7-dimensional simply-connected spin manifolds whose integral cohomology rings are isomorphic to that of CP 2 × S 3 admit round fold maps
aa r X i v : . [ m a t h . A T ] S e p -DIMENSIONAL SIMPLY-CONNECTED SPIN MANIFOLDSWHOSE INTEGRAL COHOMOLOGY RINGS ARE ISOMORPHICTO THAT OF C P × S ADMIT ROUND FOLD MAPS
NAOKI KITAZAWA
Abstract. fold maps, which are higher dimensional versions of Morse functions, on the man-ifolds. The studies have been motivated by studies of special generic maps,which are higher dimensional versions of Morse functions on homotopy sphereswith exactly two singular points, characterizing them topologically except 4-dimensional cases and the class contains canonical projections of unit spheresfor example. This class has been found to be interesting, restricting the topolo-gies and the differentiable structures of the manifolds strictly owing to studiesof Saeki, Sakuma, Wrazidlo and so on. The present paper concerns fold mapson 7-dimensional simply-connected spin manifolds whose integral cohomologyrings are isomorphic to that of C P × S . Introduction, terminologies and notation. fold maps, which are higher dimensionalversions of Morse functions, on the manifolds. The studies have been motivatedby studies of special generic maps, which are higher dimensional versions of Morsefunctions on homotopy spheres with exactly two singular points, characterizingthem topologically except 4-dimensional cases and the class contains canonical pro-jections of unit spheres for example. This class has been found to be interesting,
Mathematics Subject Classification.
Primary 57R45. Secondary 57R19.
Key words and phrases.
Singularities of differentiable maps; fold maps. Differential structures.Higher dimensional closed and simply-connected manifolds.2020
Mathematics Subject Classification : Primary 57R45. Secondary 57R19. restricting the topologies and the differentiable structures of homotopy spheres andmanifolds admitting them strictly owing to studies of Saeki, Sakuma, Wrazidlo andso on: for example 7-dimensional exotic homotopy spheres, which are not diffeo-morphic to the standard sphere S , never admit special geneirc maps into R n for n = 4 , , n = 3 ([13], [14], [15],[16], [18] and so on). It is also an important fact that construction of explicitfold maps on explicit manifolds is difficult even for fundamental manifolds and ingeneral. The present paper concerns fold maps on 7-dimensional simply-connectedspin manifolds whose integral cohomology rings are isomorphic to that of C P × S .A study on understanding the class of these manifolds is demonstrated in [17] forexample.1.1. Terminologies and notation on smooth maps and so on.
Throughoutthe present paper, manifolds and maps between manifolds are smooth (of class C ∞ ).Diffeomorphisms on manifolds are always smooth and the diffeomorphism group ofa manifold is defined as the group of all smooth diffeomorphisms on the manifold.For bundles whose fibers are manifolds, the structure groups are subgroups of thediffeomorphism groups or the bundles are assumed to be smooth unless otherwisestated. A linear bundle means a smooth bundle whose fiber is a unit sphere or aunit disc in a Euclidean space with structure group acting linearly in a canonicalway on the fiber.A singular point of a smooth map is a point in the domain at which the dimensionof the image of the differential is smaller than both the dimensions of the manifoldsof the domain and the target set. We call the set of all singular points the singularset of the map. We call the image of the singular set the singular value set of themap. The regular value set of the map is the complementary set of the singularvalue set. A singular ( regular ) value is a point in the singular (resp. regular) valueset. || x || denotes the distance between x ∈ R k and the origin 0 ∈ R k where theEuclidean space is endowed with the standard metric.1.2. The definition and fundamental properties of a fold map and ex-plicit fold maps.
Hereafter, let m > n ≥ m -dimensional smooth manifold with no boundary into an n -dimensional smoothmanifold with no boundary is said to be a fold map if at each singular point p , themap is represented as ( x , · · · , x m ) ( x , · · · , x n − , P m − ik = n x k − P mk = m − i +1 x k )for suitable coordinates and an integer 0 ≤ i ( p ) ≤ m − n +12 . For the singular point p , i ( p ) is unique and called the index of p . The set consisting of all singular pointsof a fixed index of the map is a closed submanifold of dimension n − m -dimensional manifold. The restriction map to the singular setis an immersion. A special generic map, which is explained before, is defined as afold map such that the index of a singular point is always 0.Explicit construction of fold maps on explicit manifolds are fundamental, im-portant and challenging: for example it also leads us to understand several impor-tant classes of manifolds such as 7-dimensional or higher dimensional closed andsimply-connected manifolds more in geometric and constructive ways. Throughthe challenge, we have obtained the following theorem. For classical theory on7-dimensional homotopy spheres, see [1] and [10] for example. OLD MAPS ON SIMPLY-CONNECTED MANIFOLDS COHOMOLOGICALLY C P × S Theorem . (1) Every 7-dimensional homotopy oriented sphere M of 28 types admits a fold map f : M → R such that f | S ( f ) is an em-bedding satisfying f ( S ( f )) = { x ∈ R | || x || = 1 , , } , that the index ofeach singular point is 0 or 1, and the preimage of a regular value in eachconnected component of f is, empty, diffeomorphic to S , diffeomorphic to S ⊔ S and diffeomorphic to S ⊔ S ⊔ S , respectively.(2) 7-dimensional homotopy sphere M admits a fold map f : M → R suchthat f | S ( f ) is an embedding and that f ( S ( f )) = { x ∈ R | || x || = 1 } ifand only if M is a standard sphere. A 7-dimensional homotopy sphere M admits a fold map f : M → R such that f | S ( f ) is an embedding and thatthe index of each singular pan oint is 0 or 1 as before and in addition thefollowing properties hold if and only if M is the total space of a linearbundle whose fiber is S over S (7-dimensional oriented homotopy spheresof 16 types are represented in this way).(a) f ( S ( f )) = { x ∈ R | || x || = 1 , } .(b) For any connected component C ⊂ f ( S ( f )) and a small closed tubularneighborhood N ( C ), the bundle given by the projection representedas the composition of f | f − ( N ( C )) with the canonical projection to C is trivial.(c) The preimage of a regular value in each connected component is,empty, diffeomorphic to S , and diffeomorphic to S ⊔ S , respectively.As special generic maps, fold maps of a suitable class of maps considered thereaffect the differentiable structures of the homotopy spheres. Note that fold mapshere are round fold maps: a round fold map is a fold map such that the restrictionto the singular set is an embedding and that the singular value set is concentricspheres. Note also that fold maps here have been obtained as specific cases of foldmaps on manifolds represented as connected sums of total spaces of bundles overthe standard sphere of a fixed dimension whose fibers are standard spheres in [2],[3] and [6] for example. Related to this fact, manifolds represented as connectedsums of manifolds represented as products of standard spheres admit special genericmaps into suitable Euclidean spaces whose dimensions are smaller in considerablecases: for example a manifold represented as a connected sum of finite copies of S n − × S m − n +1 admits a special generic map into R n . We can construct the mapso that the restriction to the singular set is an embedding and that the image isa manifold represented as a boundary connected sum of finite copies of S n − × I where I denotes the closed interval.We present the following recent results, showing explicit fold maps on infin-itely many 7-dimensional closed and simply-connected manifolds. Their integralcohomology rings are mutually non-isomorphic and they are not represented asconnected sums of manifolds represented as products of standard spheres as beforeor we cannot know whether they are represented so in general.A crossing of an immersion is the point in the target space such that the preimagehas at least two points. Theorem . Let { G j } j =0 be a sequence of free and finitely generatedcommutative groups such that G = G = Z , that G j is zero for j = 1 ,
6, that G j and G − j are mutually isomorphic for j = 2 , G is smallerthan or equal to that of G . NAOKI KITAZAWA (1) In this situation, there exist a closed, simply-connected and spin manifold M of dimension 7 such that the homology group is free and that H j ( M ; Z )is isomorphic to G j and a fold map f : M → R such that f | S ( f ) is em-bedding, that the index of each singular point is always 0 or 1, and thatfor each connected component of the regular value set of f , the preimageof a regular value in each connected component is, empty, diffeomorphicto S , or diffeomorphic to S ⊔ S . Furthermore, if G and G are non-trivial groups, then there exist infinitely many closed and simply-connectedmanifolds M λ of dimension 7 admitting the fold maps as before such that H j ( M λ ; Z ) is isomorphic to G j and that the cohomology rings of M λ and M λ are not isomorphic for distinct λ , λ ∈ Λ.(2) In this situation, there exist a closed, simply-connected and spin manifold M of dimension 7 such that the homology group is free and that H j ( M ; Z )is isomorphic to G j and a fold map f : M → R such that f | S ( f ) is animmersion which may have crossings, that the index of each singular pointis always 0 or 1, and that for each connected component of the regular valueset of f , the preimage of a regular value in each connected component is,empty, diffeomorphic to S , diffeomorphic to S ⊔ S , or diffeomorphicto S ⊔ S ⊔ S . Furthermore, if G and G are non-trivial groups, thenthere exist infinitely many closed and simply-connected manifolds M λ ofdimension 7 admitting the fold maps as before such that H j ( M λ ; Z ) isisomorphic to G j and that the cohomology rings of M λ and M λ are notisomorphic for distinct λ , λ ∈ Λ.This result is seen as one capturing the cohomology rings of 7-dimensional closedand simply-connected manifolds of suitable families via explicit fold maps on theminto R .The following theorem is the main theorem. Main Theorem.
There exists an infinite family { M k } k ∈ Z of closed, simply-connectedand spin (oriented) manifolds whose integral cohomology rings are isomorphic tothat of C P × S such that the 1st-Pontryagin class of M k is 4 k times a generator of H ( M k ; Z ) ∼ = Z and these manifolds admit round fold maps { f k : M k → R } . Fur-thermore, every 7-dimensional, closed, simply-connected and spin manifold whoseintegral cohomology ring is isomorphic to that of C P × S admits a round foldmap into R .Remark 1 . [8] announces that the construction of explicit fold maps on manifolds inMain Theorem has been done through the proof of Theorem 2. However, the authorfound that it contains crucial errors on construction of maps on these manifolds.This will be revised and we will remove arguments on construction on the manifolds.Most of the results and arguments are true as the author considers now. See alsoRemark 2 later, contradicting this false statement. In addition, the author believesthat the class of 7-dimensional manifolds in the latter statement in Theorem 2 iswider than that in the former statement. However, we could not see this yet.1.3. The content of the paper and acknowledgement.
The present paperconsists of two sections and the remaining section is devoted to the proof of themain theorem. We construct a round fold map on a 7-dimensional, closed andsimply-connected manifold and see that this is cohomologically C P × S . We also OLD MAPS ON SIMPLY-CONNECTED MANIFOLDS COHOMOLOGICALLY C P × S present related results. We do not need to understand technique of construction ofexplicit fold maps such as arguments in [7], [8], and so on well.1.4. Acknowledgement.
The main theorem and related results.
We will prove the main theorem. We introduce fundamental terminologies.For a finitely generated and free commutative group G . let a ∈ G be a non-zeroelement we cannot represent as a = ra ′ for any pair ( r, a ′ ) ∈ ( Z − {− , , } ) × G and G is the internal direct sum of the group generated by the one element set { a } and a subgroup of G . A homomorphism a ∗ : G → Z is said to be the dual of a if a ∗ ( a ) = 1 and for any subgroup G ′ such that G is the internal direct sum of thegroup generated by the one element set { a } and G ′ , a ∗ ( G ′ ) = { } .For a closed and orientable manifold (which is oriented), consider an integralhomology class of degree k . It is said to be represented by a closed and connectedsubmanifold of dimension k with no boundary if the class is equal to the value ofthe homomorphism canonically induced from the inclusion at the fundamental class of the submanifold with an orientation: the fundamental class is a generator of theintegral cohomology of degree k (of the k -dimensional manifold).2.1. The -dimensional complex projective space and its structure. The3-dimensional complex projective space C P is regarded as the total space of alinear bundle over S whose fiber is S . The following theorem presents severalclassical and important properties. Theorem . For the projection π C P : C P → S of the linear bundle over S whosefiber is S , the following properties hold.(1) There exists a complex projective plane C P C P ⊂ C P being a complexsubmanifold and representing a generator of H ( C P ; Z ), isomorphic to Z .(2) π C P | C P C P is a covering on the preimage of a smoothly embedded 4-dimensional standard closed disc in S .(3) There exists a complex projective line C P C P ⊂ C P C P being a complexsubmanifold and representing the generator of H ( C P ; Z ), isomorphic to Z . The complex projective line can be taken as a fiber of the bundle.(4) The square of the dual of the integral homology class represented by C P C P with an orientation is the dual of the integral homology class representedby C P C P with an orientation. The product of the dual of the integralhomology class represented by C P C P with an orientation and the dual ofthe integral homology class represented by C P C P with an orientation isa generator of H ( C P ; Z ), isomorphic to Z .2.2. A proof of the main theorem.
We consider a submersion obtained bycomposing the canonical projection from C P × S to C P with π C P : C P → S in Theorem 3. We have a 7-dimensional, closed, simply-connected and spin manifold M and a fold map f S : M → S by exchanging a restriction to the preimage of asmoothly embedded 4-dimensional standard disc D so that the following propertieshold. Let D ′ ⊂ S be a smoothly embedded 4-dimensional standard disc containingthe disc D in the interior. NAOKI KITAZAWA (1) f S is a fold map such that the singular set S ( f S ) is diffeomorphic to S ,that f S | S ( f S ) is an embedding and that f S ( S ( f S )) = ∂D ′ .(2) The preimage of a regular value of each connected component of S − f S ( S ( f S )) is S and S × S , respectively.Note that in the second property, (a copy of) S is regarded as a manifold obtainedby a so-called handle attachment to (a copy of) S × S : we attach a D × D along S × {∗} ⊂ S × S , respecting the structure of a Morse function.See [11] for example for related theory. The following show topological propertiesof M obtained by investigating the two properties of the map before.(1) There exists an oriented complex projective plane C P M ⊂ M being asubmanifold such that f S | C P M is a covering on the preimage of the disc D in S .(2) The integral cohomology ring of M is isomorphic to that of C P × S .(3) The integral homology class represented by the submanifold we can re-gard S × {∗} ⊂ S × S of the preimage of a regular value diffeomor-phic to S × S is a generator of H ( M ; Z ) ∼ = Z . The integral homologyclasses represented by the preimages of regular values are a generator of H ( M ; Z ) ∼ = Z . The square of the dual of the integral homology class rep-resented by S × {∗} before with an orientation is the dual of the integralhomology class represented by C P M ⊂ M .(4) The 1st Pontryagin class of M is four times a generator of H ( M ; Z ), iso-morphic to Z .We take the union of two smoothly and disjointly embedded 4-dimensional stan-dard closed discs D := D and D so that they are mutually in distinct con-nected components of the regular value set of f S . By composing the restrictionof f S to the preimage f S − ( D ⊔ D ) with a natural 2-fold covering over a 4-dimensional standard closed disc D , we have a trivial bundle over D whose fiberis S ⊔ S × S . We consider the projection and embed the base space as the space { x ∈ R | | x | ≤ r } for a positive integer r >
0. We denote the resulting submer-sion by ¯ f r . The restriction of f S to f S − ( S − Int( D ⊔ D )) is regarded as aproduct map of a Morse function with exactly one singular point and the identitymap on id S . On the domain of this, we can construct a product map of a Morsefunction ¯ f and the identity map on id S such that the Morse function ¯ f satisfiesthe following properties.(1) The function is a function on a manifold diffeomorphic to a manifold ob-tained by removing the interior of a 4-dimensional standard closed discsmoothly embedded in the interior of S × D .(2) The minimum is r .(3) The preimage of the minimum r of the function coincides with the bound-ary and contains no singular point.(4) The function has exactly three singular points and at distinct singularpoints, the values are distinct.We can set the target space of the function ¯ f as the half-closed interval [ r , + ∞ ) ⊂ R . Thanks to the structures of manifolds and maps, we can glue the maps ¯ f r andthe product map of ¯ f and the identity map on the boundary of { x ∈ R | | x | ≤ r } to obtain a global fold map on the original manifold M into R . The topologicalproperties of M and f S : M → S presented before together with fundamental OLD MAPS ON SIMPLY-CONNECTED MANIFOLDS COHOMOLOGICALLY C P × S Figure 1.
The image and the preimages of regular values of around fold map in Theorem 4: circles represent the singular valueset and 3-dimensional spheres.propositions on 1st Pontryagin classes enable us to construct a similar fold mapon a manifold whose integral cohomology ring is isomorphic to that of M (, whichis oriented suitably,) and whose 1st Pontryagin class is 4 k times a generator of H ( M ; Z ) ∼ = Z for an arbitrary integer k . We need to change the way we glue thetrivial bundle over D whose fiber is S . See [12] and see also Theorems 3 and 4 of[7].This yields the following main theorem. A round fold map is a fold map suchthat the restriction to the singular set is an embedding and that the singular valueset is concentric. The class of round fold maps was first defined by the author in2010s and have been systematically studied. See [2], [3], [4], [5] and [6] for moredetailed presentations. Theorem . There exist an infinite family { M k } k ∈ Z of closed, simply-connected andspin (oriented) manifolds whose integral cohomology rings are isomorphic to thatof C P × S such that the 1st-Pontryagin class of M k is 4 k times a generator of H ( M k ; Z ) ∼ = Z and a family of round fold maps { f k : M k → R } satisfying thefollowing properties.(1) The singular set of each map consists of exactly three connected compo-nents.(2) The preimage of a regular value in each connected component of the regularvalue set in the image is diffeomorphic to S , S × S and S ⊔ S × S ,respectively.See also FIGURE 1. Theorem . (1) For an ar-bitrary 7-dimensional, closed, simply-connected and spin manifold X whoseintegral cohomology ring is isomorphic to that of C P × S , consider an iso-morphism φ of integral cohomology rings from the integral cohomology ringof C P × S onto that of X and set a as a generator of H ( C P × S ; Z ) ∼ = Z .In this situation, the 1st Pontryagin class of X is represented as 4 kφ ( a ) forsome integer k . Moreover, the topology of a 7-dimensional closed, simply-connected and spin manifold whose integral cohomology ring is isomorphicto that of C P × S is determined by its 1st Pontryagin class.(2) For any pair of mutually homeomorphic 7-dimensional, closed, simply-connected and spin manifolds whose integral cohomology rings are isomor-phic to that of C P × S , one of the two manifolds is represented as aconnected sum of the other manifold and a suitable 7-dimensional homo-topy sphere. Moreover, for a 7-dimensional, closed, simply-connected andspin manifold whose integral cohomology ring is isomorphic to the integralcohomology ring of C P × S , consider a pair of manifolds each of which NAOKI KITAZAWA ↓ Figure 2.
Construction of a round fold map on the manifold rep-resented as a connected sum of the two manifolds admitting foldmaps in Corollary 1: for example manifolds represent preimages ofregular values in the connected components of the regular valuesets.is represented as a connected sum of the given manifold and a homotopysphere. They are diffeomorphic if and only if the homotopy spheres arediffeomorphic.
Corollary . A 7-dimensional, closed, simply-connected and spin manifold whoseintegral cohomology ring is isomorphic to that of C P × S always admits a roundfold map into R . Proof.
Theorem 5 implies that a 7-dimensional, closed, simply-connected and spinmanifold whose integral cohomology ring is isomorphic to that of C P × S is alwaysrepresented as a connected sum of a manifold in Theorem 4 and a 7-dimensionalhomotopy sphere. [3] and also [2], [6], and so on, show construction of a new roundfold map from a round fold map in Theorem 4 and a round fold map in Theorem2. We take a 4-dimensional, standard and closed disc in the connected componentof the regular value set of the former map in the center and remove the interiorof a connected component, regarded as the total space of a trivial bundle over thedisc whose fiber is S of the preimage. We remove (the preimage of) the union ofthe interior of the small closed tubular neighborfood of the outermost connectedcomponent of the singular value set of the latter map and the complement of theimage. We glue the two obtained maps in a suitable way to obtain a round foldmap on the desired manifold. See also FIGURE 2. This completes the proof. (cid:3) Remark 2 . The author believes that we can prove the following fact: 7-dimensionalmanifolds of Theorems 3, 4 and 5 and Main Theorem never admit special genericmaps or fold maps into R n (n=1,2,3,4,5,6) such that the index of each singularpoint is 0 or 1 and that preimages of regular values are disjoint unions of standardspheres as in Theorems 1, 2, and so on. References [1] J. J. Eells and N. H. Kuiper,
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