On non-orientable surfaces in 4-manifolds
OON NON-ORIENTABLE SURFACES IN 4-MANIFOLDS
DAVID AUCKLY, RUSTAM SADYKOV
Abstract.
We find conditions under which a non-orientable closed surface F em-bedded into an orientable closed 4-manifold X can be represented by a connectedsum of an embedded closed surface in X and an unknotted projective plane in a4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem andthe Auckly-Kim-Melvin-Ruberman-Schwartz “one is enough” theorem to the caseof non-orientable surfaces. Introduction
The goal of the present note is to determine conditions under which a non-orientableclosed surface S embedded into a closed 4-manifold X admits a splitting into theconnected sum of an embedded surface S (cid:48) and an unknotted projective plane P ina 4-sphere, i.e., there exists a diffeomorphism of pairs( X, S ) ≈ ( X, S (cid:48) ) S , P ) . (1)The important ingredient for the existence of the splitting is the existence of anembedded transverse sphere for S . We say that a sphere G embedded into X is anembedded transverse sphere for S if the Euler normal number of G is trivial and G intersects S transversally at a unique point. The surface S is G -inessential if theinduced homomorphism π ( S \ G ) → π ( X \ G ) is trivial. When there is such atransverse sphere, any P summand in S can be split off. Theorem 1.
Let S be a connected non-orientable closed surface in a closed orientable -manifold X . Suppose that S is G -inessential for a transverse sphere G . Let P bea projective plane summand of S . Then the pair ( X, S ) splits as in (1) with P unknotted, and with the surface S (cid:48) still G -inessential for the transverse sphere G .Remark . Let M ⊂ S ⊂ S be the standard M¨obius band. The boundary of M canbe pushed radially into the upper hemisphere of S where it bounds a unique disc D up to isotopy. The union of the M¨obius band and the disc D is an embedded,unknotted projective plane P in S . Depending on the sign of the half-twist of theM¨obius band, there are two non-isotopic unknotted projective planes, P and P − .These can be detected by one of two invariants: the normal Euler number, or theBrown invariant, see section 2. Remark . If the Euler characteristics of S is odd, then we may choose the projectiveplane P in S so that the surface S (cid:48) in the splitting (1) is orientable. When the Eulercharacteristics of S is even, we may split off two unknotted projective planes leavingan orientable surface. It also follows that there is a diffeomorphism of pairs( X, S ) ≈ ( X, S ) k ( S , P ) (cid:96) ( S , P − ) Date : August 28, 2018. a r X i v : . [ m a t h . G T ] A ug DAVID AUCKLY, RUSTAM SADYKOV where S is a 2-sphere embedded into X , and k + (cid:96) is the cross-cap number of S .As a consequence of the splitting theorem (Theorem 1) we show that a versionof the recent Gabai light bulb theorem (Theorem 8) holds true for non-orientablesurfaces as well. Theorem 2.
Let X be an orientable -manifold such that π ( X ) has no -torsion.Let S and S be two homotopic embedded G -inessential closed surfaces with commontransverse sphere G . Suppose that the normal Euler numbers of S and S agree.Then the surfaces are ambiently isotopic via an isotopy that fixes the transverse spherepointwise. In the orientable case considered by Gabai the normal Euler number does not playa role. However, this invariant is critical when the surfaces are non-orientable, seeRemark 3. Theorem 2 has a number of applications.One source of applications is to stabilization of smoothly knotted surfaces in 4-manifolds. If (
X, S ) is a pair consisting of a 4-manifold and an embedded surface,one has four types of stabilization: external stabilization (
X, S ) S × S , ∅ ), pairwisestabilization ( X, S ) S × S , { pt } × S ), internal stabilization ( X, S ) S , T ) andnon-orintable internal stabilization ( X, S ) S , P ). Baykur and Sunukjian provedthat a sufficient number of internal stabilizations results in isotopic surfaces [2]. In [8],S. Kamada shows that two embedded surfaces become isotopic after enough internalnon-orientable stabilizations.The hypothesis that S is G -inessential for a transverse sphere G is always satisfiedafter taking the connected sum of the pair ( X, S ) with ( S × S , {∗} × S ) whichimmediately implies the following: Corollary 3.
When π ( X ) has no -torsion, regularly homotopic embedded surfacesin X become isotopic after just one pairwise stabilizatoin. Quinn [16] and Perron [14, 15] show that topologically isotopic surfaces in simply-connected 4-manifolds become isotopic after sufficiently many external stabilizations.Auckly, Kim, Melvin and Ruberman proved that just one external stabilization wasenough for ordinary topologically isotopic orientable surfaces [1]. A second conse-quence of the splitting theorem is a non-orientable version of this “one is enough”theorem.
Theorem 4.
Let S and S be regularly homotopic (possibly non-orientable), em-bedded surfaces in a -manifold X , each with simply-connected complement. If thehomology class [ S ] = [ S ] in H ( X ; Z ) is ordinary, then S is isotopic to S in X S × S ) . If the homology class is characteristic, then the surfaces are isotopicin X S ˜ × S ) . In § § G is essential. In § S S (cid:48) in the pair ( X, S ) X (cid:48) , S (cid:48) ) may changewhen S and S (cid:48) are changed by isotopy. In contrast, in § S S (cid:48) is well defined when X (cid:48) is a sphere, see Lemma 6. Lemma 6 is essentialfor the proof of the Gabai theorem for non-orientable surfaces (Theorem 2). Another preliminary statement is proved in § § § Background A pair of manifolds ( X, S ) is a manifold X together with an embedded submanifold S . The connected sum of pairs [10] is denoted by( X , S ) X , S ) = ( X X , S S ) . Given a possibly non-orientable surface S in an oriented 4-manifold X , one maydefine the self-intersection (or, normal Euler) number e ( S ). Take a small isotopicdisplacement ˜ S of S in the normal directions and count the algebraic number ofintersection points in S ∩ ˜ S . The sign of an intersection point p is positive (respective,negative) if ( e , e , ˜ e , ˜ e ) is positively (respectively, negatively) oriented, where e , e is an arbitrary basis of the tangent space T p S and ˜ e and ˜ e the image of e and e in T p ˜ S . Remark . The normal Euler number is well defined up to regular homotopy, i.e.,homotopy through immersions.
Figure 1.
The positive un-knotted projective plane P with push-off. Figure 2.
The normal Eu-ler number of P is −
2. Notshown is the vector ˜ e di-rected into the interior of thelower half space R − .The positive unknotted projective plane P in R ⊂ S is obtained by cappingoff the gray right-handed M¨obius band in Figure 1 with a disc D in the upper halfspace R = [0 , ∞ ) × R . There is a displacement ˜ D of D in R that has an emptyintersection with D . It is bounded by the red curve in Figure 1. This curve haszero linking number with the boundary of the M¨obius band. The red curve may beextended to the lower half space R − and then capped with a red M¨obius band toobtain a displacement ˜ P . The only points of intersection are the two green points inFigure 1 and 2. Orienting the tangent space of the grey M¨obius band at a green pointby vectors e and e and taking displaced vectors ˜ e and ˜ e in the tangent space of thered M¨obius band, we can see that the two intersection points are counted negatively.Therefore, the normal Euler number of P is negative two. The negative unknotted DAVID AUCKLY, RUSTAM SADYKOV projective plane P − is obtained by capping off a left-handed M¨obius band with a discin the upper half space R . Its normal Euler number is 2. Remark . Let r be the linear transformation of R given by ( t, x, y, z ) (cid:55)→ ( − t, − x, y, z ).It takes the projective plane obtained by capping off the right-handed M¨obius bandwith a disk in the upper half-space to the projective plane obtained by capping off aleft-handed M¨obius band with a disc in the lower half space R − . Remark . There is a different invariant defined when S is a characteristic (possi-bly non-orientable) closed surface in a closed 4-manifold X . It is called the Browninvariant [3]. The Brown invariant will not play a role in this paper.3. The splitting theorem
In this section we prove the main splitting theorem. We begin with a simpleobservation that any splitting is determined by a special disk.
Lemma 5.
Let S be a closed surface embedded into a closed manifold X . Supposethat there is an open -disc U ⊂ X such that the intersection U ∩ S is a M¨obius band M and ∂U is an embedded submanifold of X intersecting S transversally. Supposethat ∂M is an unknot in ∂U ≈ S . Then ( X, S ) is diffeomorphic to the connectedsum of pairs of manifolds ( X, S (cid:48) ) and ( S , P ± ) where S (cid:48) is a surface obtained from S \ U by capping off its only boundary component with a -disc.Proof. Since ∂M is an unknot in ∂U , the boundary of each of the pairs ( X \ U, S \ U )and ( ¯ U , ¯ U ∩ S ) is diffeomorphic to the boundary of the pair ( D , D ) of standarddiscs. In other words, the boundary of each of the two pairs can be capped off by thepair of standard discs to produce pairs ( X, S (cid:48) ) and ( S , P ± ) whose connected sum isdiffeomorphic to ( X, S ). (cid:3) In practice, the open set U in Lemma 5 is constructed by taking a neighborhoodof a 2-disc D such that ∂D is the central closed curve of the M¨obius band M , theinterior of D does not contain points of S and D is nowhere tangent to S . If such adisc D exists, then we say that D is the core of the splitting of Lemma 5.Suppose a connected surface S possesses a transverse sphere G . Given anothersurface S (cid:48) ⊂ X , an intersection point p ∈ S ∩ S (cid:48) can be tubed off using G along apath γ in S from the point p to G (cid:48) ∩ S , where G (cid:48) is a parallel copy of G , see Figure 3.The result of this procedure is a new surface S (cid:48)(cid:48) obtained from S (cid:48) by taking the unionof S (cid:48) and a copy G (cid:48) , removing a disc neighborhood D G of G (cid:48) ∩ S in G (cid:48) , removing adisc neighborhood D S of p in S (cid:48) , attaching a tube S × [0 ,
1] along γ to the two newboundary components of S (cid:48) \ D S and G (cid:48) \ D G , and smoothing the corners. Proof of Theorem 1.
Let α be a simple closed orientation reversing curve in S thatis disjoint from G .Since S is G -inessential, the curve α bounds an immersed disc D in X \ G . We mayjoin any self intersection point p of D with a point in ∂D by a curve and use a fingermove to eliminate the self intersection point p of D , see Figure 4. By repeating thisprocedure we obtain an embedded disc D ⊂ X \ G that may intersect S in interiorpoints. We may then use the transverse sphere G to tube off the intersection pointsof D with S , see Figure 3. Thus there exists an embedded 2-disc D in X \ G such Figure 3.
Using a trans-verse sphere to tube off anintersection point.
Figure 4.
Using a fingermove to remove an intersec-tion point.that the intersection D ∩ S is the curve α . Furthermore, we may assume that D approaches S orthogonally (with respect to a Riemannian metric on X ). Figure 5.
A neighborhoodof the surface F , dual disc G and an embedded disc D . Figure 6.
A twist of the 1-handleA neighborhood S × D of α in X is diffeomorphic to the complement in D of aneighborhood of D . This is depicted by a dotted circle representing the boundary ofthe disc D , see Figure 5. Such a neighborhood of α already contains the M¨obius bandneighborhood of α in S . With respect to a trivialization of S × D , it twists k + 1 / k . The rest of the neighborhood of S in X is obtained from thedescribed neighborhood of α by attaching 1-handles that correspond to thickeningsof 1-handles of S and one 2-handle h S which corresponds to the thickening of the2-cell of S of cell decomposition of S corresponding to a perfect Morse function. Theneighborhood of a transverse sphere contributes a 2-handle attached along a meridianof S with zero framing. The neighborhood of the disc D also contributes a 2-handle h D attached along a circle that passes over the 1-handle h once and which, a priori ,could be linked with the attaching circle of h S . Using the transverse sphere G , theattaching circle of h D can be unlinked from the attaching circle of h S . We denote by m and n the framings of the attaching circles of the 2-handles h D and h S respectively. DAVID AUCKLY, RUSTAM SADYKOV
Figure 7.
Sliding the 2-handle h D over the 1-handle links the attach-ing sphere h D with the attaching sphere h S and changes the framingby ± h D with h S as well as decreasing simultaneously m and k by 1 and n by 4, see Figure 6. (A right rotation has the opposite effect.) Thus,we may assume that k is 0 or −
1, corresponding to ± / m is even. If k = 0 (respectively, k = −
1) and m is odd, then a full − m and k = − h D along h , links the attaching circles of h D and h S and changes the framing m by ±
2, see Figure 7. In view of the transverse sphere, wemay again unlink the attaching circles h D from h S . To summarize, we may assumethat in Figure 5, the twisting number k is 0 or −
1, and the framing m is 0.Theorem 1 now follows from Lemma 5. (cid:3) Problems with splitting and sums
In the absence of a transverse sphere the splitting surgery along arbitrary 1-sidedcurves may not be possible, see Example 1.
Example . According to the Massey theorem, there exists an embedding of a closednon-orientable surface S ⊂ S of Euler characteristics χ with normal Euler number ν = 2 χ − , χ, ..., − χ . We may choose an embedding so that ν (cid:54) = ±
2. Suppose thatthere exists a splitting surgery representing ( S , S ) by a connected sum of ( S , S (cid:48) )and ( S , P ± ) where S (cid:48) is a closed orientable surface. Since the normal Euler numberof P ± is ∓ S istrivial, we deduce that the normal Euler number of their connected sum S is ± ν (cid:54) = ±
2. Therefore, such a splitting surgerydoes not exist.In fact it may be the case that no splitting is possible as in the following example.
Example . The rational elliptic surface has a cusp fiber F . In Kirby calculus, aneighborhood of this fiber is obtained by attaching a 0-framed 2-handle to D alonga right-handed trefoil. The right handed trefoil bounds an obvious M¨obius band.Capping the M¨obius band with the core of the 2-handle results in an embedded P representing the fiber class [ F ]. Notice that this class is characteristic. A splitting ofthe form ( E (1) , P ) ∼ = ( E (1) , S ) S , P )would imply the existence of a smoothly embedded sphere representing the fiber class[ F ] in E (1) contradicting the Kervaire-Milnor theorem. When splitting is possible, it need not be unique. Indeed changing the homotopyclass of the core of the splitting disk can change the homology class of the summands.
Example . Suppose that D is a core of splitting in a pair ( X, S ). Suppose that thesplitting results in the decomposition(
X, S ) (cid:39) ( X, S (cid:48) D ) S , P ± ) . Let A be an embedded sphere in X \ ( S ∪ D ) with trivial normal bundle. Considerthe splitting with the core D A ,( X, S ) (cid:39) ( X, S (cid:48) D A ) S , P ± ) . Then [ S (cid:48) D A ] = [ S (cid:48) D ] + 2[ A ], where the orientation of A agrees with the orientation of D A .In the last example, the Z homology classes of the surfaces S (cid:48) D A and S (cid:48) D in thedecomposition agree. However, the integral homology classes of the surfaces are notthe same. Thus, Example 3 shows that the connected sum decomposition of pairsis not unique. This should not be surprising as the connected sum decomposition ofmanifolds is not unique in 4-dimensions. We now give one more example where bychanging the core of the splitting disk we are able to change from splitting off a copyof P + to splitting off a copy of P − . Example . The projective plane in (
X, S × { } ) S , P + ) has transverse sphere {∗} × S in the connected factor S × S , where X = C P S × S ). We claim thatwe can split off either P + or P − . Indeed, the P + -splitting is obvious. To describe the P − -splitting, let D denote the core of the P + -splitting. We note that this correspondsto k = 0 and m = 0 in Figure 5 using the trivialization of D . Replacing D with itsconnected sum with C P ⊂ C P , results in a model corresponding to k = 0 and m = 1, see Figure 8 where the 2-handle corresponding to C P is denoted by u . Toview the neighborhood in the trivialization of the new disk we apply a twist to the1-handle. This model corresponds to k = − m = 0, see Figure 9. Now wemay slide the handle D + u along the 2-handle − G twice to obtain the core for a P − -summand. In other words,( X, S × { } ) S , P + ) (cid:39) ( X, S (cid:48) ) S , P − ) . The normal Euler number of P − is 2, while the normal Euler number of P + is − S (cid:48) is −
4. In fact, [ S (cid:48) ] = [ S × {∗} ] +2[ C P ] − {∗} × S ] in the homology group H ( C P S × S ).The connected sum of two manifolds X and Y is defined by removing coordinateopen balls D X ⊂ X and D Y ⊂ Y , and then identifying the new boundaries in X \ D X and Y \ D Y appropriately.The connected sum of pairs of manifolds is defined similarly by means of pairsof balls. We say that ( D X , D S ) ⊂ ( X, S ) is a pair of coordinate open balls if ∂D X intersects S along ∂D S and the pair ( D X , D S ) is parametrized by a diffeomorphismfrom the standard pair of unit balls in ( R , R ). The connected sum of pairs ( X, S )and (
Y, R ) is defined by removing pairs of coordinate open discs ( D X , D S ) ⊂ ( X, S )and ( D Y , D R ) ⊂ ( Y, R ) and then identifying the new boundaries appropriately.In this paper we are interested in internal sums, which are defined by taking theconnected sum with (
Y, R ) where R is a surface in Y = S . In this case, without loss DAVID AUCKLY, RUSTAM SADYKOV
Figure 8.
The handle D + u of the connected sum ofthe disc D with C P . Figure 9.
A twist of the 1-handleof generality we may assume that D Y is the lower hemisphere, and identify Y \ D Y with a closed coordinate ball. Then X S is canonically diffeomorphic to the originalmanifold X as the connected sum operation replaces the coordinate open ball D X with the interior of the coordinate ball S \ D Y . We will write S i R for the resultedsurface in X = X S when i : D X → X is a specified inclusion and D Y is an openlower hemisphere in Y = S .To motivate Lemma 6, we note that in the case of connected sums of pairs ofmanifolds there is an additional subtelety. Namely, let S and S be two isotopicsurfaces in X that agree in a coordinate open ball D X . Furthermore, suppose that( X, S ) and ( X, S ) share the same pair of coordinate balls ( D X , D S ). Then theambient space X Y in the pair ( X, S ) Y, R ) coincides with the ambient space inthe pair (
X, S ) Y, R ). However, in general, the surface S R may not be isotopicto S R in X Y , as the isotopy of X \ { } , where { } is the center of the coordinateball D X , may not admit an extension to an isotopy of X Y . Given a pair (
X, S ), we say that a tuple of vectors v , ..., v at a point x ∈ S is anadapted frame if it is a basis for the tangent space T x X and if the vectors v and v form a basis for the tangent space T x S . If X and S are oriented, then we additionallyrequire that the basis { v , ..., v } for T x X and { v , v } for T x S are positively oriented.We note that up to isotopy the pair of the coordinate open balls ( D X , D S ) in ( X, S )determines and is determined by the standard coordinate adopted frame e , ..., e in T D X ⊂ T X . Lemma 6.
Let S and S be connected isotopic surfaces in an oriented connectedclosed -manifold X , and R be a surface in S . Let i and i be possibly differentorientation preserving embeddings of an open coordinate -ball D into X such that ( i k D , i k D ) is a pair of coordinate open discs in ( X, S k ) for k = 0 , . In the casewhere S and S are oriented, suppose that the isotopy from S to S is orientationpreserving, and the frames associated with the pairs ( i k D , i k D ) of coordinate ballsare adapted. Then the surface S i R is isotopic to the surface S i R in X .Proof. The ambient isotopy taking S to S takes the connected sum S i R to S j R for some embedding j : D → X . Let { v i } and { w i } denote the frames over( X, S ) corresponding to the embeddings j and i respectively. Since the framesare adapted, by applying an ambient isotopy of X that fixes S setwise, we mayassume that v = w and v = w . Since both { v i } and { w i } are positively orientedframes over X , the coincidences v = w and v = w imply that there is an ambientisotopy of X that fixes S setwise and takes the frame { v i } to the frame { w i } . Thus,the surface S i R is isotopic to the surface S R . This completes the proof ofLemma 6. (cid:3) Regularly homotopic surfaces
At the first step in the Gabai’s proof of the 4-dimensional light bulb theorem [5] onemodifies a given homotopy between orientable surfaces into a regular homotopy. Inthis section we will show that the hypotheses on the surfaces in the non-orientable ver-sion of the Gabai theorem (Theorem 2) guarantee that the (possibly non-orientable)surfaces are still regularly homotopic.By the Smale-Hirsch theorem, the space of immersions of a manifold S into a man-ifold X is weakly homotopy equivalent to the space Imm F ( S, X ) of smooth injectivebundle homomorphisms
T S → T X provided that dim S ≤ dim X or that S is open.There is a natural fibration: Imm F ( S, X ) → C ∞ ( S, X ) , where C ∞ ( S, X ) is the space of smooth maps f : S → X . Its fiber over the pathcomponent of f is the space Γ( T S, f ∗ T X )) of sections of the bundle V ( T S, f ∗ T X ) → S of injective bundle homomorphisms T S → f ∗ T X . When the dimension of S and X is 4, the fiber of the fiber bundle V ( T S, f ∗ T X ) over S is homotopy equivalent to O (4). The following theorem is known. The orientable case is used in [5]. Since weneed the non-orientable case, we give a quick outline of its proof here. Theorem 7.
Suppose that f and g are two homotopic embeddings of a closed con-nected surface S into X . If the surface S is non-orientable, suppose, in addition,that the normal Euler numbers of f and g agree. Then the embeddings f and g areregularly homotopic. Proof.
Choose a handle decomposition of S with a unique 2-cell. By a general positionargument, we may assume that a homotopy of f to g restricts to an isotopy of aneighborhood of the 1-skeleton of S . Furthermore, the isotopy of this neighborhoodextends to an isotopy of the ambient manifold X . Thus, we may assume that f and g agree in the neighborhood of the 1-skeleton of S . Consequently, the normal bundles N ( f ) and N ( g ) agree over the same neighborhood.Since f and g are homotopic, their normal Euler numbers agree when S is ori-entable. In particular, in both cases, when S is orientable or non-orientable, underthe hypotheses of Theorem 7, the normal bundle of the immersion f is isomorphicto that of g . Even more is true, the isomorphism already given over the 1-skeletonextends to an isomorphism N ( f ) ≈ N ( g ). Let ˆ f denote the inclusion of N = N ( f )into X , and ˆ g : N → X denote the composition of the isomorhism N ≈ N ( g ) and theinclusion. Since π ( O ) = 0 we can extend the constant path from d ˆ f to d ˆ g definedover the 1-skeleton to a path defined over all of S . It follows that ˆ f is regularlyhomotopic to ˆ g , hence f is regularly homotopic to g . (cid:3) Remark . By a general position argument, one may assume that the regular homo-topy in the conclusion of Theorem 7 restricts to an isotopy away from any disk in thesurface S .6. The Gabai light-bulb theorem for non-orientable surfaces
Recently Gabai proved the following theorem, see [5, Theorem 9.7].
Theorem 8 (Gabai, [5]) . Let X be an orientable -manifold such that π ( X ) hasno -torsion. Two homotopic embedded G -inessential orientable surfaces S and S with common transverse sphere G are ambiently isotopic via an isotopy that fixes thetransverse sphere pointwise. We prove the Gabai result is still true for non-orientable surfaces as well, providedthat the normal Euler numbers of the surfaces agree.
Remark . If a surface S has a transverse sphere, then S is ordinary. Thus the Browninvairant does not play a role in this theorem.The idea of the proof is to use the splitting theorem (Theorem 1) to reduce thegeneral case of possibly non-orientable surfaces S and S to the case of orientablesurfaces by representing S and S as internal connected sums of orientable surfaces S (cid:48) and S (cid:48) with unknotted projective planes. As examples in § S (cid:48) and S (cid:48) may not even be homotopic. Lemma 9 below shows that we mayassume that the surfaces agree away from an open ball. For surfaces meeting theconclusion of Lemma 9, we prove Lemma 10 ensuring that there is a splitting suchthat the surfaces S (cid:48) and S (cid:48) are homotopic. Lemma 9.
Let S and S be regularly homotopic surfaces in a -manifold X . Thenthere exist a surface S in X and a neighborhood U of a point in S such that S agrees with S on the complement of U , and S is regularly homotopic to S by ahomotopy that is constant on the complement of U , and S is isotopic to S .Proof. Let R : I × S → X be a regular homotopy with R ( S ) = S and R ( S ) = S .By Remark 6, we may assume that R restricts to an isotopy in a neighborhood of a S with complement an open disk U . By the isotopy extension theorem,there is an ambient isotopy J : I × X → X that agrees with R restricted to S \ U .Clearly, the surface S := J − ( S ) is isotopic to S , and S agrees with S in thecomplement to U . The required regular homotopy of S to S fixing the complementof U is given by ˆ R t ( x ) := J − t ◦ R t ( x ). (cid:3) Lemma 9 is the first step in the proof of Theorem 2. It establishes an isotopyof S to S so that S and S agree away from a neighborhood of a point. Thefollowing lemma establishes that there are core disks such that the surfaces S (cid:48) and S (cid:48) obtained by splitting off unknotted projective planes from S and S respectivelyare still homotopic. Lemma 10.
Let S be a G -inessential orientable surface with D a core of a splitting.Let S be a surface that agrees with S away from a neighborhood U ⊂ S of a pointsuch that ∂D ∩ U = ∅ . Suppose that S is regularly homotopic to S relative to thecomplement to U . Then there exists a core D of a splitting for S that agrees with D in a neighborhood of ∂D and such that D ∪ D is null-homotopic.Proof. If the intersection of the interior of D and S is empty, then D = D is adesired core of a splitting for S . Otherwise, if the intersection is not empty, then,as in the proof of the splitting theorem, we may tube the intersection points off tocopies of G to remove them. A bit of care is necessary to insure that the resultingdisc D is homotopic to D relative to the boundary. To establish this homotopy,it will suffice to establish that a certain lift of D ∪ D is zero homologous in theuniversal cover of X \ G .By a slight perturbation of D with support in the interior, we may assume that S is transverse to the interior of D . By assumption a neighborhood of G is diffeo-morphic to G × D . Pick a collection { G k } of distinct parallel spheres; one sphere G k for each intersection point in S ∩ D . Since S is G -inessential and G k and D are simply connected, all lift to copies in the universal cover (cid:93) X \ G . Pick referencelifts S , G k and D so that ∂D ⊂ S and G k ∩ S (cid:54) = ∅ . Let S τ , G τk and D τ denotethe translates of these chosen lifts by an element τ of the deck group. The regularhomotopy relative to the complement of U lifts to homotopies of each S τ to a surface S τ .There will now be two cases. The first is when the complement of ∂D in S isorientable, the second is when it is non-orientable. Suppose the complement to ∂D is orientable. By choosing an orientation on S \ ∂D as well as on X , all of thelifts inherit orientations and we may associate a sign to each intersection point. Wenow wish to show that the algebraic count of the intersection points between S and D τ is zero. (By equivariance this will show that the algebraic count of intersectionsbetween S α and D β is zero as well.) There is a problem with the usual intersectiontheory argument because intersection points can run off the boundary of D τ . We fixthis by adding a correction term.Consider what takes place in a regular homotopy between S and S . Such ahomotopy is comprised of isotopy and finger moves introducing oppositely orientedpairs of self intersections of the surface as well as Whitney moves removing pairs ofintersections. By general position these finger moves and Whitney moves may beassumed to take place away from ∂D τ . There will also be finger moves and Whiteny moves introducing and removing pairs of intersections between S and translates S α as well as between the surface S and the disk D τ . Finally, an intersection point (self-intersection point if τ = 1) in S ∩ S τ may slide past ∂D τ . We explore this posibilityin Figure 10. Here we use the vertical red line to represent a portion of D τ and thehorizontal black line to represent a portion of the surface S τ . Both of these extendforward and backward in time. The thimble shape represents a portion of the surface S existing in the present. A positive intersection point (recall the complement of ∂D τ in S is oriented) becomes a negative intersection point and a new intersectionpoint is formed between the surface and the disk. We take our orientation conventionfor the disk so that this point will be counted positively. Figure 10.
Sliding a self intersection point past the boundary of thecore diskLet S t be the surfaces in a regulary homotopy parametrized by t ∈ [0 , t we define the following invariant obtaind as a combination of algebraicintersection numbers ψ ( S t , D τ ) := S t · D τ + 12 S t · S τt . Since S · D τ = S · S τ = 0, we see that ψ ( S , D τ ) = 0. As ψ ( S t , D τ ) is invariantunder the basic moves and is equal to zero at t = 0, we conclude that it is equal tozero at t = 2. Since S is embedded S · S τ = 0 and we conclude that S · D τ = 0.The same may be said for all translates, e.g., S τ · D = 0Tube each intersection point p k ∈ S ∩ D τ to a G τk . The arc that each tube followsprojects to an embedded arc because S is G -inessential. Thus we can tube in theuniversal cover and in the base at the same time. Denote the resulting embeddeddisks D τ in the cover and D in the base. Since the algebraic intersection number of D with each lift S τ is zero, an algebraically trivial number of copies of each G τk isadded. It follows that D ∪ ∂ D is null-homologous and D ∪ ∂ D is null-homotopic.Now turn to the case where the complement of ∂D is non-orientable. The function ψ ( S t , D τ ) is no longer well-defined since a self-intersection point may move around anorientation-reversing loop in the complement of the boundary of the core disk. (Suchan isotopy would change the sign of the intersection point.) However the parity of ψ ( S t , D τ ) is well-defined. We conclude that the parity of each intersection number S τ ∩ D ) is even. The result of tubing an intersection point p k to G τk along onepath γ will change from [ D ∪ D ] + [ G τk ] to [ D ∪ D ] ± [ G τk ] in H ( (cid:93) X \ G ); Z ) whenusing a second path δ depending on whether γ ∪ δ is an orientation preserving ororientation reversing loop. Thus by making the correct path choices we may assumethat D ∪ D is null-homotopic. (cid:3) Proof of Theorem 2.
We first prove the result for surfaces of odd Euler characteristic.By Lemma 9 we may isotope S to agree with S away from a neighborhood of a point,and have a further regular homotopy relative to the complement of this neighborhood taking S to S . Since the Euler characteristic of S is odd there is a simple closedcurve in S having orientable complement. By the splitting theorem (Theorem 1) weknow that there is an embedded disk D with boundary this curve that forms thecore of a splitting of S . By Lemma 10 we also know that there is a core for S sothat D ∪ ∂ D is null-homologous and D ∪ ∂ D is null-homotopic.Let S (cid:48) j denote the result of splitting surgery of S j along D j . One sees that S (cid:48) j ishomotopy equivalent to S j ∪ D j and S ∩ D is homotopic to S ∩ D . If followsthat S (cid:48) and S (cid:48) are orientable surfaces in X satisfying the hypothesis of the light-bulb theorem. They are therefore isotopic. Now ( X, S ) ∼ = ( X, S (cid:48) ) S , P ± ) and( X, S ) ∼ = ( X, S (cid:48) ) S , P ± ) implies that S is isotopic to S as proved in Lemma 6.Turn now to the case where S j is non-orientable with even Euler charasteric. Theproof is similar to the odd Euler characteristic case. The difference is that the com-plement in S of ∂D must be non-orientable. Lemma 10 still generates a disk D forming the core of a splitting so that D is homotopic rel boundary to D . Let S (cid:48) k be the surfaces that result after the splitting. They have odd Euler characteristicand are homotopic. The normal Euler numbers of S and S agree since they areregularly homotopic. It follows that the normal Euler numbers of S (cid:48) and S (cid:48) agree,so they are regularly homotopic. Thus by the odd Euler characteristic case we knowthat S (cid:48) is isotopic to S (cid:48) which using Lemma 6 completes the proof. (cid:3) Isotopy of surfaces in -manifolds. In view of the Gabai’s theorem for non-orientable surfaces (Theorem 2), we are inposition to extend a recent theorem of [1] to the case of non-orientable surfaces.
Proof of Theorem 4.
In the case where S and S are oriented surfaces, Theorem 4is established in [1]. In view of Theorem 2, the proof of Theorem 4 in the non-orientable case follows in the same way as in the orientable case with one minorchange. We provide an abbreviated outline referring to [1] for longer exposition. As X \ S is simply-connected, it contains an immersed disk bounded by a meridian of S .Capping the immersed disc with a fiber of the normal bundle results in an immerseddual Σ to S . If the homology class of S is ordinary, then, after taking the sum withan immersed sphere disjoint form S , we may assume that the self-intersection of Σis even.In contrast to the orientable case, in the present case only the parity of the inter-section number Σ · S is well-defined up to regular homotopy. It follows that Σ · S is odd. We can reduce the geometric intersection number ∩ S ) down to one.Indeed, if there is excess intersection, pick a pair of intersection points. Given anarc in S joining the two intersection points, one may associate a relative sign tothe intersection points via an orientation of a neighborhood of the path. Since S is non-orientable, one may arrange that the relative sign is negative via a suitablechoice of path.The proof now continues as in the orientable case in[1]. Namely, as X \ S is simply-connected, the union of the chosen path in S and a path in Σ bounds an immerseddisk in the complement of S . Via finger moves intersections between the disk and S may be removed. The correct framing may be obtained by boundary twisting along the portion of the boundary of the disk meeting Σ. The result is an immersedWhitney disk. Sliding Σ across this disk removes a pair of intersection points.If follows that one may assume that Σ intersects S and S , each at exactly onepoint. In the ordinary case the self-intersection of Σ is even. It follows that by takingthe connected sum of pairs ( X, Σ) S × S , { pt } × S ) = ( X S × S ) , (cid:101) Σ) andtubing with copies of S × { pt } one may eliminate the self-intersections of (cid:101) Σ andadjust the square to zero. The result now follows from the orientable version of thelight bulb theorem.In the characteristic case the self-intersection number of Σ is odd. Here onetakes the sum ( X, Σ) S (cid:101) × S , zero section) = ( X S (cid:101) × S ) , (cid:101) Σ) to obtain an im-mersed dual with even square. Tubing with copies of the fiber will remove the self-intersections and adjust the framing to zero. (cid:3)
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