Branched covering simply-connected 4-manifolds
David Auckly, R. Inanc Baykur, Roger Casals, Sudipta Kolay, Tye Lidman, Daniele Zuddas
aa r X i v : . [ m a t h . G T ] F e b BRANCHED COVERING SIMPLY-CONNECTED -MANIFOLDS DAVID AUCKLY, R. ˙INANC¸ BAYKUR, ROGER CASALS,SUDIPTA KOLAY, TYE LIDMAN, AND DANIELE ZUDDAS
Abstract.
We prove that any closed simply-connected smooth 4-manifold is 16-foldbranched covered by a product of an orientable surface with the 2-torus, where theconstruction is natural with respect to spin structures. We also discuss analogous resultsfor other families of 4-manifolds with infinite fundamental groups. Introduction
The work in this note was prompted by the following natural question: “Does every closed –manifold admit a branched covering by a symplectic –manifold?”, which was studied by the authors at the 2018 American Institute of Mathematics Work-shop on “Symplectic four-manifolds through branched coverings”, motivated by a con-jecture of Eliashberg [9, Conjecture 6.2]. We provide a fairly strong answer to the abovequestion in the case of simply connected 4–manifolds: Theorem 1.
Let X be a closed oriented simply-connected smooth –manifold. Then thereexists g ∈ N and a degree branched covering f : X ′ → X such that X ′ is the smooth4-manifold T × Σ g and the branched locus of f is a smooth closed orientable surfaceself-transversally immersed in X . In addition, if the 4–manifold X is spin, the branchedcovering f is natural with respect to a spin structure on T × Σ g . Note that the smooth 4-manifold T × Σ g admits a symplectic structure. It follows fromTheorem 1 that if instead X is a closed (possibly non-orientable) connected smooth4–manifold with finite π ( X ), then there is a branched covering T × Σ g → X of degree16 | π ( X ) | , which factors through the universal covering e X → X .The above do not generalize to 4–manifolds with arbitrary fundamental groups; forinstance, no Σ g –bundle over Σ h with g, h ≥ X = g ( S × S ), with π ( X ) ∼ = ∗ Z g , Mathematics Subject Classification. is degree 4 branched covered by X ′ = S × Σ g by [23, Theorem 1.2]. In addition, thebranched virtual fibering theorem of Sakuma [27, Addendum 1] implies the following: Proposition 2.
For any closed connected oriented product –manifold X = S × Y ,there is a double branched covering X ′ → X , where X ′ is a symplectic –manifold whichis a Σ g –bundle over T , for some g . Indeed, [27] shows that any closed oriented 3–manifold Y is double branched covered bya surface bundle over a circle, from which Proposition 2 is immediately deduced; thisprovides yet another class of 4–manifolds with infinite fundamental group for which a(symplectic) branched cover can be readily described. Here, we recall that the productof a fibered 3–manifold and the circle is symplectic [28].It is worth noting that with a little more information on the smooth topology of X ,one can easily determine the topology of the branched coverings X ′ → X in Theorem 1and Proposition 2. For the former, one only needs to know the number of stabilizationsby taking the connected sum with S × S that are required before the simply-connected4–manifold X completely decomposes into a connected sum of copies of CP , S × S and the K3 surface, taken with either orientation. This of course can always be achievedby a classical result of Wall [29], and for vast families of simply-connected 4–manifolds,one stabilization is known to be enough [2]. Similarly, for Proposition 2, one just needsto know a Heegaard decomposition of the 3–manifold factor Y [27], or any open book onit [20]. See Remark 6 for some explicit examples.In all the results we have discussed above, the covering symplectic 4–manifold X ′ is notof general type, in contrast with the symplectic domination results of Fine–Panov [10]; seeRemark 7 below. It would be interesting to find more general families of non-symplectic4–manifolds branched covered (with universally fixed degree) by specific families of sym-plectic 4–manifolds like ours, say by Σ g –bundles over Σ h , for arbitrary h . Acknowledgments.
This project was started at the 2018 AIM workshop on “Symplec-tic four-manifolds through branched coverings”, and was resumed following the 2020BIRS Workshop on “Interactions of gauge theory with contact and symplectic topologyin dimensions 3 and 4.”
The authors would like to thank the American Institute ofMathematics and the Banff International Research Station, and the other organizers ofthese workshops. D. A. was partially supported by the Simons Foundation grant 585139and NSF grant DMS 1952755. R. I. B. was partially supported by the NSF grants DMS-200532 and DMS-1510395. R. C. is supported by the NSF grant DMS-1841913, the NSFCAREER grant DMS-1942363 and the Alfred P. Sloan Foundation. T. L. was partiallysupported by the NSF grant DMS-1709702 and a Sloan Fellowship. D. Z. was partiallysupported by the 2013 ERC Advanced Research Grant 340258 TADMICAMT; he ismember of GNSAGA, Istituto Nazionale di Alta Matematica “Francesco Severi”, Italy.
RANCHED COVERING 4-MANIFOLDS 3 Proof of Theorem 1
Henceforth all the manifolds and maps we consider are assumed to be smooth. Wedenote by X the oriented 4–manifold X with the reversed orientation, and by a X b Y the smooth connected sum of a copies of X and b copies of Y . We denote by Σ bg a closedconnected oriented surface of genus g with b boundary components, and we drop b fromthe notation when there is no boundary.2.1. Preliminaries.
Let us briefly recall the definition of a branched covering.
Definition 3.
Let X and X ′ be compact connected smooth manifolds (possibly withboundary) of the same dimension, and let f : X ′ → X be a smooth proper surjectivemap. We say that f is a branched covering if it is finite-to-one and open, and moreoverthe (open) subset of X ′ where f is locally injective coincides with the subset of X ′ where f is a local diffeomorphism. ✷ The subset B ′ f ⊂ X ′ where f fails to be locally injective is called the branch set of f , and its image B f = f ( B ′ f ) ⊂ X is called the branch locus of f . By a result ofChurch [7, Corollary 2.3], either B ′ f = ∅ or dim B ′ f = dim B f = dim X −
2, and thenthe restriction of f over the complement of B f is an ordinary connected covering space X ′ \ f − ( B f ) → X \ B f .Moroever, for every smooth point of B ′ f at which f | B ′ f : B ′ f → X is a local smoothembedding, the map f is topologically locally equivalent to the map p d : C × R n − → C × R n − defined by p d ( z, x ) = ( z d , x ), for some d ≥
2, where n = dim X ′ = dim X .However, the branched coverings f i that we consider below turn out to be smoothly locally equivalent to p , while their composition, which will be indicated by f , has thisproperty away from the double points of B ′ f . Notice that every finite composition ofbranched coverings is a branched covering, and the restriction to the boundary of abranched covering is a branched covering as well. Throughout, we assume that branchedcoverings between oriented manifolds are orientation-preserving.2.2. The argument.
Let X be a closed oriented simply-connected smooth 4–manifold.We will describe the branched covering in the statement of Theorem 1, that is f : T × Σ g → X , as a composition of four simpler double branched coverings f , f , f , f .While all the latter will be branched over embedded orientable surfaces, the branch locusof the composition will typically be a singular surface. That said, up to perturbing thebranched coverings f i , we can assume that the only singularities (if any) are transversaldouble points.For the clarity of the exposition, we will not explicitly keep track of how the topologyis growing at each step, but instead, we will illustrate with some examples in Remark 6how one can deduce this information. D. AUCKLY, R. ˙I. BAYKUR, R. CASALS, S. KOLAY, T. LIDMAN, AND D. ZUDDAS
Step 1:
By Wall [29], the connected sum of X with a certain number of copies of S × S is diffeomorphic to a connected sum of copies of the standard 4–manifolds CP , S × S and the K3 surface, taken with either orientations. Note that when X is spin, thedecomposition has only spin connected summands, and also that the resulting 4–manifolddoes satisfy 11 / m is large enough.Moreover, since we have K3 ∼ = ( S × S ) and CP S × S ) ∼ = CP CP [12, Page 344], the complete decomposition as above can be written as a K3 b ( S × S )or a K3 b ( S × S ) when X is spin (depending on the sign of the signature σ ( X )),and as a CP b CP when X is non-spin, for some non-negative integers a and b whichare not both zero. In the spin case, we can guarantee that b ≥ a by taking sufficientlymany stabilizations.The conjugation map ( z , z ) (¯ z , ¯ z ), which is an anti-holomorphic involution on CP × CP ∼ = S × S , induces a double branched covering S × S → S , where thebranch locus is the unknotted T ⊂ S (bounding a handlebody). Taking equivariantconnected sum of m copies of it, we get an involution on m ( S × S ), which induces adouble covering m ( S × S ) → S branched along an unknotted Σ m , for every m ≥ X branched along an unknotted Σ m in X (viewing X ∼ = X S , take an unknotted Σ m in S ), which we denote by f : X → X ,where clearly X ∼ = X X m ( S × S ). We choose m ≥ X m ( S × S )completely decomposes, so does X (as one gets at least m copies of S × S afterdecomposing X m ( S × S )). Then X is diffeomorphic to one of the standard connectedsums we listed above. Step 2:
We would like to obtain a double branched covering of X by some g ( S × S ).We will describe this covering in essentially two different ways, depending on whether X (and thus X ) is spin or not.The K3 surface can be obtained as a holomorphic double covering of S × S branchedalong a curve of bi-degree (4 ,
4) in CP × CP ∼ = S × S [12, Page 262]. Reversingthe orientations, we see that K3 is also a double branched covering of S × S (recallthat S × S admits an orientation-reversing diffeomorphism). By taking equivariantconnected sums, we can then express both n K3 and n K3 as branched double coveringsof n ( S × S ). Taking n = 2 a , we then conclude that a K3 b ( S × S ) admits a doublebranched covering by a K3 a K3 b − a ) ( S × S ). Since K3 ∼ = ( S × S ), wehave obtained the desired double branched cover g ( S × S ), for g = 40 a + 2 b . Mirroringthe same argument, we see that a K3 b ( S × S ) is also double branched covered bysome g ( S × S ). This concludes the construction in the spin case.The following variation can be run for both spin and non-spin manifolds. Switching thetwo factors ( z , z ) ( z , z ), which is a holomorphic involution on CP × CP ∼ = S × S ,induces a double branched covering S × S → CP , where the branch locus is thequadric (this may be interpreted as the map taking a pair of numbers to the quadraticequation having those roots). Reversing the orientations, we obtain a double branched RANCHED COVERING 4-MANIFOLDS 5 covering over CP . Taking equivariant connected sums once again, we then deduce that a ( S × S ) b ( S × S ) is a double branched covering of a CP b CP . So in thenon-spin case, we arrive at the desired double covering g ( S × S ) as well.We let f : X → X denote the double branched covering we described in either case. Step 3:
We next show that S × g ( S × S ) is a double branched covering of g ( S × S ),which will prescribe our next covering f : X → X . A similar double branched coveringover g ( CP CP ) was described by Neofytidis in [22, Theorem 1].The hyperelliptic involution on T induces a double branched covering p : T → S with four simple branch points x , x , x , x ∈ S . Taking its product with the identitymap on S yields a double branched covering p × id S : T × S → S × S with branchlocus { x , x , x , x } × S . Note that if g = 1, we can stop here and skip Step 4.Let D ⊂ S be a 2–disk containing exactly two branch points of p . So A = p − ( D ) ⊂ T is an equivariant annulus that contains two fixed points of the hyperelliptic involution.Moreover, D can be chosen such that A is a union of fibers of the trivial S –bundle T = S × S → S given by the canonical projection onto the second factor.Let S ∼ = S × { y } ⊂ S × S be a fiber sphere, for a certain y ∈ S . Let D ′ ⊂ S bea disk centered at y . Then, U = D × D ′ ⊂ S × S is a fibered bidisk, whose preimage V = ( p × id S ) − ( U ) ∼ = A × D ′ is a fibered neighborhood of a fiber of the trivial S –bundle T × S = S × ( S × S ) → S × S . By taking two copies of the branched covering T × S → S × S , and performingequivariant fiber sum upstairs along V and connected sum downstairs along D × D ′ ∼ = D ,and repeating the construction for every g ≥
2, we finally get a branched double covering S × g ( S × S ) → g ( S × S ).We can also describe this branched covering as follows: start with a double covering q : S × D → D branched over two points in Int D (this is the above branched cov-ering A → D ), so the product q × id D : S × D × D → D × D yields a doublecovering q ′ : S × D → D branched over the union of two parallel proper trivial arcsin D (this fills the above branched covering p : T → S ), up to the identifications S × D × D ∼ = S × D and D × D ∼ = D . Then, we get a double branched covering q ′′ = q ′ × id S : S × D × S → D × S . Let D ⊂ S be a 2–disk. Up to the iden-tification D × D × S ∼ = D × S , we consider the bidisks C − = D × {− } × D and C + = D × { } × D ⊂ ∂ ( D × S ), each of which intersects the branch locus of q ′′ along the union of two parallel proper trivial 2–disks. Consider g copies of q ′′ , say q ′′ i : ( S × D × S ) i → ( D × S ) i , and let C − i , C + i ⊂ ∂ ( D × S ) i be the correspondingbidisks. Thus, we obtain a double branched covering q ′′′ = q ′′ ∪ · · · ∪ q ′′ g : ∪ i ( S × D × S ) i → ∪ i ( D × S ) i , where ( D × S ) i is attached to ( D × S ) i +1 by identifying C + i with C − i +1 and( S × D × S ) i is attached to ( S × D × S ) i +1 by identifying ( q ′′ i ) − ( C + i ) with D. AUCKLY, R. ˙I. BAYKUR, R. CASALS, S. KOLAY, T. LIDMAN, AND D. ZUDDAS ( q ′′ i +1 ) − ( C − i +1 ) in the obvious way, for all i = 1 , . . . , g −
1. This in turn is a doublebranched covering q ′′′ : S × ♯ g ( D × S ) → ♯ g ( D × S ) , as it can be easily realized by looking at the attaching maps, where ♯ denotes the bound-ary connected sum. Finally, the desired branched covering S × g ( S × S ) → g ( S × S )can be obtained by restricting q ′′′ to the boundary. Step 4:
Our final double branched covering f : T × Σ g → S × g ( S × S ) is aspecial case of Proposition 2 and can be obtained by taking the product of the identitymap on the S factor with a double branched covering S × Σ g → g ( S × S ). Thelatter can be derived from the work of Sakuma [27] we mentioned in the introduction,or from Montesinos’ alternative construction [20], which is quicker to describe here: theinvolution ( z, t ) → (¯ z, − t ) on the annulus A = S × [ − , ⊂ C × R induces a doublecovering q : A → D branched at two points (this is same as the double branched coverdescribed in Step 3), so we get a double branched covering q × id S : A × S → D × S .Then, for any open book decomposition of a closed connected oriented 3-manifold Y with pages Σ mk and monodromy φ , we can get a double covering h : Y ′ → Y branchedover two parallel copies of the binding, where Y ′ is now a surface bundle whose fiberand the monodromy are the doubles of Σ mk and φ . Indeed, by lifting the usual splitting Y = ( D × ∂ Σ mk ) ∪ ∂ T ( φ ) that gives the open book decomposition of Y , with the branchlink contained in D × ∂ Σ mk , and where T ( φ ) denotes the mapping torus of φ , one obtainsa splitting Y ′ = ( A × ∂ Σ mk ) ∪ ∂ ( T ( φ ) ∪ T ( φ ) ), with the annulus A instead of D , where T ( φ ) and T ( φ ) are two disjoint copies of T ( φ ) (the branched covering h : Y ′ → Y istrivial over T ( φ )). By looking at the attaching maps, it is immediate to get the bundlestructure on Y ′ as above.In our case, since g ( S × S ) admits a planar open book with pages Σ g +10 and φ = id,we obtain the desired covering. (The covering produced by the arguments of both Sakumaand Montesinos in this simple setting is equivalent to the one given in [16, Proposition 4].)The composition f = f ◦ f ◦ f ◦ f : T × Σ g → X gives the desired covering. The spin case:
Let us conclude by observing that our construction is natural with respectto the spin structures, when X is spin, and then briefly discuss the topology of the branchlocus of f .Recall that a spin structure on a 4–manifold is the same as a trivialization of thetangent bundle over the 1–skeleton that extends over the 2–skeleton [19, 14]. We mayuse a handlebody decomposition in this definition. Given an unramified cover over a spin4–manifold the trivialization will lift to the tangent bundle of the cover restricted tothe 1–skeleton and any extension to the 2–skeleton, so there is a natural lift of a spinstructure to a covering space.Now consider a 2–fold branched covering with branch locus B . We may build a handledecomposition of the base in the following way. Start with a handle decomposition of B . RANCHED COVERING 4-MANIFOLDS 7
This extends to a handle decomposition of a tubular neighborhood of B with only zero,one and two handles. Now extend this to a handle decomposition of the rest of the base X . Finally turn the entire handle decomposition over. Notice that all of the 1–handlesof this new handle decomposition are in the exterior of B . Each of these handles lifts tothe cover of the exterior and the restriction of the spin structure to the exterior lifts tothe cover. We now complete the handle decomposition of the total space of the branchedcover as follows. Use the identification of the inverse image e B with B to construct adecomposition of e B which is then extended to a decomposition of the normal bundle of e B . Turn this upside down and add it the to decomposition of the inverse image of theexterior. This only adds 2-, 3- and 4-handles to the decomposition. It is not necessarilytrue that the trivialization of the tangent bundle over the 1–skeleton will extend over the2–skeleton. It will extend precisely when the mod two reduction of the integral homologyclass [ B ] / Z coefficients [4, 21]. Notethat the class of [ B ] is necessarily divisible by 2 due to the existence of the doublebranched cover. So, a spin structure does not have to lift to the total space of a 2–foldbranched covering, but if it does, there is a natural lift.It is now straightforward to check that each double cover f i that we employed in ourconstruction when X is spin satisfies the above criterion, so for the initial spin structure s on X , there is a spin structure s ′ on X ′ ∼ = T × Σ g constructed this way. (Note thatthere are 2 g +1) different spin structures on X ′ .) Thus the branched covering X ′ → X is compatible with the spin structures s on X and s ′ on X ′ . The branch locus:
The branch locus B f ⊂ X of f is given by B f = B f ∪ f (cid:16) B f ∪ f (cid:0) B f ∪ f ( B f ) (cid:1)(cid:17) , where B f i ⊂ X i − denotes the branch locus of f i , for i = 1 , , ,
4, with X = X . Each B f i is a smooth embedded closed orientable surface in X i − . By taking into account thateach covering f i is two-to-one and its tangent map has a 2-dimensional kernel along thebranch set, an easy transversality argument based on perturbing the f i ’s up to isotopy,shows that B f can be assumed to be a smooth closed orientable surface self-transversallyimmersed in X . (cid:3) Ancillary Remarks
Let us list a few comments in relation to Theorem 1, its proof and related works.
Remark 4 (Variations) . In Step 1 above we could have stabilized by taking connectedsums with copies of CP and CP so that we get a double covering g : a CP b CP → X branched over a genus m non-orientable surface, which is trivially embedded in X , forcertain integers a, b and m (once again by Wall [29]). The complex conjugation on CP induces this double covering CP → S branched over the standard smooth RP ⊂ S D. AUCKLY, R. ˙I. BAYKUR, R. CASALS, S. KOLAY, T. LIDMAN, AND D. ZUDDAS [18, 17]. Now, we can invoke Theorem 1.2 in [23] to conclude that there exists a 4–foldsimple branched covering g : Σ h × Σ g → a CP b CP for every given a, b, h ≥
0, andfor some g large enough. Thus, the composition g = g ◦ g : Σ g × Σ h → X is a degree 8branched covering, whose branch locus can be assumed to be a smooth self-transversallyimmersed non-orientable surface in X .Again by Theorem 1.2 in [23] (see also Remark 2 therein), there exist degree 4 branchedcoverings T = T × T → X , with X = m CP n CP and X = n ( S × S ), forevery m, n ≤
3. Note that the case X = S × S is straightforward by taking theproduct p × p : T × T → S × S , and the case X = ( S × S ) was previouslyobtained by Rickman [26]. Branched coverings from the n –dimensional torus are relevantin connection with the theory of quasiregularly elliptic manifolds, see Bonk and Heinonen[3]. In this direction, a result by Prywes [24, Theorem 1.1] implies that if there is abranched covering T → X , then b ( X ) ≤ b ( X ) ≤
6, so in Theorem 1 we cannottake g ≤ b ( X ) ≥ Remark 5 (Branched cover geometries) . Theorem 1 and our subsequent remark inthe introduction imply that any X with finite π ( X ) is branched covered by T × Σ g ,where it is easy to see from our proof that we can always assume g ≥
2. In terms of4–dimensional geometries [13], this shows that all such X can be branched covered by a4–manifold with E × H geometry. However, if we replace the double branched covering h : Y ′ → g ( S × S ) we used in the construction of f = id S × h with the one built byBrooks in [5], we can also get Y ′ to be a Σ g –bundle over S with hyperbolic total space.Therefore, any X with finite π ( X ) can also be branched covered by a 4–manifold with E × H geometry. Similarly, one can modify the construction in Proposition 2 to get adouble branched cover of any product 4–manifold S × Y by a 4–manifold with E × H geometry. Remark 6 (Topology of the branched coverings) . Here we will try to demonstrateby way of example how one can control the topology of the branched coverings inTheorem 1. For some variety, we will run our construction for two infinite families ofirreducible 4–manifolds which are not completely decomposable: Dolgachev surfaces,which are non-spin complex surfaces of general type, and knot surgered K3 surfacesof Fintushel–Stern, which include spin 4–manifolds that do not admit any symplecticstructures [11]. The members of either one of these two families of simply-connected4–manifolds completely decompose after a single stabilization by S × S ; see e.g. [1].Now, if X is a Dolgachev surface, we can take the branch locus of f as an unknotted T , and get X = CP CP . The branched covering f performed along the con-nected sum of 22 quadrics (each coming from distinct copies of CP and CP ) then gives X = ( S × S ), and the last two coverings yield X ′ = T × Σ . If we take X to be a RANCHED COVERING 4-MANIFOLDS 9 knot surgered K3 surface instead, thinking ahead of the second step, we take the branchlocus of f this time as Σ , so X = K3 ( S × S ). The next double cover f is takenalong a connected sum of four bi-degree (4 ,
4) curves (each coming from distinct copiesof S × S , with non-complex orientation), and we get X = K3 K3 ∼ = ( S × S ).The last two coverings this time yield X ′ = T × Σ . Remark 7 (Symplectic domination) . A recent article of Fine–Panov provides a symplec-tic domination result [10, Theorem 1] which is worth mentioning here. Their beautifulconstruction is very general: for any closed oriented even dimensional smooth manifold M , the authors build a closed symplectic manifold S of the same dimension with a pos-itive degree map f : S → M . In dimension 4, where we can compare their result withours in Theorem 1, their symplectic manifold S is constructed as a Donaldson hypersur-face in the 6–dimensional symplectic twistor space Z of a negatively pinched manifold N , where the latter admits a degree one map g : N → M . The construction of N , withsectional curvature arbitrarily close to − Z of N isa symplectic 6–manifold.) Secondly, the construction of a symplectic hypersurface S in N , which is built through asymptotically holomorphic techniques of [8], is also implicitand the smooth topology of S is effectively impossible to control. Hence, one does nothave any information on the smooth topology of the dominating symplectic 4–manifold S , other than that it is of general type, i.e. of Kodaira dimension 2 [10]. Besides the veryimplicit nature of this construction, since the map f : S → M factors through the de-gree one map g above, the authors’ domination is essentially never a branched covering.Moreover, because the symplectic twistor space Z is in fact known to be non-K¨ahler [25],the dominating symplectic 4–manifold S has a priori no reason to be a K¨ahler surface.On the other hand, the dominating symplectic 4–manifold X ′ = T × Σ g of Theorem 1 isobviously a K¨ahler surface, and X ′ in both Theorem 1 and Proposition 2 is of Kodairadimension −∞ , 0 or 1, depending on whether this (possibly trivial) Σ g –bundle over T ,has fiber genus g = 0, 1 or ≥
2, respectively.Domination is certainly distinct from branched covering as the following exampleshows. There is a degree one map from Σ × Σ to Σ × Σ given by the extensionof the natural collapse of a copy of Σ × Σ to Σ × Σ . However there can be nobranched covering from Σ × Σ to Σ × Σ since the Gromov norm of the former is24(4 − −
1) = 72, the Gromov norm of the latter is 24(3 − −
1) = 48 and theGromov norm is super multiplicative with respect to degree [6].
References [1] R. I. Baykur,
Dissolving knot surgered -manifolds by classical cobordism arguments, J. KnotTheory Ramifications, 27:5 (2018), 1871001.[2] R. I. Baykur and N. Sunukjian,
Round handles, logarithmic transforms and smooth -manifolds ,J. Topol. 6 (2013), no. 1, 49–63.[3] M. Bonk and J. Heinonen, Quasiregular mappings and cohomology , Acta Math. 186 (2001), no.2, 219–238.[4] N. Brand,
Necessary conditions for the existence of branched coverings,
Invent. Math. 54 (1979),1–10.[5] R. Brooks,
On branched coverings of -manifolds which fiber over the circle, J. Reine Angew.Math. 362 (1985), 87–101.[6] M. Bucher-Karlsson,
The simplicial volume of closed manifolds covered by H × H , J. Topol. 1(2008), no. 3, 584–602.[7] P. T. Church, Differentiable open maps on manifolds , Trans. Amer. Math. Soc. 109 (1963),87–100.[8] S. K. Donaldson,
Symplectic submanifolds and almost-complex geometry , J. Differential Geom.44 (1996), no. 4, 666–705.[9] Y. Eliashberg,
Recent advances in symplectic flexibility , Bull. Amer. Math. Soc. (N.S.) 52 (2015),no. 1, 1–26.[10] J. Fine and D. Panov,
Symplectic domination , Bull. London. Math. Soc. doi:10.1112/blms.12402,2020.[11] R. Fintushel and R. Stern,
Knots, links, and -manifolds , Invent. Math. 134 (1998), no. 2,363–400.[12] R. Gompf and A. Stipsicz, 4 –manifolds and Kirby calculus , Graduate Studies in Mathematics,vol. 20, American Mathematical Society, Providence, RI. 1999.[13] J. Hillman, Four-manifolds, geometries and knots , Geom. Topol. Monographs, 5 (2002), 379 pp.[14] R. Kirby,
The topology of 4-manifolds , Lecture Notes in Mathematics, 1374. (1989), 108 pp.[15] D. Kotschick and C. L¨oh,
Fundamental classes not representable by products,
J. London Math.Soc. (2) 79 (2009), 545–561.[16] D. Kotschick and C. Neofytidis,
On three-manifolds dominated by circle bundles,
Math. Z. 274(2013) 21–32.[17] N. H. Kuiper,
The quotient space of CP by complex conjugation is the -sphere , Math. Ann.208 (1974), 175–177.[18] W. S. Massey, The quotient space of the complex projective plane under conjugation is a 4-sphere , Geometriae Dedicata 2 (1973), 371–374.[19] J. Milnor,
Spin structures on manifolds , Enseign. Math. 9 (1963), 198–203.[20] J. M. Montesinos, On -manifolds having surface bundles as branched coverings , Proc. Amer.Math. Soc. 101 (1987), no. 3, 555–558.[21] S. Nagami, On spin structures of double branched covering spaces , JP J. Geom. Topol. 14 (2013),no. 2, 119–147.[22] C. Neofytidis,
Branched coverings of simply-connected manifolds , Topology Appl. 178 (2014),360–371.[23] R. Piergallini and D. Zuddas,
Branched coverings of CP and other basic -manifolds , to appearin Bull. London Math. Soc., https://arxiv.org/abs/1707.03667 .[24] E. Prywes, A bound on the cohomology of quasiregularly elliptic manifolds , Ann. of Math. 189(2019), 863–883.[25] A. G. Reznikov,
Symplectic twistor spaces , Ann. Global Anal. Geom. 11 (1993), no. 2, 109–118.
RANCHED COVERING 4-MANIFOLDS 11 [26] S. Rickman,
Simply connected quasiregularly elliptic 4-manifolds , Ann. Acad. Sci. Fenn. Math.31 (2006), no. 1, 97–110.[27] M. Sakuma,
Surface bundles over S which are -fold branched cyclic coverings of S , Math.Sem. Notes 9 (1981), 159–180.[28] W. P. Thurston, Some simple examples of symplectic manifolds , Proc. Amer. Math. Soc. 55(1976), no.2, 467–468.[29] C. T. C. Wall,
On simply-connected -manifolds, J. London Math. Soc. 39 (1964) 141–149.
Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
Email address : [email protected] Department of Mathematics and Statistics, University of Massachusetts, Amherst,MA 01003-9305, USA
Email address : [email protected] University of California Davis, Dept. of Mathematics, Shields Avenue, Davis, CA95616, USA
Email address : [email protected] School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA
Email address : [email protected] Department of Mathematics, North Carolina State University, Raleigh, NC, 27607,USA
Email address : [email protected] Dipartimento di Matematica e Geoscienze, Universit`a di Trieste, Via Valerio 12/1,34127 Trieste, Italy
Email address ::