Stable existence of incompressible 3-manifolds in 4-manifolds
aa r X i v : . [ m a t h . G T ] A p r STABLE EXISTENCE OF INCOMPRESSIBLE3-MANIFOLDS IN 4-MANIFOLDS
QAYUM KHAN AND GERRIT SMITH
Abstract.
Given a separating embedded connected 3-manifold in a closed 4-manifold, the Seifert–van Kampen theorem implies that the fundamental groupof the 4-manifold is an amalgamated product along the fundamental groupof the 3-manifold. In the other direction, given a closed 4-manifold whosefundamental group admits an injective amalgamated product structure alongthe fundamental group of a 3-manifold, is there a corresponding geometric-topological decomposition of the 4-manifold in a stable sense? We find analgebraic-topological splitting criterion in terms of the orientation classes anduniversal covers. Also, we equivariantly generalize the Lickorish–Wallace the-orem to regular covers. Introduction
In this paper, we examine the correspondence between algebraic topology andthe stable geometric topology of 4-dimensional manifolds. Two 4-manifolds are stably equivalent if they become diffeomorphic after forming the connected sumwith finitely many copies of S × S . Note this does not change the fundamentalgroup, signature, or spin of a 4-manifold, but does change the second Betti number.As in stable homotopy theory, computations are more tractable and still distinguishmany spaces. The Whitney trick, which in higher dimensions allows for mirroringbetween topology and algebra, cannot be used in 4-manifolds since a disc mayintersect itself. By stabilization of the 4-manifold, self-intersections of a disc maybe removed, by a modification called the Norman trick [CS71, 2.1].The Kneser conjecture in 3-manifold topology states that if the fundamentalgroup of a closed 3-manifold X is a free product G − ∗ G + , then X ∼ = X − X + where X ± have fundamental group G ± respectively. This conjecture was proved byJ Stallings in his dissertation (see [Sta71, 1.B.3, 2.B.3]). Later, C Feustel [Feu73]and G A Swarup [Swa73] proved a generalized version of the conjecture when thefundamental group admits an injective amalgamated product along a surface group.Following Hillman’s work [Hil95] on the 4-dimensional version of the Kneser con-jecture, Kreck–L¨uck–Teichner proved the 4-dimensional conjecture is false [KLT95a]but is true if one allows for stabilization [KLT95b]. We investigate the problem ofstably realizing injective amalgamated product decompositions of the fundamentalgroup of a 4-manifold via separating embedded codimension-one submanifolds.1.1. Bistable results.
Our results on stable embeddings vary according to the so-called w -type of the 4-manifold. So we first consider a weaker equivalence relation.We call 4-manifolds bistably diffeomorphic if they become diffeomorphic afterconnecting sum each with finitely many copies of the complex-projective plane C P
21 Q KHAN AND G SMITH (nonspin) and its orientation-reversal C P . For any oriented 4-manifold X , denote X ( r ) := X r ( S × S ) X ( a, b ) := X a ( C P ) b ( C P ) . Stable implies bistable, as ( S × S ) C P ) ≈ C P ) C P ) [Wal64, Cor 1,Lem 1].Given a nonempty connected CW-complex A , by a continuous map u : A −→ B Γ classifying the universal cover e A , we mean the induced map u on fundamentalgroups is an isomorphism, for some basepoints. The map u is uniquely determinedup to homotopy and composition with self-homotopy equivalences Bα : B Γ −→ B Γfor α an automorphism of Γ. By a connected subcomplex being incompressible ,we shall mean that the inclusion induces a monomorphism on fundamental groups. Theorem 1.1.
Let X be a oriented closed smooth 4-manifold. Let c : X −→ BG classify its universal cover. Let X be a connected oriented closed 3-manifold withfundamental group G . Suppose G = G − ∗ G G + with G ⊂ G ± . There exists anincompressible embedding of X in some bistabilization X ( a, b ) inducing the giveninjective amalgamation of fundamental groups, if and only if there exists a map d : X −→ BG classifying its universal cover and satisfying the equation (1.1) d ∗ [ X ] = ∂c ∗ [ X ] ∈ H ( G ; Z ) , with ∂ the boundary in a Mayer–Vietoris sequence in group homology [Bro94, III:6a] . The simplest case of G = 1 was done transparently by J Hillman [Hil95], whosehands-on approach with direct manipulation of handles we generalize in this paper. Corollary 1.2 (Hillman) . Let X be a connected orientable closed smooth 4-manifoldwhose fundamental group is a free product G − ∗ G + . Some bistabilization X ( a, b ) is diffeomorphic to a connected sum X − X + with X ± having fundamental group G ± respectively. Similarly, when G = Z , note X is bistably diffeomorphic to some X − ∪ S × S X + . Proof.
Here G = 1, hence H ( G ) = 0. Take X = S and d the constant map. (cid:3) The proof of Theorem 1.1 generalizes Hillman’s strategy for proving Corollary 1.2and employs an equivariant generalization of the Lickorish–Wallace theorem ( § SO = 0).1.2. Stable results.
Shortly after Hillman’s result, Kreck–L¨uck–Teichner offeredan alternative proof, using Kreck’s machinery of modified surgery theory [Kre99].They were able to replace bistabilization with stabilization, due to a careful analysisof w -types and triviality of 3-plane bundles over embedded 2-spheres in certain 5-dimensional cobordisms [KLT95b]. In general, stabilization is required [KLT95a].Regarding the removal of S × S factors (destabilization), see [HK93] and [Kha17].Recall that a (stable) spin structure s on a smooth oriented manifold M is ahomotopy-commutative diagram (reduction of structure groups in [LM89, II:1.3]): B Spin (cid:15) (cid:15) M τ M / / s ; ; ①①①①①①①① BSO.
NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 3
Theorem 1.3 (totally nonspin) . Let X be a oriented closed smooth 4-manifoldwhose universal cover has no spin structure. Let c : X −→ BG classify this cover.Let X be a connected oriented closed 3-manifold with fundamental group G . Sup-pose G = G − ∗ G G + with G ⊂ G ± . There exists an incompressible embedding of X in some X ( r ) inducing the given injective amalgam of fundamental groups, ifand only if there is d : X −→ BG classifying its universal cover satisfying (1.1) . Any oriented manifold has a spin structure if and only if w of its tangent bundlevanishes [LM89]. So any oriented 3-manifold has a spin structure (as w = v = 0). Theorem 1.4 (spinnable) . Let X be a oriented closed smooth 4-manifold thatadmits some spin structure. Let c : X −→ BG classify the universal cover. Let X be a connected oriented closed 3-manifold with fundamental group G . Suppose G = G − ∗ G G + with G ⊂ G ± . There exists an incompressible embedding of X ina stabilization X ( r ) inducing the given injective amalgam of fundamental groups,if and only if there exist a map d : X −→ BG classifying its universal cover andspin structures s on X and t on X satisfying: (1.2) [ X , t, d ] = ∂ [ X, s, c ] ∈ Ω Spin3 ( BG ) , with ∂ the boundary map in a Mayer–Vietoris sequence in spin bordism [CF64, 5.7] . Observe that (1.2) is a lift of (1.1), via the cobordism-Hurewicz homomorphismΩ
Spin3 ( BG ) epi −−−→ Ω SO ( BG ) iso −−−→ H ( BG ) . Finally, we generalize Theorem 1.4 to only require that e X admits a spin structure.In order to understand the more delicate criterion, we state a lemma and definition. Lemma 1.5.
Let u : Y −→ B Γ classify the universal cover of an oriented connectedsmooth manifold Y . The universal cover e Y admits a spin structure if and only ifthere is a class w u ∈ H (Γ; Z / satisfying the equation w ( T Y ) = u ∗ ( w u ) . Whensuch a class exists it is unique. The secondary characteristic class w u will vanish if Y admits a spin structure.The following definition is rather delicate due to two explicit choices of homotopies.For H : A × [0 , −→ B and a ∈ A , the a -track is H a := ( t H ( a, t )) ∈ B [0 , . Definition 1.6.
Let Y be an oriented connected smooth manifold whose universalcover e Y admits a spin structure. Let u : Y −→ B Γ classify the universal cover. Fixhomotopy representatives τ Y : Y −→ BSO, w : BSO −→ K ( Z / , , w u : B Γ −→ K ( Z / , η from w ◦ τ Y to w u ◦ u . SupposeΓ = Γ − ∗ Γ Γ + with Γ ⊂ Γ ± . Write i : Γ −→ Γ for the inclusion homomorphism.Assume Bi : B Γ −→ B Γ is the inclusion of a bicollared subspace, with u trans-verse to B Γ . If there exists a nulhomotopy θ of the map w u ◦ Bi , then we definethe induced spin structure s θη on the submanifold N := u − ( B Γ ) of Y by s θη : N −→ B Spin ; x (cid:16) τ Y ( x ) , η x ∗ θ u ( x ) (cid:17) , where we identify B Spin with the homotopy fiber of w and ∗ denotes join of paths.We arrive at a generalization of Theorem 1.4 which further requires i ∗ ( w c ) = 0. Theorem 1.7 (pre-spinnable) . Let X be a oriented closed smooth 4-manifold whoseuniversal cover admits a spin structure. Let c : X −→ BG classify this cover. Let X be a connected oriented closed 3-manifold with fundamental group G . Suppose Q KHAN AND G SMITH G = G − ∗ G G + with G ⊂ G ± . There exists an incompressible embedding of X in some X ( r ) inducing the given injective amalgam of fundamental groups, if andonly if there exist a map d : X −→ BG classifying its universal cover and a spinstructure t on X and a nulhomotopy θ of w c ◦ Bi satisfying, with M := c − ( BG ) : (1.3) [ X , t, d ] = (cid:2) M, s θη , c | M (cid:3) ∈ Ω Spin3 ( BG ) . The special case [KLT95b] is a consequence of Theorems 1.3 and 1.7.
Corollary 1.8 (Kreck–L¨uck–Teichner) . Let X be a nonempty connected orientableclosed smooth 4-manifold whose fundamental group is a free product G − ∗ G + . Some X ( r ) is diffeomorphic to a sum X − X + with each X ± of fundamental group G ± .Proof. Here G = 1, so H ( G ) = 0 = Ω Spin3 ( BG ). Take X = S , d constant. (cid:3) Albeit that Kreck’s modified surgery theory [Kre99] is a powerful formalism, bywhich we were inspired and against which we checked our progress, we sought towrite this paper from first principles, to be accessible to low-dimensional topologists.In particular, we avoid ‘subtraction of solid tori’ and ‘stable s -cobordism theorem.’2. Surgery on a link and regular covers
We generalize the notion of classifying a universal cover. For a nonempty con-nected CW-complex A , a continuous map u : A −→ B Γ classifies a regular cover means that the induced map u on fundamental groups is an epimorphism, for achoice of basepoints. The (connected) regular cover b A corresponds to the kernel of u , and its covering group is identified with Γ, which acts transitively on the fibers.2.1. Oriented version.
This development is used to prove Theorems 1.1 and 1.3.
Theorem 2.1.
Let M and M ′ be connected oriented closed 3-manifolds. Let f : M −→ B Γ and f ′ : M ′ −→ B Γ classify regular covers. Then there exists a framedoriented link L in M that transforms ( M, f ) into ( M ′ , f ′ ) by surgery if and only if (2.1) f ∗ [ M ] = f ′∗ [ M ′ ] ∈ H (Γ; Z ) . In other words, this is an algebraic-topological criterion for whether or not thereis a link in M whose preimage in c M has a Γ-equivariant surgery resulting in c M ′ .The original version is simply without reference maps; see [Wal60] and [Lic62]. Corollary 2.2 (Lickorish–Wallace) . Any nonempty connected oriented closed 3-manifold N is the result of surgery on a framed oriented link L in the 3-sphere. Lickorish also obtained each component is unknotted with ± Proof.
Here Γ = 1 , M = S , M ′ = N . Note B Γ is a point, hence H (Γ) = 0. (cid:3) Here is a more general, technical version of Theorem 2.1 that we shall use later.
Lemma 2.3.
Let M and M ′ be connected oriented closed 3-manifolds, and let B bea connected CW-complex. Suppose f : M −→ B and f ′ : M ′ −→ B are continuousmaps that induce epimorphisms on fundamental groups, for some basepoints. Then (2.2) f ∗ [ M ] = f ′∗ [ M ′ ] ∈ H ( B ; Z ) if and only if there is a 4-dimensional smooth connected oriented compact bordism ( F ; f, f ′ ) : ( W ; M, M ′ ) −→ ( B × [0 , B × { } , B × { } ) NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 5 such that W has no 1-handles with respect to M and no 1-handles with respect to M ′ , for a certain handle decomposition of the 4-dimensional cobordism ( W ; M, M ′ ) .Proof of Theorem 2.1. By Lemma 2.3, use only 2-handles: surger along a link. (cid:3)
The argument below, after the preliminary three paragraphs, can be perceivedin two geometric steps, even though it is combined into a single surgical move. Thefirst step is to slide 1-handles , along with the map data, so that they become trivial.The second step is a reference-maps version of Wallace’s trick to exchange oriented1-handles for trivial 2-handles [Wal60, 5.1]. (If dim
M >
3, see [RS72, 6.15] andsubsequent remark to replace 1-handles for 3-handles in certain cobordisms on M .) Proof of Lemma 2.3. ⇐ = is due to Ω SO ( B ) ∼ = H ( B ). Consider the = ⇒ direction.Clearly Ω SO = Z and Ω SO = Ω SO = 0; recall that Ω SO = 0 by Rohlin–Thom[Tho54, IV.13]. Then note, for the CW-complex B , by the Atiyah–Hirzebruchspectral sequence, that the cobordism-Hurewicz map is an isomorphism:Ω SO ( B ) −→ H ( B ) ; [ M, f : M → B ] f ∗ [ M ] . Thus the criterion (2.2) transforms into the equation: [
M, f ] = [ M ′ , f ′ ] ∈ Ω SO ( B ).In other words, there exists a 4-dimensional smooth oriented compact bordism( F ; f, f ′ ) : ( W ; M, M ′ ) −→ ( B × [0 , B × { } , B × { } ) . Since M and M ′ are connected, by joining their two possibly different compo-nents in W via connected sum and ignoring the rest, we may assume that W is con-nected. Hence, in the handle decomposition of a Morse function ( W ; M, M ′ ) −→ ([0 , { } , { } ), W has no 0-handles with respect to M and no 0-handles withrespect to M ′ . Therefore, it remains to eliminate the 1-handles of W with respectto M and M ′ . For simplicity of notation, we assume that W has a single 1-handle.Let h : ( D × D , S × D ) −→ ( W , M ) be the 1-handle, preserving orientation.Since M is connected, there is a path α : [ − , −→ M with α ( ±
1) = h ( ∓ , β = h ( − , ∗ α : S −→ W . Since f is an epimor-phism, there exists a loop β : ( S , −→ ( M, α (1)) such that f [ β ] = F [ β ]. Bygeneral position, there is a normally framed embedded arc α : [ − , × D −→ M such that α ( ± ,
0) = h ( ∓ ,
0) and α ( − ,
0) is homotopic rel boundary to α ∗ β − .Push off the 1-handle core h ( − ,
0) to obtain a normally framed embedded arc h : [ − , × D −→ M ′ . A matching isotopy takes α to α ′ : [ − , × D −→ M ′ with α ′ ( ±
1) = h ( ∓ , λ := { h ( − , r ) ∗ α ′ ( − , r ) } r ∈ D : S × D −→ M ′ . Write W := M ′ × [0 , ∪ λ D × D for the trace of the surgery along λ in M ′ .Since F ◦ λ ( − ,
0) is nulhomotopic, choose a nulhomotopy to yield a bordism( F ; f ′ , f ′′ ) : ( W ; M ′ , M ′′ ) −→ ( B × [0 , B × { } , B × { } ) . Observe that W is the trace of surgery along the framed belt sphere S × D ֒ → M ′′ .By the cancellation lemma [RS72, 6.4], W ∪ M ′ W is diffeomorphic to M × [0 , M × { } . In particular, there is δ : M ≈ M ′′ with f ′′ ◦ δ ≃ f . Write W ′ := M × [0 , ∪ δ W . So we have a new bordism with h replaced by a 2-handle:( δ ∪ f ′′ F ; f, f ′ ) : ( W ′ ; M, M ′ ) −→ ( B × [0 , B × { } , B × { } ) . By iteration, we kill all 1-handles of W ′ relative to M . Similarly, repeat relativeto M ′ . Thus, we obtain the desired bordism ( W, F ) with only 2-handles rel M . (cid:3) Q KHAN AND G SMITH
Spin version.
We shall need this development to prove Theorem 1.4.
Theorem 2.4.
Let ( M, s ) and ( M ′ , s ′ ) be spin closed 3-manifolds. Let f : M −→ B Γ and f ′ : M ′ −→ B Γ classify regular covers. There exists a framed oriented link L in M that transforms ( M, s, f ) into ( M ′ , s ′ , f ′ ) by a spin bordism if and only if (2.3) [ M, s, f ] = [ M ′ , s ′ , f ′ ] ∈ Ω Spin3 ( B Γ) . Lemma 2.5.
Let ( M, s ) and ( M ′ , s ′ ) be connected spin closed 3-manifolds, andlet B be a connected CW-complex. Suppose f : M −→ B and f ′ : M ′ −→ B arecontinuous maps that induce epimorphisms on fundamental groups. Then (2.4) [ M, s, f ] = [ M ′ , s ′ , f ′ ] ∈ Ω Spin3 ( B ) if and only if there is a 4-dimensional smooth connected spin compact bordism ( F ; f, f ′ ) : ( W, t ; M, s, M ′ , s ′ ) −→ ( B × [0 , B × { } , B × { } ) such that W has no 1-handles with respect to M and no 1-handles with respect to M ′ , for a certain handle decomposition of the 4-dimensional cobordism ( W ; M, M ′ ) .Proof of Theorem 2.4. By Lemma 2.5, use only 2-handles: surger along a link. (cid:3)
Proof of Lemma 2.5.
The ⇐ = implication is obvious. Consider the = ⇒ implication.Recall the proof of Lemma 2.3. We reconstruct W to admit a spin structureextending the spin structure s ′ on ∂ − W = M ′ , since by gluing along M ′′ this willinduce a spin structure on W ′ extending the spin structure s ⊔ s ′ on ∂W ′ = M ⊔ M ′ .Since H i +1 ( M ′ ; π i (Spin )) = 0 for all i >
0, by obstruction theory, the spinstructure s ′ lifts to a framing φ of the tangent bundle T M ′ . The sole obstructionto extending the stable framing φ ⊕ id of T M ′ ⊕ R to the tangent bundle τ of W is o ( τ ) ∈ H ( W , M ′ ; π ( SO )) = H ( D , S ; π ( SO )) = π ( SO ) ∼ = Z / . Let η ∈ π ( SO ) ∼ = Z . Reframe the normal bundle of the surgery circle λ ( − ,
0) as λ η : S × D −→ M ′ ; ( z, r ) λ ( z, η z ( r )) . Write W η := M ′ × [0 , ∪ λ η D × D with tangent bundle τ η . By [KM63, Lemma 6.1], o ( τ η ) = o ( τ ) + σ ( η ) ∈ π ( SO ) , where σ : SO −→ SO denotes the inclusion. Since the induced map σ onfundamental groups is surjective, find η so that τ η has a framing extending φ ⊕ id.Hence W η has a spin structure extending s ′ on its lower oriented boundary M ′ .By gluing, we obtain an induced spin structure on W ∪ M ′ W η ≈ M × [0 , M × { } . Modifying Proof 2.3, redefine W ′ := M × [0 , ∪ δ η W η withspin structure the union of this one and the orientation-reversal of the one on W η .Therefore, the spin structure on W ′ restricts to s ⊔ s ′ on ∂W ′ = M ⊔ M ′ . (cid:3) Ambient surgery on pairs of points
Let G = G − ∗ G G + be an injective amalgam of groups. The corresponding double mapping cylinder model of its classifying space is the homotopy colimit(3.1) BG := BG − ∪ BG ×{− } BG × [ − , +1] ∪ BG ×{ +1 } BG + with respect to the maps BG −→ BG ± induced from the inclusions G −→ G ± .Akin to Stalling’s thesis, here is a folklore fact proven in [Bow99, 1.1] (cf. [BS90]). Theorem 3.1 (Bowditch) . If G and G are finitely presented, so are G − and G + . NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 7
Instead of G being finitely presented, the proof of the next statement can workassuming G − , G , G + are finitely generated, but we prefer the former hypothesis. Proposition 3.2.
Let X be a connected oriented closed smooth 4-manifold. Sup-pose f : X −→ BG classifies a regular cover. Assume G is finitely presented.Then f can be re-chosen up to homotopy so that: f is transverse to the bicollaredsubspace BG × { } in the model (3.1) , the 3-submanifold preimage M in X isconnected, and the restriction f : M −→ BG also classifies a regular cover. This is proven after three lemmas. The first is an apparatus to recalibrate paths.
Lemma 3.3.
Let X be a connected oriented smooth n -manifold with n > . Con-sider a space B = B − ∪ B B + with B, B ± − B path-connected and B = B − ∩ B + .Suppose f : X −→ B is π -surjective. Assume π ( B ± − B ) are finitely generated,by r ± elements. There are disjoint 1-handlebodies Λ ± ≈ r ± ( S × D n − ) ⊂ X and f ′ : X −→ B homotopic to f having π -surjective restrictions f ′ : Λ ± −→ B ± − B .Proof. We may homotope f so that its image contains some points b ± in B ± − B .Then there are x ± ∈ X such that f ( x ± ) = b ± . There are based loops µ ± , . . . , µ r ± ± :( S , −→ ( B ± − B , b ± ) whose based homotopy classes generate π ( B ± − B , b ± ).Since f : π ( X, x ± ) −→ π ( B, b ± ) is surjective and n >
2, there exist disjointsmoothly embedded based loops λ ± , . . . , λ r ± ± : ( S , −→ ( X, x ± ) and a basedhomotopy H i ± : S × [0 , −→ B from f ◦ λ i ± to µ i ± . Since X is oriented, for each i , there is a tubular neighborhood Λ ± i of λ ± i ( S ) and a diffeomorphism Λ ± i ≈ S × D n − . Taking the radii of the tubes sufficiently small, we find that the Λ ± i pairwiseintersect in a fixed D n -neighborhood of x ± . Thus we obtain disjoint embeddings of r − ( S × D n − ) and r + ( S × D n − ) in X , say with images called Λ − and Λ + .Finally, using these NDR neighborhoods Λ ± of W i λ i ± and specific homotopies W i H i ± as the data for the homotopy extension property [Bre97, Theorem VII:1.5],we obtain a homotopy H : X × [0 , −→ B from f to a map f ′ such that f ′ ◦ λ i ± = µ i ± . Hence f ′ (Λ ± ) ⊂ B ± − B and ( f ′ | Λ ± ) π (Λ ± , x ± ) = π ( B ± − B , b ± ). (cid:3) Given a continuous map f : X −→ B from a smooth manifold X to a topologicalspace B , and given a subspace B that admits a tubular neighborhood E ( ξ ) ⊂ B ,W Browder defines f to be transverse to B to mean that the conclusion ofthe implicit-function theorem holds: the preimage X = f − ( B ) is a smoothsubmanifold of X with normal bundle ν ( X ֒ → X ) = ( f | X ) ∗ ( ξ ) [Bro72, II: § ξ = R . Lemma 3.4.
Let f : X −→ B be a continuous map from a smooth manifold to aspace B . For any bicollared subspace B of B (i.e., B has a neighborhood in B homeomorphic to B × R ), there exists a map f ′ : X −→ B transverse to B andhomotopic to f , relative to the complement of an open neighborhood of f − ( B ) .Proof. We have an open embedding β : B × R −→ B with β ( B × { } ) = B ⊂ B .Write N ⊂ B for the image of β , and write π : B × R −→ R for the projection.Note f − ( N ) ⊂ X is a smooth manifold, since it is an open set in a smoothmanifold. By Whitney’s approximation theorem [Bre97, II:11.7], the C function π ◦ β − ◦ f : f − ( N ) −→ R is 0 . C ∞ function g : f − ( N ) −→ R . Define H : f − ( N ) × [0 , −→ B ; ( x, t ) β (cid:0) ( π β − f )( x ) , (1 − t )( π β − f )( x ) + tg ( x ) (cid:1) . Q KHAN AND G SMITH
Note H is a homotopy from H ( x,
0) = f ( x ) to a map f ′ := H ( − ,
1) : f − ( N ) −→ B .Then f ′ is transverse to B with ( f ′ ) − ( B ) = g − { } a smooth submanifold of X ;where by Sard’s theorem and a tiny homotopy, we assume 0 is a regular value of g .It remains to extend H to X × [0 ,
1] so that H ( x, t ) = f ( x ) for all x ∈ X − N .Using explicit formulas derived from the tubular neighborhood structure β , this isachieved by the homotopy extension property for the neighborhood deformationretract f − β ( B × [ − , ⊔ ( X − N ) closed in the T space X ; see [Bre97, Theo-rem VII:1.5]. The desired map f ′ : X −→ B is again H ( − ,
1) of this extension. (cid:3)
We perform by an obstruction-theoretic ar-gument. This is not in the literature, but see [Hem76, p67] and [Cap76, Lemma I:3].Recall the frontier Fr X ( A ) := Cl X ( A ) ∩ Cl X ( X − A ) for A ⊂ X , a topological space. Lemma 3.5.
Let f : X −→ B be a continuous map from a smooth 4-manifold toa path-connected space B , transverse to a path-connected separating subspace B of B = B − ∪ B B + . Decompose X = X − ∪ X X + by the f -preimages. Let α :( D , ∂D ) −→ ( X ± , X ) be a smoothly embedded arc with [ f ◦ α ] = 0 ∈ π ( B ± , B ) .Suppose π ( B ∓ ) = 0 = π ( B ) . Then f is homotopic to a B -transverse map g : X −→ B whose preimage of B is the result of adding a 1-handle with core α .Namely, for some open-tubular neighborhood U ≈ D × ˚ D of the arc α in X ± : g − ( B ) = ( X ∪ Fr X U ) − ( X ∩ U ) . Proof.
Let T be a closed-tubular neighborhood of α ( D ) in X ± . There is a framingdiffeomorphism φ : ( D × D , ∂D × D ) −→ ( T, T ∩ X ) with φ ( s,
0) = α ( s ). Define O := φ (cid:8) ( s, x ) ∈ D × D | k x k (cid:9) M := φ (cid:8) ( s, x ) ∈ D × D | k x k (cid:9) I := φ (cid:8) ( s, x ) ∈ D × D | k x k (cid:9) , which is a decomposition of T = O ∪ M ∪ I into three closed subsets. Define a map g : O −→ B ± ; φ ( s, x ) ( f ◦ φ )( s, (3 k x k − x ) . Since [ f ◦ α ] = 0 ∈ π ( B ± , B ), there exists a map H : D × [0 , −→ B ± such that H ( s,
1) = ( f ◦ α )( s ) ∀ s ∈ D H ( ± , t ) = α ( ± ∀ t ∈ [0 , H − ( B ) = ∂D × [0 , ∪ D × { } . By the pasting lemma, we can extend g from O to O ∪ M by g : M −→ B ± ; φ ( s, x ) H ( s, k x k − . Next, there exist both a neighborhood C of the attaching 0-sphere α ( ∂D ) in X ∓ and a diffeomorphism ψ : ∂D × D − −→ X ∓ such that ψ | ∂D × D = φ | ∂D × D .Extend g from Fr X T = φ ( ∂D × ∂D ) to Fr X ∓ C = ψ ( ∂D × ∂ − D − ) by g = f .Then g is defined on the ‘riveted’ 3-sphere S := Fr X ( C ∪ I ). Since g ( S ) ⊂ B ∓ and π ( B ∓ ) = 0, we may extend g to the ‘riveted’ 4-disc C ∪ I ; using the collar of B in B ∓ , we can guarantee that g ( C ∪ I − S ) ⊂ B ∓ − B . Lastly, extend g to thecomplement X − ( C ∪ T ) by g = f . Therefore g : X −→ B is transverse to B with g − ( B ) = ( X − I ) ∪ ( M ∩ I ) . Finally, note Fr X ( C ∪ T ) is a 3-sphere in X , so we obtain a 4-sphere X × [0 , C ∪ T ) × { } ∪ Fr X ( C ∪ T ) × [0 , ∪ ( C ∪ T ) × { } . NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 9
Since π ( B ) = 0, we may fill in ( f ◦ proj X ) | Σ to obtain a homotopy from f to g . (cid:3) We adapt to dimension 4, and simplify, ‘arc-chasing’ arguments of [Hem76, p67]and [Cap76, p88]. Further, we generalize the sliding of 1-handles trick of Proof 2.3.
Proof of Proposition 3.2.
By Theorem 3.1 and by Lemma 3.3 with respect to themodel (3.1), we homtope f so that there are disjointly embedded 1-handlebodiesΛ ± ≈ r ± ( S × D ) ⊂ X satisfying f (Λ ± ) ⊂ BG ± − BG and f π (Λ ± , x ± ) = G ± .Hence f (Λ − ⊔ Λ + ) is disjoint from the bicollar neighborhood BG × [ − ,
1] in BG .Next, by Lemma 3.4, we further re-choose f up to homotopy relative to Λ − ⊔ Λ + so that f is also transverse to BG × { } , say with f -preimage K . Write V ± for the X -closure of the path-component neighborhood of Λ ± in the open subset X − K .Assume that V + ∩ K has at least two components, say K ⊔ K . Since V + is connected, there is a properly and smoothly embedded arc α : [0 , −→ V + satisfying: α ( i ) ∈ K i if 0 i α ( ) is near-but-not x + , and α − (˚ V + ) = (0 , π (Λ + ) f −−−→ π ( BG + ) −→ π ( BG + , BG ) is surjective,upon midpoint-concatenation of some based loop ( S , −→ ( ∂ Λ + , α ( )), we mayassume that [ f ◦ α ] = 0 ∈ π ( BG + , BG ). Then, by Lemma 3.5, we re-choose f upto homotopy relative Λ − ⊔ Λ + so that the new component neighborhood V + of Λ + in X − f − ( BG ) contains K K . Since X is compact, so is K , so we repeat finitelymany steps until V + ∩ K becomes connected. Similarly, make V − ∩ K connected.Write L := V − ∩ V + , a connected 3-submanifold of X . Let x ∈ L . Assumethere exists x ∈ K − L . Since X is connected, there exists a path γ : [0 , −→ X from x to x . Define s := sup γ − ( V − ∪ V + ). Since 0 < s <
1, we must have γ ( s ) ∈ Fr X ( V − ) ∪ Fr X ( V + ) = L . Then γ ( s ) is in the interior of V − ∪ V + . Sothere exists s > s with γ ( s ) also in the interior of V − ∪ V + . This contradicts themaximality of s . Therefore K − L is empty. Hence K = L and so it is connected.Finally, since G ⊂ G + is finitely generated and since ( f | Λ + ) : π (Λ + ) −→ G + is surjective, there exist based loops δ , . . . , δ r : ( S , −→ ( V + , x ) such that G = h f [ δ ] , . . . , f [ δ r ] i . In particular, each f ◦ δ i is based homotopic into BG .Since each f ◦ δ i represents 0 in π ( BG + , BG ), by Lemma 3.5 applied r times, were-choose f so that, further, its restriction to M := f − ( BG ) is π -surjective. (cid:3) Proofs of the embedding theorems
Proof of Theorem 1.1.
Clearly (1.1) is a necessary condition. So now, assume (1.1).Consider the double mapping cylinder model (3.1). Since G − , G , G + are finitelygenerated and c : X −→ BG classifies a regular cover, by Proposition 3.2, we mayre-choose c up to homotopy so that: c is transverse to BG , the 3-submanifoldpreimage M is connected, and the restriction c : M −→ BG classifies a regularcover. Write X = X − ∪ M X + and c = c − ∪ c c + with restrictions c ± : X ± −→ BG ± .Next, consider the commutative square, with horizontal maps being connectinghomomorphisms induced from (3.1) and with vertical maps being of Hurewicz type:Ω SO ( BG ) ∂ / / (cid:15) (cid:15) Ω SO ( BG ) (cid:15) (cid:15) H ( BG ) ∂ / / H ( BG ) ; [ X, c ] ✤ / / ❴ (cid:15) (cid:15) [ M, c ] ❴ (cid:15) (cid:15) c ∗ [ X ] ✤ / / c ∗ [ M ] . Hence the criterion (1.1) implies: there is a classifying map d : X −→ BG with d ∗ [ X ] = c ∗ [ M ] ∈ H ( G ) . Since d and c are surjective, by Lemma 2.3, there is a 4-dimensional orientedsmooth bordism e : V −→ BG from ( M, c ) to ( X , d ) made with only 2-handles.Since V is obtained from X using only 2-handles, the inclusion j : X −→ V induces an epimorphism on fundamental groups. Since d = e ◦ j is a monomor-phism, note that j is also a monomorphism. So both j and e are isomorphisms.Now, we obtain a connected 5-dimensional oriented compact smooth cobordism T := X × [0 , ∪ M × [ − , V × [ − , M × [ − ,
1] in X × { } and we smooth the corners at M × {− , } .The resultant 4-manifold and map are X ′ := ∂T − X × { } and c ′ := D | X ′ , where D : T −→ BG ; ( ( x, t ) ∈ X × [0 , c ( x )( v, s ) ∈ V × [ − , ( e ( v ) , s ) . Decompose the space X ′ = X ′− ∪ X X ′ + with X ′± = X ± ∪ M V × {± } ∪ X × [ ± , c ′ = c ′− ∪ d c ′ + : X ′ −→ BG with c ′± = c ± ∪ c e : X ′± −→ BG ± .Write i : M −→ X for the inclusion. Since V is the trace of a surgery ona framed oriented link L in M , correspondingly note T is the trace of a surgeryon i ◦ L in X . By a similar argument as earlier, we find that the kernel of c equals the kernel of the map induced by the inclusion M −→ V , which is generatedby the (unbased) components L k of L , upon anchoring them to the basepointwith choices of connecting paths. In addition, since c = c ◦ i and c is anisomorphism, the kernel of c equals the kernel of i . In particular, each embeddedcircle L k is nulhomotopic in X , bounding an immersed disc with tranverse doublepoints, which can be isotoped away using finger-moves [FQ90, 1.5]; thus each L k bounds an embedded disc in X . Another consequence is that D ≃ c ∪ c e : X ∪ M V −→ BG induces an isomorphism on fundamental groups. Then, by an argumentwith alternating words, each c ′± : X ′± −→ BG ± also does so. So, since d is anisomorphism, c ′ induces an isomorphism on fundamental groups.Finally, we show that the embedding solution X ′ is bistably diffeomorphic to X . For each L k , consider embedded in ˚ T the 2-sphere S k with equator L k , withnorthern hemisphere the core of the bounding 2-handle in V × { } , and with south-ern hemisphere the bounding 2-disc in X × { } . Write N k for the 5-dimensionalclosed-tubular neighborhood of S k in ˚ T . Observe that T is diffeomorphic to theboundary-connected sum ( X × [0 , ♮ ( F k N k ). Each N k is diffeomorphic to either D × S or D ⋊ S , where the latter is the nontrivial (nonspin) disc bundle. Thus,we obtain X ′ ≈ X p ( S × S ) q ( S ⋊ S ) for some p > r and q >
0. Since( S × S ) C P ) ≈ C P ) C P ) and S ⋊ S ≈ ( C P ) C P ) [Wal64, C1, L1], X ′ (1 , ≈ X (1 + p + q, p + q ) . (cid:3) Proof of Theorem 1.3.
Since w ( e X ) = 0, by the Hurewicz theorem, there exists aspherical class e α : S −→ e X such that h w ( e X ) , e α ∗ [ S ] i 6 = 0. Write p : e X −→ X for the covering map, and write α := p ◦ e α : S −→ X . Since on tangent bundles T e X = p ∗ ( T X ), as one obtains the smooth structure on e X by even-covering, note h w ( X ) , α ∗ [ S ] i = h w ( X ) , p ∗ e α ∗ [ S ] i = h p ∗ w ( X ) , e α ∗ [ S ] i = h w ( e X ) , e α ∗ [ S ] i = 1 . NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 11
Do the same as in the proof of Theorem 1.1, until the construction of the 2-sphere S k . In the case that the normal bundle of S k is nontrivial, replace the southernhemisphere with its one-point union with α , smoothed rel L k into immersion thenan embedding by finger-moves, to obtain S ′ k . Since [ S ′ k ] = [ S k ] + [ α ] ∈ π ( X ), note h w ( X ) , S ′ k ∗ [ S ] i = h w ( X ) , S k ∗ [ S ] + α ∗ [ S ] i = 1 + 1 = 0 ∈ Z / . Hence the normal 3-plane bundle of the new embedded 2-sphere S ′ k in ˚ T is trivial.So T is diffeomorphic to the boundary-connected sum ( X × [0 , ♮ ( F rk =1 D × S ).Therefore, we obtain X ′ is diffeomorphic to X ( r ) = X r ( S × S ). (cid:3) Proof of Theorem 1.4.
Do the same as in the proof of Theorem 1.1, except using(1.2) and Lemma 2.5 instead of (1.1) and Lemma 2.3, until the construction ofthe 2-sphere S k . Here, the spin structure s M on M is the restriction of the spinstructure s on X × { } , where the spin structure on the normal line bundle isinduced from its pullback orientation [LM89, II:2.15]. Since s M is the restrictionof the spin structure on V × { } , we obtain that T has an induced spin structure.Then, since w ( T ) = 0, the normal 3-plane bundle of each S k in ˚ T is trivial.So T is diffeomorphic to the boundary-connected sum ( X × [0 , ♮ ( F rk =1 D × S ).Therefore, we obtain X ′ is diffeomorphic to X ( r ) = X r ( S × S ). (cid:3) For clarity, we repeat the following proof from [KLT95b, p258] and [Kre99, p713].The statement shall be applied in Proof 1.7 for manifolds Y of dimensions 3, 4, 5. Proof of Lemma 1.5.
Since e Y is 1-connected, by the Leray–Serre spectral sequencefor the homotopy fibration sequence e Y p −−→ Y u −−→ B Γ, we obtain an exact sequence(4.1) 0 / / H ( B Γ; Z / u ∗ / / H ( Y ; Z / p ∗ / / H ( e Y ; Z / G . Then, since w ( T e Y ) = w ( p ∗ ( T Y )) = p ∗ ( w ( T Y )), the oriented smooth manifold e Y admits a spin structure if and only if there exists w u ∈ H ( B Γ; Z /
2) such that u ∗ ( w u ) = w ( T Y ). Further by exactness, this class w u is unique if it exists. (cid:3) For r >
0, the pinch map p : X ( r ) −→ X ∨ r ( S × S ) gives a degree-one map k := (id ∨ const) ◦ p : X ( r ) −→ X. The π -isomorphism c : X −→ BG induces the π -isomorphism c ◦ k : X ( r ) −→ BG . Proof of Theorem 1.7: necessity of (1.3) . Assume for some r > j : X −→ X ( r ) such that ( c ◦ k ◦ j ) ( π X ) = G .Then X ( r ) = X ′− ∪ X X ′ + with inclusions j ± : X ′± −→ X ( r ). Since X ( r ) and X areconnected, so are X ′± . Furthermore, since ( c ◦ k ) and ( c ◦ k ◦ j ) are isomorphisms,by a basic observation on normal form [Smi17], so are ( c ◦ k ◦ j ± ) : π ( X ′± ) −→ G ± .Consider the double mapping cylinder model (3.1) of BG , where Bi : BG −→ BG is the inclusion of a bicollared subspace. Since X is a CW-complex, thereis a homotopically unique map d : X −→ BG such that Bi ◦ d ≃ c ◦ k ◦ j .Furthermore, since X ± are CW-complexes, d extends to maps c ± : X ′± −→ BG ± ∪ BG × [0 , ±
1] with Bi ± ◦ c ± ≃ c ◦ k ◦ j ± . Therefore, c ◦ k is homotopic to a BG -transverse map c ′ := c ′− ∪ d c ′ + : X ( r ) −→ BG satisfying ( c ′ ) − ( BG × { } ) = X .Next, since e X admits a spin structure, by Lemma 1.5, there is a unique class w c ∈ H ( BG ; Z /
2) such that w ( T X ) = c ∗ ( w c ). Since S is stably parallelizable,so is S × S . Then the tangent bundle T X ( r ) is stably isomorphic to the pullback k ∗ T X . (The corresponding statement is false for a bistabilization X ( a, b ) unless a = 0 = b .) Hence w ( T X ( r )) = k ∗ w ( T X ). Note d ∗ ( i ∗ w c ) = ( i ◦ d ) ∗ ( w c ) = ( c ◦ k ◦ j ) ∗ ( w c ) = j ∗ k ∗ ( c ∗ w c )= j ∗ k ∗ ( w ( T X )) = j ∗ w ( T X ( r )) = w ( T X ) = v ( X ) = 0 , with v = w + w the second Wu class [MS74, 11.14] and Sq = 0 on H ( X ; Z / X , so Ker( d ∗ ) = 0 hence i ∗ ( w c ) = 0.Thus, there is a nulhomotopy θ of w c ◦ Bi . By Lemma 3.4, we may assume c istransverse to BG in model (3.1), with 3-submanifold M := c − ( BG × { } ) of X .Now, k : X ( r ) −→ X extends to a retraction K : X [ r ] ≃ X ∨ rS −→ X , where X [ r ] := ( X × [0 , ♮ r ( D × S ) is the canonical cobordism from X to X ( r ). Since c is transverse to BG , so is c ◦ K . Recall there is homotopy H : X ( r ) × [1 , −→ BG such that H ( − ,
1) = c ◦ k and H ( − ,
2) = c ′ . These unite to a B -transverse map C := ( c ◦ K ) ∪ c ◦ k H : W := X [ r ] ∪ ( X ( r ) × [1 , −→ BG.
The preimage 4-manifold V := C − ( BG × { } ) fits into an oriented bordism( V, C | V ) from ( M, c | M ) to ( X , d ). Furthermore, this enhances to a spin bordism,as Definition 1.6 produces a spin structure s θµ on V defined by the formula s θµ : V −→ B Spin = hofib( w ) ; x (cid:16) τ V ( x ) , µ x ∗ θ C ( x ) (cid:17) , with µ : W × [0 , −→ K ( Z / ,
2) a homotopy from w ◦ τ W to w C ◦ C . Indeed, w C exists by Lemma 1.5, since f W has a spin structure as T f W ∼ = e K ∗ T e X ⊕ R . Define η : X × [0 , −→ K ( Z / ,
2) as a restriction of µ . So s θµ on V restricts to spinstructures s θη on M and t := s θµ | X on X . Therefore, Equation (1.3) holds. (cid:3) Recall the homotopy fiber of a map f : A −→ B with respect to b ∈ B ishofib( f ) := { ( a, p ) ∈ A × B [0 , | p (0) = f ( a ) and p (1) = b } . Proof of Theorem 1.7: sufficiency of (1.3) . Assume Equation (1.3) holds, wherethe transverse 3-submanifold M := c − ( BG ) of X exists by Lemma 3.4, upon al-tering c by a homotopy. Furthermore, by Proposition 3.2, we can further homotope c so that M is connected and its restriction c : M −→ BG is a π -epimorphism.Then the spin bordism ( V, Σ , e ) from ( M, s θη , c | M ) to ( X , t, d ), by Lemma 2.5, canbe assumed to only have 2-handles relative to X . From the proof of Theorem 1.1, e : V −→ BG is a π -isomorphism, and the map D ≃ c ∪ c e : T −→ BG is also.Observe that the spin structure Σ : V −→ B Spin = hofib( w ) is of the formΣ = (cid:16) τ V : V −→ BSO, σ : V −→ K ( Z / , [0 , (cid:17) , with σ ( x ) ∈ K ( Z / , [0 , a path from w ( τ V ( x )) to the basepoint ω of K ( Z / , θ : BG × [0 , −→ K ( Z / ,
2) is a homotopy from w c ◦ Bi to const ω .Then define a homotopy ξ : V × [0 , −→ K ( Z / ,
2) from w ◦ τ V to w c ◦ Bi ◦ e by ξ x := σ ( x ) ∗ θ e ( x ) . Recall that η : X × [0 , −→ K ( Z / ,
2) is a homotopy from w ◦ τ X to w c ◦ c . Thisrestricts to a homotopy η : M × [0 , −→ K ( Z / ,
2) from w ◦ τ M to w c ◦ Bi ◦ c .Note ξ extends η , since τ V extends τ M and e extends c . Thus, since T ≃ X ∪ M V ,we obtain a homotopy η ∪ η ξ from w ◦ τ T to w c ◦ D . Since D classifies the universalcover of the 5-manifold T , by Lemma 1.5, the universal cover e T has a spin structure. NCOMPRESSIBLE HYPERSURFACES STABLY IN 4-MANIFOLDS 13
Consider the embedded 2-spheres S k : S −→ ˚ T , in the proof of Theorem 1.1.Write P : e T −→ T for the universal covering map. As S is simply connected, bythe lifting theorem, there is an embedding f S k : S −→ e T with S k = P ◦ f S k . Note h w T, S k ∗ [ S ] i = h w T, P ∗ f S k ∗ [ S ] i = h P ∗ ( w T ) , f S k ∗ [ S ] i = h w e T , f S k ∗ [ S ] i = 0 . Then, although T need not be spin, nonetheless the normal 3-plane bundle ofeach S k in ˚ T is trivial. So T is diffeomorphic to the boundary-connected sum ( X × [0 , ♮ ( F rk =1 D × S ). Therefore X ′ is diffeomorphic to X ( r ) = X r ( S × S ). (cid:3) A final remark on (1.1) is that H ( G ) = Z if X is irreducible with infinite fun-damental group, as X models BG , a consequence of the sphere theorem [Hem76]. Acknowledgements.
The second author is a doctoral student of the first authorand thanks him for equal involvement on this stable existence project. His disserta-tion solves the stable uniqueness problem, using instead the power of Kreck’s mod-ified surgery machine [Smi17]. The first author is grateful to his low-dimensionaltopology professors at U Illinois–Chicago, Louis Kauffman and Peter Shalen, whoinstilled in him the beauty of knots and of incompressible surfaces in 3-manifolds.This project is a ‘boyhood dream’ to see if that philosophy works up one dimension.
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Department of Mathematics Indiana University Bloomington IN 47405 USA
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