them around by an ambient isotopy for β t ) butwe should not expect that β • = β • . We can, however, assume that each component of ∂α • transversely intersects the corresponding component of β • at one fixed point inthe interior of each B , since β and ∂α • are meridians of each other. Thus we knowthat ∂α • and β • are now in cancelling position, and we still have that α • and ∂β • arein cancelling position. Now assume that β • t = β • and α • t = α • for t ∈ [1 − (cid:15), β • − (cid:15) back to β • (one could use the original forward isotopyin reverse, or any other isotopy) which can be extended to an ambient isotopy and thenused on the interval [1 − (cid:15), − (cid:15)/
2] to move both β • t and α • t , discarding completely thegiven β • t and α • t on [1 − (cid:15), ∂β • t iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins and α • t and the pair β • t and ∂α • t are in cancelling position. Now, when t = − (cid:15)/ β • t = β • but the only thing we know about ∂α • t is that it is in cancellingposition with respect to β • t = β • , and also that ∂α • is also in cancelling position withrespect to β • t = β • . But now we can use Gabai’s 4–dimensional lightbulb theoremto isotope ∂α • − (cid:15)/ to α • remaining in cancelling position, and use this isotopy to get ∂α • t for t ∈ [1 − (cid:15)/ ,
1. Note that we have lost track of, but do not really care about,the embeddings α • t for t ∈ [1 − (cid:15)/ , Z built as afamily of cobordisms Y t , we first cancel the 2– and 3–handles for all t ∈ [1 − (cid:15), α ’s to the β ’s for all t ∈ [0 , − (cid:15) ], thus getting a newcobordism Z (cid:48) from S × S to S × φ S . This Z (cid:48) is built from cobordisms Y (cid:48) t whichstart with [0 , × S , then experience the birth of n t , with 1–handles described by the dotted balls β • t and the 2–handles attached alongthe circles ∂α • t , and all end up cancelling at time t = − (cid:15) . Then the cancellationcan be postponed until t = − (cid:15)/ ∂α • t to return to their startingpositions. Finally, wrapping from t = − (cid:15)/ t = ∼
0, we can merge thedeaths and births so that in the end the cobordisms Y t are described by the loop of maps ∂α • t (cid:113) β • t : ( (cid:113) n S ) (cid:113) ( (cid:113) n B ) → S , with β • t being dotted balls describing 1–handlesand ∂α • t being attaching circles for 2–handles.Finally, a further ambient isotopy can now be used to arrange that β • t is independentof t , and thus the entire construction is governed by the loop of framed circles ∂α • t . – pairs to a single – pair We now know that diffeomorphisms of S that can be realized by a family of cobordismsbuilt with n n (cid:113) n S into n ( S × S ) lies in (cid:99) S n . Before we get to Montesinos twins, we need now to showthat every diffeomorphism of S that can be a realized by a family of cobordisms builtwith n Theorem 12
For any n , F H n ( π ( F S n ( S , = F H ( π ( F S ( S , . David T Gay
Proof
We begin again with a cobordism Z from S × S to S × φ S built as afamily of cobordisms Y t , each Y t built by attaching n fixed standard 1–handles to[0 , × S followed by n “moving” 2–handles governed by a loop of embeddings α t : (cid:113) n ( S × B ) (cid:44) → n ( S × S ). Cancelling the 1–2 pairs at time t = ∼
1, we revertto the Cerf theoretic perspective to get a family ( f t , V t ) of Morse functions with gradient-like vector fields on [0 , × S interpolating from f , which is projection onto [0 , f , which is the pullback of f via some pseudoisotopy Φ : [0 , × S → [0 , × S from id S to φ . The graphic now looks like Figure 5, exactly as in Figure 4 except thatnow the critical points are of index 1 and 2; there are still no handle slides.Theorem 2.1.1 of Chenciner’s thesis [4], restated as Hatcher and Wagoner’s Proposi-tion 1.4 on p.177 of [5], asserts that, given a 1–parameter family f t of Morse functionson [0 , × X where X is an m –manifold, if the Cerf graphic contains a swallowtailinvolving critical points of index i and i + i ≤ m − m = i = = −
3. We use this to reduce thenumber of 1–2 pairs using the main idea of Proposition 4 on p.217 of [5], as in thefigure on the top of p.218 of [5]. We essentially reproduce this figure here in Figure 7which shows how to reduce a nested pair of birth-deaths of 1–2 handles to a singlepair. (The other elementary moves are introducing a swallowtail, which can alwaysbe done, and merging a death with a birth, which can always be done if level sets areconnected, which they are in our case.)Repeating this we can turn n nested 1–2 “eyes” into a single nested 1–2 “eye”, and iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins iii + i + i + David T Gay then we can merge the birth again at t = ∼
1. Note that in fact we could have left thislast (bottom-most) index 1 critical point completely unchanged in this whole process,we even did not need to cancel it with its cancelling 2–handle at the beginning. Thuswe can easily arrange that this last 1–handle is still stationary, i.e. its attaching mapdoes not move with t . This shows that this cobordism can be built with a single fixedstandard 1–handle followed by a single moving 2–handle whose attaching map is givenby a loop of embeddings S × B (cid:44) → S × S . Therefore [ φ ] ∈ F H ( π ( F S ( S , Remark 13
In case it is needed in another context, the general version of this theoremis that, if i ≤ m − X is an m –manifold, then for any n , F H n ( π ( F S n ( X , i ))) = F H ( π ( F S ( X , i ))). As a consequence of the preceding two theorems we now know that any diffeomorphismof S arising as the monodromy of the top of a cobordism constructed as above froma loop of n n ( S × S ) which remain disjoint from a parallel copy (cid:113) n ( S ×{ p (cid:48) } ) of the basepoint embedding (cid:113) n ( S ×{ p } ) is isotopic to a diffeomorphismarising from a loop of embeddings of a single circle in S × S . This is summarized as: Corollary 14 H ∞ ( ı ∗ ( π ( (cid:98) S ∞ ))) = F H ( π ( F S ( S , F H ( π ( F S ( S , Proof of Theorem 4
Recall that
F S ( S , S in S × S while S ( S , = Emb ( S , S × S ) is the space of unframed embed-dings of S in S × S . As noted at the end of Section 2, we might worry that thehomomorphism π ( F S ( S , → π ( S ( S , π ( Emb ( S , S × S )) all of which can be seento lift to framed loops of embeddings, and thus the map is surjective so we do not needto worry about framings anymore. (There is probably some other more direct way to iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins see this, the point being that there is no loop of embeddings of S in S × S whichswitches the two framings of S .)For the remainder of this proof, we will use the less obscure notation Emb ( S , S × S )to refer to the space S ( S , π , the puncture is irrelevantsince loops of circles are 2–dimensional while homotopies of loops of circles are 3–dimensional, so since the ambient space is 4–dimensional everything can be assumedto miss a point.In fact [2] shows that every class g ∈ π ( Emb ( S , S × S )) can be represented by a loop γ t : S (cid:44) → S × S of embeddings such that the associated map Γ : S × S → S × S given by Γ ( t , s ) = γ t ( s ) is itself an embedding. Thus we have an embedded torus Γ : S × S (cid:44) → S × S such that Γ ( { }× S ) is the basepoint embedding C = S ×{ p } .Surgery along C applied to the pair ( S × S , Γ ) yields ( S , R ) for some embedded2–sphere R ⊂ S , and the 2–sphere S dual to the surgery circle is an unknottedsphere S ⊂ S such that ( R , S ) forms a Montesinos twin. Furthermore, the boundary ∂ν ( R ∪ S ) of a neighborhood of this twin in S is the same as the boundary of atubular neighborhood of Γ ( S × S ) in S × S . When this 3–torus is parametrized as S l × S R × S S as in the introduction, we see that the S l parameter corresponds to the t parameter in Γ ( t , s ) = γ t ( s ), that the S R direction corresponds to the s –parameter,and that the S S direction corresponds to the boundary of the disk factor in the tubularneighborhood ν ( Γ ( S × S )) ∼ = D × S × S .Because Γ is embedded, it is relatively easy to see what H ([ γ t ]) looks like. We needan ambient isotopy φ t of S × S with φ = id, φ t ◦ γ = γ t and φ equal to theidentity on a neighborhood of C . (This is the “circle pushing” map we get by draggingthe circle around the embedded torus and back to its starting position.) This can happenentirely in a tubular neighborhood D × S × S of Γ , by spinning in the t directionmore and more as we move towards the center of D , which we state explicitly asfollows: Let ( r , θ ) be polar coordinates on D , and let ( t , s ) be coordinates as beforeon S × S . Choose a smooth non-increasing function T : [0 , → [0 , π ] which is1 on [0 , / / , φ t ( r , θ, t , s ) = ( r , θ, t + T ( r ) , s ). From this it isclear that φ is the identity on r ∈ [0 , /
4] and r ∈ [3 / , / , / × S × S × S is equal to a Dehn twist on [1 / , / × S crossed withthe identity in the remaining S × S direction. Back in S this is exactly the twist τ W along the twin W = ( R , S ). David T GayC
Figure 8: The embedded torus T (3) in S × S , the obvious next member of the family of toridescribed in Figure 4 of [2]. The top is glued to the bottom, and horizontal slices are S ’s, withthe “time” coordinate indicated in red/blue shading, as in Figure 1. Here we have exaggeratedcertain features of this torus and deformed somewhat from the drawings in Figure 4 of [2] sothat the connection with the Montesinos twin W (3) = ( R (3) , S ) in Figure 1 is visually apparent.Surgering along the red circle C collapses the vertical cylinder on the right into a ball (the tailof the snake), with the dual sphere to C becoming the tail-piercing sphere S . In fact [2] establishes an isomorphism W × W : π ( Emb ( S , S × S ) , C ) → Z ⊕ Λ where Λ is a free abelian group on a countably infinite generating set. The Z factorin Z ⊕ Λ is given by the loops of S –reparametrizing embeddings γ t ( s ) = γ ( s + nt ),and it is easy to see that H applied to such a loop of embeddings is isotopic to id S ,i.e. this Z factor is in the kernel of H . Modulo this Z factor, Figure 4 in [2] givesthe first two tori T (1) and T (2) in an obvious family T ( i ) of tori in S × S which givethe countably infinite generating set corresponding to Λ . We draw T (3) in Figure 8.In this figure, the circle C ⊂ S × S is represented as a red vertical line on the farright side of the torus. The torus T ( n ) is just like this but wraps n times around the S direction. Surgering along C yields our Montesinos twins W ( i ) = ( R ( i ) , S ). iffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins References [1] Ryan Budney. A family of embedding spaces. In
Groups, homotopy and configura-tion spaces , volume 13 of
Geom. Topol. Monogr. , pages 41–83. Geom. Topol. Publ.,Coventry, 2008.[2] Ryan Budney and David Gabai. Knotted 3-balls in S , 2019. arXiv:1912.09029.[3] Jean Cerf. La stratification naturelle des espaces de fonctions diff´erentiables r´eelles etle th´eor`eme de la pseudo-isotopie. Inst. Hautes ´Etudes Sci. Publ. Math. , (39):5–173,1970.[4] Alain Chenciner.
Sur la ge´eom´etrie des strates de petites codimensions de l’espace desfonctions diff´erentiables r´elles sur une vari´et´e . PhD thesis, 1971.[5] Allen Hatcher and John Wagoner.
Pseudo-isotopies of compact manifolds . Soci´et´eMath´ematique de France, Paris, 1973. With English and French prefaces, Ast´erisque,No. 6.[6] Michel A. Kervaire and John W. Milnor. Groups of homotopy spheres. I.
Ann. of Math.(2) , 77:504–537, 1963.[7] J. P. Levine. Lectures on groups of homotopy spheres. In
Algebraic and geometrictopology (New Brunswick, N.J., 1983) , volume 1126 of
Lecture Notes in Math. , pages62–95. Springer, Berlin, 1985.[8] Jos´e M. Montesinos. On twins in the four-sphere. I.
Quart. J. Math. Oxford Ser. (2) ,34(134):171–199, 1983.[9] Richard S. Palais. Local triviality of the restriction map for embeddings.
Comment.Math. Helv. , 34:305–312, 1960.[10] Tadayuki Watanabe. Some exotic nontrivial elements of the rational homotopy groupsof Diff( S4