Explicit fold maps on 7-dimensional closed and simply-connected manifolds of new classes
aa r X i v : . [ m a t h . A T ] J un EXPLICIT FOLD MAPS ON 7-DIMENSIONAL CLOSED ANDSIMPLY-CONNECTED MANIFOLDS OF NEW CLASSES
NAOKI KITAZAWA
Abstract.
Closed (and simply-connected) manifolds whose dimensions arelarger than 4 are classified via sophisticated algebraic and abstract theorysuch as surgery theory and homotopy theory. It is difficult to handle 3 or4-dimensional closed manifolds in such ways. However, the latter work per-formed in geometric and constructive ways is not so difficult: the fact that thedimensions are not high enables us to handle the manifolds via diagrams forexample. It is difficult to study higher dimensional manifolds in these ways,although it is natural and important.In the present paper, we present such studies via fold maps, higher dimen-sional versions of Morse functions. The author previously constructed foldmaps on 7-dimensional closed and simply-connected manifolds satisfying ad-ditional conditions on cohomology rings, including exotic homotopy spheres.This paper presents fold maps on such manifolds of new classes including man-ifolds whose integral cohomology rings are isomorphic to that of the productof a complex projective plane and a 3-dimensional sphere. Introduction and fold maps.
Closed (and simply-connected) manifolds whose dimensions are larger than 4are classified via classical and sophisticated algebraic and abstract theory such assurgery theory, homotopy theory and so on.It is difficult to handle 3 or 4-dimensional closed manifolds in such ways. How-ever, this work is, in geometric and constructive ways, not so difficult. The factthat the dimensions are not so high enables us to handle the manifolds via diagramssuch as Heegaard diagrams and Kirby’s diagrams for example.It is difficult to study higher dimensional manifolds in these ways, although it isnatural and important. This paper presents studies via Morse functions and fold maps, which are higher dimensional versions of Morse functions on 7-dimensionalclosed and simply-connected manifolds into the 4-dimensional Euclidean space R .1.1. Notation on differentiable maps and bundles.
Throughout this paper,manifolds and maps between manifolds are fundamental objects and they are smoothand of class C ∞ . Diffeomorphisms on manifolds are always smooth. The diffeo-morphism group of a manifold is the group of all diffeomorphism on the manifolds.For bundles whose fibers are manifolds, the structure groups are subgroups of thediffeomorphism groups or the bundles are smooth unless otherwise stated. A linear bundle is a smooth bundle whose fiber is regarded as a unit sphere or a unit discand whose structure group acts linearly in a canonical way on the fiber. Mathematics Subject Classification.
Primary 57R45. Secondary 57N15.
Key words and phrases.
Singularities of differentiable maps; fold maps. Differential structures.Higher dimensional closed and simply-connected manifolds. A singular point p ∈ X of a differentiable map c : X → Y is a point at which therank of the differential dc of the map is smaller than the dimension on the targetmanifold: rank dc p < dim Y holds where dc p denotes the differential at p . Theset S ( c ) of all singular points is the singular set of c . We call c ( S ( c )) the singularvalue set of c . We call Y − c ( S ( c )) the regular value set of c . A singular ( regular ) value is a point in the singular (resp. regular) value set of c .1.2. Fold maps and investigating algebraic topological and differentialtopological properties of manifolds via fold maps on them.
Let m > n ≥ m -dimensional smooth manifold with noboundary into an n -dimensional smooth manifold with no boundary is said to be a fold map if at each singular point p , the map is represented as( x , · · · , x m ) ( x , · · · , x n − , m − i X k = n x k − m X k = m − i +1 x k )for suitable coordinates and an integer 0 ≤ i ( p ) ≤ m − n +12 We have the following three fundamental properties for this map. • For any singular point p , i ( p ) is unique. • The set consisting of all singular points of a fixed index of the map is aclosed submanifold with of dimension n − m -dimensional manifold. • The restriction map to the singular set is an immersion. i ( p ) is said to be the index of p .In the present paper, we consider fold maps such that the immersions in thethird property satisfy the following two conditions.(1) The preimage of a point in the target space consists of at most two points.(2) For a preimage consisting of exactly two points as before, we denote thepreimage by { p , p } and the value at these points by q in the target space Y . We consider the subspace of the tangent vector space of the target spaceat q represented as the intersection of the images of the differentials of theimmersions at p and p and the dimension is dim Y − . An immersion satisfying the two conditions is said to be normal . Apoint in the target space whose preimage consists of two points is called a crossing .We extend these notions for a family of (smooth) immersions in a canonical way.Fold maps have been important tools in studying geometric properties of mani-folds in the branch of the singularity theory of differentiable maps and applicationto geometry of manifolds as Morse functions are in so-called Morse theory. Stud-ies related this was essentially started in 1950s by Thom and Whitney ([26] and[28]). These studies are on smooth maps on manifolds whose dimensions are largerthan 2 into the plane. After various studies, recently, as an important topic, Saeki,Sakuma and so on, have been studying fold maps satisfying appropriate conditionsand manifolds admitting them in [18], [19], [20], [22], [23] and so on. Several studiessuch as [2], [3], [4] and [5] of the author are motivated by these studies. The au-thor has been developing these studies mainly to handle wider classes of manifoldsadmitting explicit fold maps. As closely related studies, the author also have been
EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 3 studying the topologies of
Reeb spaces of fold maps in [6], [7] and [9] and so on:the
Reeb space of a fold map is the polyhedron whose dimension is equal to thedimension of the target space, defined as the quotient space of the domain obtainedby the following rule: two points in the domain are equivalent if and only if they arein a same connected component of the preimage of a point. These objects inheritmuch information on homology groups, cohomology rings, and so on, of manifoldsadmitting the maps in considerable cases and fundamental and important tools inthe studies.1.3.
Explicit fold maps on -dimensional closed and simply-connectedmanifolds of a class and the main purpose of the present paper. Theorem . Let A , B and C be free finitely generated commutative groupsof rank a , b and c , respectively. Let { a j } aj =1 , { b j } bj =1 and { c j } cj =1 be bases of A , B and C , respectively. For each integer 1 ≤ i ≤ b , let { a i,j } aj =1 a sequence ofintegers given. Let p ∈ B ⊕ C . In this situation, there exist a 7-dimensional closedand simply-connected spin manifold M and a fold map f : M → R satisfying thefollowing properties.(1) H ( M ; Z ) is isomorphic to A ⊕ B and H ( M ; Z ) is isomorphic to B ⊕ C . H ( M ; Z ) is free.(2) There exist suitable isomorphisms φ : A ⊕ B → H ( M ; Z ) and φ : B ⊕ C → H ( M ; Z ) and we can define the duals a j ∗ , b j, ∗ , b j, ∗ and c j ∗ of φ (( a j , φ ((0 , b j )), φ (( b j , φ ((0 , c j )), respectively, and canonically obtainbases of H ( M ; Z ) and H ( M ; Z ). We denote the composition of φ withan isomorphism mapping the elements of the basis to their Poincar´e dualsby φ , : B ⊕ C → H ( M ; Z ). We denote the composition of φ with anisomorphism mapping the elements of the basis to their Poincar´e duals by φ , : A ⊕ B → H ( M ; Z ). In this situation, the following properties hold.(a) (i) The product of a j ∗ and a j ∗ and that of b j , ∗ and b j , ∗ vanishfor any j and j .(ii) The product of a j ∗ and b j , ∗ is a j ,j b j , ∗ ∈ H ( M ; Z ). forany j and j .(b) (i) The product of a j ∗ and φ , ((0 , c )) vanishes for any j and c ∈ C .(ii) The product of b j, ∗ and φ , ((0 , c )) vanishes for any j and c ∈ C .(iii) The product of b j , ∗ and φ , (( b j , j , j ) of distinct numbers.(iv) The product of a j ∗ and φ , (( b j , a j ,j φ , ((0 , b j )) ∈ H ( M ; Z ). for any j and j .(v) The product of b i, ∗ and φ , (( b i , aj =1 a i,j φ , (( a j , ∈ H ( M ; Z ).(c) (i) The product of a j ∗ and φ , (( a j , H ( M ; Z ).(ii) The product of a j ∗ and φ , (( a j , j , j )of distinct numbers.(iii) The product of a j ∗ and φ , ((0 , b )) vanishes for any j and b ∈ B .(iv) The product of b j, ∗ and φ , (( a, j and a ∈ A .(v) The product of b j, ∗ and φ , ((0 , b j )) yields a generator of H ( M ; Z ).(vi) The product of b j , ∗ and φ , ((0 , b j )) vanishes for any pair( j , j ) of distinct numbers. NAOKI KITAZAWA (vii) The product of b j, ∗ and φ , (( b j , H ( M ; Z ).(viii) The product of b j , ∗ and φ , (( b j , j , j ) of distinct numbers.(ix) The product of b j, ∗ and φ , ((0 , c )) vanishes for any j and c ∈ C .(x) The product of c j ∗ and φ , (( b, j and b ∈ B .(xi) The product of c j ∗ and φ , ((0 , c j )) yields a generator of H ( M ; Z ).(xii) The product of c j ∗ and φ , ((0 , c j )) vanishes for any pair ( j , j )of distinct numbers.(3) The 3rd and the 5th Stiefel-Whitney classes of M vanish.(4) Let d be the isomorphism d : H ( M ; Z ) → H ( M ; Z ) mapping the elementsof the basis to their duals. The first Pontryagin class of M is regarded as4 d ◦ φ ( p ) ∈ H ( M ; Z ). The 4th Stiefel-Whitney class of M vanishes.(5) f | S ( f ) is an embedding.(6) The index of each singular point is always 0 or 1.(7) Preimages of regular values are empty or diffeomorphic to S or S ⊔ S .For a point p in a Euclidean space, || p || denotes the distance between the origin0 and p . Theorem . Every 7-dimensional homotopy sphere admits a foldmap into R .(1) f | S ( f ) is embedding and f ( S ( f )) = { x ∈ R | || x || = 1 , , } .(2) The index of each singular point is always 0 or 1.(3) For each connected component of the regular value set of f , the preimageof a regular value is, empty, diffeomorphic to S , diffeomorphic to S ⊔ S and diffeomorphic to S ⊔ S ⊔ S , respectively.We will add precise explanations on Theorem 2 in the next section. The presentpaper shows construction of fold maps into the 4-dimensional Euclidean space on7-dimensional closed and simply-connected manifolds of a wider class. We willconnect the present study and closely related studies to understand these manifoldsin geometric and constructive ways.As an additional remark, the class of 7-dimensional closed and simply-connectedmanifolds still produces attractive topics. [14] is one of recent studies on explicitalgebraic topological understandings of 7-dimensional closed and simply-connectedmanifolds.1.4. The content of the paper and acknowledgement.
The organization ofthe paper is as the following. In the next section, we remark on subsection 1.3: weexplain that fold maps of suitable classes affect the differentiable structures of themanifolds admitting them. After that, we perform construction of new fold mapson 7-dimensional closed, simply-connected and spin manifolds of a new class widerthan the class of [8] or Theorems 1 and 2 and this is the main theorem (Theorem5). Key methods are based on [6], [7] and [9]. Note that these articles are mainlyfor Reeb spaces and investigating the homology groups and the cohomology ringsof manifolds are different from investigating those of the Reeb spaces and moredifficult. Example 3 of the section and Theorem 7 in the last section are on foldmaps obtained in Theorem 5 for cases where the integral cohomology rings of the 7-dimensional manifolds are isomorphic to the integral cohomology ring of C P × S .We can obtain Theorem 7 thanks to [27], in which such 7-dimensional manifoldsare classified, or Theorem 6. EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 5
Construction of new family of fold maps on -dimensional closedand simply-connected manifolds of a new class. Throughout this paper, M is a closed manifold of dimension m , n < m is apositive integer and f : M → R n is a smooth map unless otherwise stated.2.1. Additional explanations on subsection 1.3: fold maps of suitableclasses affect the differentiable structures of the manifolds admittingthem. A special generic map is a fold map such that the index of each singular pointis 0. The class of special generic maps contains the class of Morse functions on closedmanifolds with exactly two singular points, which are central objects in Reeb’stheorem, characterizing spheres topologically except the case where the manifoldis 4-dimensional: in this case, a standard 4-dimensional sphere is characterized asthis. It also contains canonical projections of unit spheres. Example 2 also presentssimplest special generic maps, extending the class of the special generic maps intoEuclidean spaces whose dimensions are larger than 1 just before. Saeki, Sakuma,Wrazidlo, and so on, discovered that special generic maps restrict the topologiesand the differentiable structures of manifolds admitting them in considerable cases.For example, their studies revealed that 7-dimensional homotopy spheres which arenot diffeomorphic to the standard sphere admit no special generic msp into R , R and R . See [18], [20], [29] and so on. Theorem . In the situation of Theorem 1, if at least one of the following hold,then M never admits a special generic map into R .(1) p ∈ B ⊕ C is not zero.(2) In { a i,j } aj =1 , at least one non-zero number exists,Definition 2 . For a fold map f : M → R as in Theorem 2 where f ( S ( f )) = ⊔ lr =1 { x ∈ R | || x || = r } instead of the original condition and the preimage ofa point in the exactly one connected component of the regular value set which isdiffeomorphic to an open disc, is the disjoint union of l copies of S for an integer l >
0: originally l = 3. We also assume that for each connected component C of thesingular value set and its small closed tubular neighborhood N ( C ), the compositionof f | f − ( N ( C )) f − ( N ( C )) → N ( C ) with a canonical projection to C gives a trivialbundle over C . We say that such f is an l normal round fold map with standardspheres .We can show the following theorem related to Theorem 2, thanks to the theoremwith [1], [15] and so on: it is a classical important fact that there are 28 types oforiented homotopy spheres. Theorem . Every 7-dimensional homotopy sphere M admits a3 normal round fold map with standard spheres f : M → R as in Theorem 2.Moreover, for a 7-dimensional homotopy sphere M , we have the following charac-terization.(1) M admits a 1 normal round fold map with standard spheres f : M → R if and only if M is a standard sphere. NAOKI KITAZAWA (2) M admits a 2 normal round fold map with standard spheres f : M → R ifand only if the homotopy sphere is one of 16 types of the 28 types (orientedhomotopy spheres of these exactly 16 types are represented as total spacesof linear S -bundles over S and a standard sphere is one of these 16 types).In a class which is not the class of special generic maps, the following fact wasexplicitly found: the differential topological properties of fold maps affect the dif-ferentiable structures of the homotopy spheres. Note also that every 7-dimensionalhomotopy sphere M is represented as a connected sum of finitely many homotopyspheres represented as total spaces of linear S -bundles over S .2.2. Reeb spaces.
For a continuous map c : X → Y , we can define an equivalencerelation ∼ c on X by the following rule: p ∼ c p if and only if p and p are ina same connected component of a preimage c − ( q ) for some q ∈ Y . We call thequotient space W c := X/ ∼ c the Reeb space of c . For a fold map satisfying thetwo conditions on the restriction to the singular set in subsection 1.2, the Reebspace is a dim Y -dimensional polyhedron. This property holds for smooth mapsof wider classes such that the dimensions of the domains are greater than those ofthe target spaces. See [13] and [24] for example. The following example presents afundamental and important fact in the next subsection.Example 1 . For a special generic map, the Reeb space is a manifold we can smoothlyimmerse into the target space.2.3.
Construction.
Surgery operations to manifolds and maps to construct new fold maps.
Definition 3 . For a fold map f : M → N , let P be a connected component of ( W f − q f ( S ( f ))) T ¯ f − ( N − f ( S ( f ))), regarded as a manifold diffeomorphic to ¯ f ( P ) ⊂ N .Let l > l ′ ≥ { S j } lj =1 offinitely many standard spheres and { N ( S j ) } lj =1 of total spaces of linear bundlesover these spheres whose fibers are unit discs. We also denote by S j the image ofthe section obtained by choosing the origin for each fiber diffeomorphic to a unitdisc for each N ( S j ). Assume that the dimensions of N ( S j ) are always n and thatthere exist immersions c j : N ( S j ) → P satisfying the following properties.(1) f | f − ( S lj =1 c j ( N ( S j ))) f − ( S lj =1 c j ( N ( S j ))) → S lj =1 c j ( N ( S j )) gives a trivial S m − n -bundle.(2) The family { c j | ∂N ( S j ) : ∂N ( S j ) → P } lj =1 is normal and the number ofcrossings is finite.(3) The family { c j | S j : S j → P } lj =1 is normal and the number of crossings isfinite.(4) We denote the set of all the crossings of the family { c j | S j : S j → P } lj =1 ofthe immersions by { p j ′ } l ′ j ′ =1 . For each p j ′ , there exist one or two integers1 ≤ a ( j ′ ) , b ( j ′ ) ≤ l and small standard closed discs D j ′ − ⊂ S a ( j ′ ) and D j ′ ⊂ S b ( j ′ ) satisfying the following four properties.(a) dim D j ′ − = dim S a ( j ′ ) and dim D j ′ = dim S b ( j ′ ) .(b) p j ′ is in the images of the immersions p j ′ ∈ c a ( j ′ ) (Int D j ′ − ) and p j ′ ∈ c b ( j ′ ) (Int D j ′ ).(c) If a ( j ′ ) = b ( j ′ ), then D j ′ − T D j ′ is empty. EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 7 (d) If we restrict the bundle N ( S a ( j ′ ) ) over the sphere to D j ′ − and thebundle N ( S b ( j ′ ) ) over the sphere to D j ′ , then the images of the totalspaces of these resulting bundles by c a ( j ′ ) and c b ( j ′ ) agree as subsets in R n : the restrictions of the immersions to these spaces are embeddings.(e) The set of all the crossings of the family { c j | S j : S j → P } lj =1 is thedisjoint union of the l ′ corners of the subsets just before each of whichis for 1 ≤ j ′ ≤ l ′ .In this situation, the family { ( S j , N ( S j ) , c j : N ( S j ) → P ) } lj =1 is said to be a normalsystem of submanifolds compatible with f .In the situation of Definition 3, let { N ′ ( S j ) ⊂ N ( S j ) } lj =1 be a family of totalspaces of subbundles of { N ( S j ) } lj =1 over the manifolds whose fibers are standardclosed discs. We assume that the diameters are all 0 < r <
1. If we take r suitably,then same properties as presented in Definition 3 hold: we can obtain anotherfamily { ( S j , N ′ ( S j ) , c j | N ′ ( S j ) : N ′ ( S j ) → P ) } lj =1 , regarded as a normal system ofsubmanifolds compatible with f . We can identify each fiber, which is a standardclosed disc of diameter r with a unit disc via the diffeomorphism mapping t to r t .Definition 4 . The familiy { ( S j , N ( S j ) , c j : N ( S j ) → P ) } lj =1 is said to be a widernormal system supporting the normal system of submanifolds { ( S j , N ′ ( S j ) , c j | N ′ ( S j ) : N ′ ( S j ) → P ) } lj =1 compatible with f .Definition 5 . For a fold map f : M → N and an integer l >
0, let P be a con-nected component of ( W f − q f ( S ( f ))) T ¯ f − ( N − f ( S ( f ))) and let { ( S j , N ( S j ) , c j : N ( S j ) → P ) } lj =1 be a normal system of submanifolds compatible with f . Let { ( S j , N ′ ( S j ) , c j ′ : N ′ ( S j ) → P ) } lj =1 be a wider normal system supporting this.Assume that we can construct a stable fold map f ′ on an m -dimensional closedmanifold M ′ into R n satisfying the following properties.(1) Q := f − ( S lj =1 c j ′ ( N ′ ( S j ))) and M − Int Q is realized as a compact subman-ifold of M ′ of dimension m via a suitable smooth embedding e : M − Int Q → M ′ .(2) f | M − Int Q = f ′ ◦ e | M − Int Q holds.(3) f ′ ( S ( f ′ )) is the disjoint union of f ( S ( f )) and S nj =1 c j ( ∂N ( S j )).(4) The indices of points in the preimage of new connected components in theresulting singular value set are all 1.(5) The preimage of each regular value p sufficiently close to the union S lj =1 c j ( S j )is a disjoint union of standard spheres.This enables us to define a procedure of constructing f ′ from f and we call this an ATSS operation to f . We call the union S lj =1 c j ( S j ) the generating image of theoperation. Proposition . In the situation of Definition 3 (Definition 5), for the normal systemof submanifolds compatible with f , we can perform an ATSS operation to f suchthat the S lj =1 c j ( S j ) is the generating image of the operation as in Definition 5. Proof.
We show a local fold map around p j ′ in Definition 3. This is also presentedin [9] with FIGURE 2 and [10]. S is the two point set with the discrete topology.Set k , k > D k × S k over D k .We also set D k as the set of all points x ∈ R k satisfying || x || ≤ . We also set a NAOKI KITAZAWA
Morse function ˜ f k , on a manifold obtained by removing the interior of a standardclosed disc of dimension k + 1 embedded smoothly in the interior of S k × [ − , , ] ⊂ (0 , + ∞ ) ⊂ R satisfying the following four properties.(1) The preimage of the minimum coincides with the disjoint union of twoconnected components of the boundary.(2) The preimage of the maximum coincides with one connected component ofthe boundary.(3) There exists exactly one singular point, and the singular point is in theinterior of the manifold.(4) The value at the singular point is 1.We glue the projection of the product bundle and the map ˜ f k , × id S k − : [ , + ∞ ) × S k − identifying the base space D k with { x ∈ R k | || x || ≤ } . By gluing thesemaps suitably, we have a desired smooth map onto { x ∈ R k | || x || ≤ } . See also[2] and [5] for this map. We can decompose the domain into two compact manifoldswith boundaries and the restriction to each manifold is as follows.(1) A surjection giving a trivial bundle whose fiber is D k .(2) A surjection such that the regular value set consist of two connected com-ponents. The preimage of a regular value in each connected component isdiffeomorphic to the following manifolds.(a) D k .(b) D k ⊔ S k .We denote the latter map by ˜ f SD k ,k .We can take a sufficiently small standard closed disc D ′ j ′ − ⊃ D j ′ − of di-mension dim D j ′ − so that the relation S a ( j ′ ) ⊃ D ′ j ′ − ⊃ Int D ′ j ′ − ⊃ D j ′ − holds. Similarly we can take a sufficiently small standard closed disc D ′ j ′ ⊃ D j ′ of dimension dim D j ′ so that the similar relation holds.We can construct a product map of the composition of ˜ f SDdim D ′ j ′− ,m − n witha diffeomorphism onto c a ( j ′ ) ( D ′ j ′ − ) and the identity map id c b ( j ′ ) ( D ′ j ′ ) . We canconstruct a product map of the composition of ˜ f SDdim D ′ j ′ ,m − n with a diffeomor-phism onto c b ( j ′ ) ( D ′ j ′ ) and the identity map id c a ( j ′ ) ( D ′ j ′− ) . We glue these twomaps suitably to obtain a local map onto c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ ): the targetspace is same as those of the two local maps.After constructing this local map around each p j ′ , we can construct for theremaining part easily. Around new singular values whose preimages consist ofexactly one point, we construct maps represented as product maps of ˜ f m − n, andidentity maps on suitable manifolds for suitable coordinates. Around new regularvalues, we construct trivial bundles.This completes the proof. (cid:3) Homology and cohomology groups of manifolds admitting the fold maps ob-tained by the operations.
The following proposition is an extension of an importantproposition in [8]. We can show this in an almost similar way.
Proposition . We consider a situation as explained in Definition 3. Let m > n ≥ n be even. Let M be an m -dimensional closed and connectedmanifold. For the normal system of submanifolds compatible with f , assume thatdim S j = n for any j . We also assume the relations 0 < n − n = n < m − n, n In this situation, we can perform an ATSS operation to f such that the S lj =1 c j ( S j )is the generating image of the operation and have a new map f ′ : M ′ → R n satis-fying the following three properties.(1) H i ( M ′ ; Z ) is isomorphic to H i ( M ; Z ) ⊕ Z l for i = n , m − n, n, m − n .(2) H i ( M ′ ; Z ) is isomorphic to H i ( M ; Z ) for i = n , m − n, n, m − n .(3) The total Stiefel-Whitney class of M ′ is 1 ∈ H ( M ′ ; Z / Z ) if that of M isso.Furthermore, for the resulting Reeb space, we can construct the map satisfyingthe following three properties.(1) H i ( W f ′ ; Z ) is isomorphic to H i ( W f ; Z ) ⊕ Z l for i = n , m − n, n, m − n . Undersuitable identifications of H i ( M ′ ; Z ) with H i ( M ; Z ) ⊕ Z l and H i ( W f ′ ; Z )with H i ( W f ; Z ) ⊕ Z l , the homomorphism q f ′ ∗ between the homology groupsinduced from the quotient map q f ′ maps an element (0 , p ) ∈ H i ( M ; Z ) ⊕ Z l to (0 , p ) ∈ H i ( W f ; Z ) ⊕ Z l . Under the identifications, q f ′ ∗ maps an element( p, ∈ H i ( M ; Z ) ⊕ Z l to ( q f ∗ ( p ) , ∈ H i ( W f ; Z ) ⊕ Z l : the homomorphism q f ∗ is induced from q f : M → W f in a canonical way.(2) H i ( W f ′ ; Z ) is isomorphic to H i ( W f ; Z ) for i = n , m − n, n, m − n .We present a short sketch of the proof explaining about homology and cohomol-ogy classes essential in the discussions later. For a commutative group G and anelement g which is of infinite order, which we cannot represent as kg ′ for an integersatisfying | k | > g ′ ∈ G and which yields an internal direct sum decomposition < g > ⊕ G ′ of G ⊃ G ′ where < g > is the group generated by the set { g } , we candefine the dual D( g ) of g as a homomorphism into Z satisfying D( g )( g ) = 1 andD( g )( G ′ ) = { } . Moreover, if the group is a homology group, then we can regardthe dual as a cohomology class in a canonical way and we regard the dual as thecohomology class obtained in this way. Sketch of the proof. The proof for the case where the immersion obtained as therestriction to the singular set is an embedding is already done in [8]. The construc-tion and existence of self-intersections of the immersion do not influence on theoriginal argument considerably.In Definition 5, consider a fiber D F ,j of the bundle N ′ ( S j ) over S j such that c j ( D F ,j ) contains no crossing of the family { c j | ∂N ( S j ) } of the immersions, set Q F ,j := f − ( c j ( D F ,j )) and consider the restriction of f to Q F ,j . Note that Q F ,j isa compact manifold diffeomorphic to a manifold obtained in the following way. Seealso [2] and [5] for this.(1) E := S n × S m − n .(2) Choose an equator E S of S n , diffeomorphic to S n − .(3) We consider a small closed tubular neighborhood of E S × {∗ E S } ⊂ E andremove its interior.(4) The resulting compact manifold E D is the desired manifold. We can con-struct and need to construct two relations E S × {∗ E S ′ } ⊂ E D ⊂ E and {∗ E S ′′ } × S m − n ⊂ E D ⊂ E hold where ∗ E S ′ and ∗ E S ′′ are suitable points.We also need to construct that {∗ E S ′′ } × S m − n is a connected componentof the preimage of some regular value. We can construct a desired map f ′ : M ′ → R n on a suitable manifold M ′ . Weexplain a generator of the j -th summand of Z l in H i ( M ; Z ) ⊕ Z l , isomorphic to H i ( M ′ ; Z ), for i = n , n, m − n, m − n .For i = n , it is represented by E S × {∗ E S ′ } before for each j . For i = n , it isrepresented by the product of the image of an embedding of S j into f − ( c j ( S j ))such that the composition with f and the immersion c j agree and E S × {∗ E S ′ } before for each j : we can take an embedding of S j by the structure of the manifoldand the map. For i = m − n , it is represented by a connected component ofthe preimage of a regular value, diffeomorphic to S m − n , before, for each j . For i = m − n , it is represented by the product of the image of an embedding of S j into f − ( c ( S j )) such that the composition with f and the immersion c j agree anda connected component of a preimage, diffeomorphic to S m − n , before for each j :we can take an embedding of S j by the structure of the manifold and the map.We can see that the remaining properties hold by the way of this construction. (cid:3) Remark 1 . On Proposition 2, originally, we consider a situation as explained inDefinition 3 as the following. Let m > n ≥ M be an m -dimensional closed and connected manifold. The normal system of submanifoldscompatible with f satisfies the following.(1) The immersions of standard spheres are embeddings and that the family ofthe immersions has no crossings.(2) dim S j = k ≤ n for any j .(3) The relations 0 < k < m − n, n < m − n + k < m and m − n = n hold.In this situation, we have an essentially similar result. This will be used in theproof of the main theorem or Theorem 5.2.3.3. The main theorem. The following proposition plays essential roles in theproof of the main theorem. Proposition . We consider a situation as explained in Proposition 2. Let m > n ≥ n is divisible by 4.For the normal system of submanifolds compatible with f , assume that dim S j = n for any j as in Proposition 2. We also assume the relation on the dimensions0 < n − n < m − n < n < m − n + n < m as in the proposition. Furthermore, let H := ( h j ,j ) be an l × l matrix whose entries are integers.We also assume the following.(1) For any single immersion c j and any pair ( c j , c j ) of the immersions, thereexist sufficiently many crossings.(2) Identifications of homology groups used in Proposition 2 are given.For H n ( M ; Z ) ⊕ Z l , let e j be the element represented as (0 , e ,j ) ∈ H n ( M ; Z ) ⊕ Z l : e ,j is the sequence of l integers whose j -th entry is 1 and the other entries are 0.For H n ( M ; Z ) ⊕ Z l , let e j ′ be the element represented as (0 , e ,j ′ ) ∈ H n ( M ; Z ) ⊕ Z l : e ,j ′ is the sequence of l integers whose j -th entry is 1 and the other entries are 0.In this situation, we can perform an ATSS operation to f such that the S lj =1 c j ( S j )is the generating image of the operation and have a new map f ′ : M ′ → R n satisfying the same properties as ones in Proposition 2 and the following additionalproperties where q f ′ ∗ : H i ( W f ′ ; Z ) → H i ( M ′ ; Z ) is the homomorphism inducedfrom the quotient map q f ′ : M ′ → W f ′ for each i . Note that for e j and e j ′ , we can EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 11 define duals D( e j ) and D( e j ′ ), respectively. Note also that by virtue of Proposition2, we can define the duals D( q f ′ ∗ ( e j )) and D( q f ′ ∗ ( e j ′ )).(1) Two relations q f ′ ∗ (D( q f ′ ∗ ( e j ))) = D( e j ) and q f ′ ∗ (D( q f ′ ∗ ( e j ′ ))) = D( e j ′ )hold.(2) The product of D( e j ) and D( e j ) is h j,j D( e j ′ ).(3) The product of D( e j ) and D( e j ) is h j ,j D( e j ′ ) + h j ,j D( e j ′ ) for distinct j and j . Proof. Main ingredients of the proof are essentially in the proofs of important ormain propositions in [9] and [10]. We will concentrate on explanations of severalessential parts.We can see the properties in Proposition 2 and the first property additionallypresented here by the way of the construction together with the observation of thetopologies of the original Reeb space W f and the resulting one W f ′ , we will explainhere. W f ′ is simple homotopy equivalent to the polyhedron obtained by attaching apolyhedron to a subpolyhedron ¯ f − ( S lj =1 c j ( S j )) ⊂ W f . The dimension of thesubpolyhedron is n .We explain about the polyhedron. We prepare l copies of a manifold diffeomor-phic to S n × S n . We denote each copy by S B,j × S F,j for 1 ≤ j ≤ l where S B,j and S F,j are diffeomorphic to S n .We choose pairs (( D B, j ′ − × D F, j ′ − ⊂ S B,p (2 j ′ − × S F,p (2 j ′ − ) , ( D B, j ′ × D F, j ′ ⊂ S B,p (2 j ′ ) × S F,p (2 j ′ ) )) of products of standard closed discs whose dimensionsare n where p is a function mapping the integer to an integer 1 ≤ i ≤ l : j ′ is anarbitrary positive integer smaller than a sufficiently large integer. We also need tochoose the products of discs disjointly. We identify the two products of discs foreach pair. We can perform an ATSS operation attaching the resulting polyhedronto the subpolyhedron ¯ f − ( S lj =1 c j ( S j )) ⊂ W f so that S j S B,j ×{∗ j } ( ⊂ S B,j × S F,j ),represented via notation respecting the original manifolds diffeomorphic to S n × S n ,is attached to this subpolyhedron and that the following properties hold.(1) q f ′ ∗ ( e j ) is represented by {∗ ′ j } × S F,j ⊂ S B,j × S F,j , after the manifold isattached to W f , in the resulting Reeb space W f ′ .(2) S B,j × {∗ j } is, after attached to W f , in the resulting Reeb space W f ′ ,a subpolyhedron representing the class ( q f ∗ ( c M,j ) , Σ lj ′′ =1 h j,j ′′ q f ′ ∗ ( e j ′′ )) ∈ H n ( W f ; Z ) ⊕ Z l where c M,j is a suitable class.This yields the fact that the second and third additional properties hold. Moreprecise explanations on this with the argument on the explicit ATSS operation justbefore will be also seen in the proofs of important propositions and theorems in [9]and [10]. (cid:3) The following special generic maps also play important roles in the construction.Example 2 . Let m > n ≥ l > M of dimension m represented as a connected sum of manifolds of afamily { S l j × S m − l j } lj =1 satisfying 1 ≤ l j ≤ n − f : M → R n such that the following properties hold.(1) W f is represented as a boundary connected sum of manifolds of a family { S l j × D n − l j } .(2) f | S ( f ) is an embedding. (3) We can obtain a trivial linear D m − n +1 -bundle over ∂W f by restricting f to the preimage of a small collar neighborhood and considering the compo-sition of this restriction with the canonical projection to ∂W f .(4) We can obtain a trivial smooth S m − n -bundle over the complement of theinterior of the collar neighborhood before in W f by considering the restric-tion to f to the preimage of this interior of the collar neighborhood.(5) The homomorphism q f ∗ between the homology groups maps the class rep-resented by S l j × {∗ } ⊂ S l j × S m − l j in the connected sum to the classrepresented by S l j × {∗ } ⊂ S l j × Int D n − l j ⊂ S l j × D n − l j in the boundaryconnected sum W f .The following theorem is the main theorem. Theorem . Let A , B and B ′ be free finitely generated commutative groups of rank a , b and b ′ , respectively. Let { a j } aj =1 , { b j } bj =1 and { b j ′ } b ′ j =1 be bases of A , B and B ′ , respectively. For each integer 1 ≤ i ≤ b , let { a i,j } aj =1 be a sequence of integers.Let H := ( h j ,j ) be a b × b matrix whose entries are integers.In this situation, there exist a 7-dimensional closed and simply-connected spinmanifold M and a fold map f : M → R satisfying the following properties.(1) H ( M ; Z ) is isomorphic to A ⊕ B and H ( M ; Z ) is isomorphic to B ⊕ B ′ . H ( M ; Z ) is free.(2) H j ( M ; Z ) is isomorphic to H j ( M ; Z ). Via suitable isomorphisms φ : A ⊕ B → H ( M ; Z ) and φ : B ⊕ B ′ → H ( M ; Z ), we can identify H ( M ; Z )with A ⊕ B and H ( M ; Z ) with B ⊕ B ′ . Furthermore, we can define theduals a j ∗ , b j, ∗ , b j, ∗ and b j ′∗ of φ (( a j , φ ((0 , b j )), φ (( b j , φ ((0 , b j ′ )), respectively, and we can canonically obtain bases of H ( M ; Z )and H ( M ; Z ). We denote the composition of φ with an isomorphismmapping the elements of the basis to their Poincar´e duals by φ , : B ⊕ B ′ → H ( M ; Z ). We denote the composition of φ with an isomorphismmapping the elements of the basis to their Poincar´e duals by φ , : A ⊕ B → H ( M ; Z ). For the products in H i ( M ; Z ) for i = 4 , , 7, propertiespresented in Theorem 1 (2) hold except the following cases: we explainfacts on the products which are satisfied instead of the original properties.Note also that ” C ” in Theorem 1 is replaced by B ′ and ” c j ” in Theorem 1is replaced by b j ′ for example.(a) The first list of (2a). Instead, the product of a j ∗ and a j ∗ vanishesand that of b j , ∗ and b j , ∗ is h j ,j b j , ∗ + h j ,j b j , ∗ for any j and j .(b) The fifth list of (2b). Instead, the product of b i, ∗ and φ , (( b i , aj =1 a i,j φ , (( a j , bj =1 h i,j φ , ((0 , b j )) ∈ H ( M ; Z ).(3) The 3rd and the 5th Stiefel-Whitney classes of M vanish.(4) Let d be the isomorphism d : H ( M ; Z ) → H ( M ; Z ) defined by cor-responding the duals. The first Pontryagin class of M is regarded as4 d ◦ φ ( p ) ∈ H ( M ; Z ). The 4th Stiefel-Whitney class of M vanishes.(5) The immersion f | S ( f ) is normal.(6) The index of each singular point is 0 or 1.(7) Preimages of regular values in the image of f are diffeomorphic to S , S ⊔ S or S ⊔ S ⊔ S . EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 13 Note that the statement is same as that of Theorem 1 except several properties.The proof is also similar to the original theorem. New ingredients are discussionsusing Proposition 3 for example. See also [7] and [8] to understand the proof morerigorously, for example. Proof. We consider a special generic map f presented in Example 2 into R on amanifold represented as a connected sum of a ≥ S × S . We identify a j ∈ A in the assumption with a generator of the 2nd homology group of j -th copyof S × S and a suitable class in the Reeb space W f presented in the example.We can choose a family { c j : S j → Int W f } bj =1 of immersions of 2-dimensionalstandard spheres into the interior of the image of f or W f so that the family isnormal and that for any single immersion and any pair of the immersions, thereexist sufficiently many crossings as assumed in Proposition 3.We can also choose b ′ points embedded disjointly in the complement of the unionof the images of the immersions before in the interior of the image of f or W f .We can take them so that the following properties hold.(1) We denote a generator of the 2nd homology group of the j -th copy of D × S by a j in the image, represented as a boundary connected sum of a copies of D × S , identified with the element a j ∈ A of the assumption. We denotea fundamental class of S i by [ S i ] and c i ∗ ([ S i ]) = Σ aj =1 a i,j a j ∈ H ( W f ; Z ): c i ∗ is the homomorphism between the homology groups induced from c i .We can perform an ATSS operation so that the resulting map and the 7-dimensionalmanifold satisfy the fifth, sixth and seventh properties.We prove the first property and statements on 2nd and 4th homology groups ofthe second property. We use Proposition 2 setting ( m, n ) = (7 , 4) in Proposition 2and Remark 1 setting ( m, n, k ) = (7 , , E S × {∗ E S ′ } in (the proof of)Proposition 2 for each S i . The class represented by this is identified with b i ∈ B inthe assumption as we have done for a j ’s and we thus obtain a desired isomorphism φ : A ⊕ B → H ( M ; Z ). We can apply Proposition 2 for the 4-th homology group.In addition, for the b ′ points p i , we apply Remark 1. We can consider similaridentifications for b i ′ ’s. This yields a desired isomorphism φ : B ⊕ B ′ → H ( M ; Z ).We prove properties on products of cohomology classes in the second propertyof the seven properties.We prove the properties for cases where products are in H ( M ; Z ).For a i ∗ ’s, we can see by the identification in the beginning. Consider the productof a cocycle representing the dual b i, ∗ of the class b i and a cocycle representing thedual a j ∗ of the class a j and value at the cycle represented as the tensor product of acycle in the class Σ aj =1 a i,j a j and a cycle in the class represented by E S × {∗ E S ′ } inthe proof of Proposition 2 (we consider a cohomology class in the product M × M defined in a canonical way and the product is obtained as the pull-back via thediagonal map from M into M × M ). For b i ∗ ’s, apply Proposition 3: this is a newingredient in the proof.We prove the properties for cases where products are in H ( M ; Z ).We investigate several homomorphisms. φ , ◦ φ − maps the 4-th homologyclass φ (( b j , φ ((0 , b j ′ )) before to the dual of the homology class representedby a connected component of the preimage of a suitable regular value. We explainabout φ , ◦ φ − . This maps each ( n − dim S i )-th homology class represented by E S × {∗ E S ′ } in the proof of Proposition 2 to the dual of the homology classrepresented by a product of {∗ E S ′′ } × S m − n ⊂ E D ⊂ E in the proof of Proposition2 and the image of an embedding of S i into f − ( c i ( S i )) such that the compositionwith f and the immersion c i agree in the proof of the proposition.The first three products in Theorem 1 (2b) vanish. In fact, for the pair ofthe cohomology classes, we can take corresponding cycles, representing the classeswhose duals are these cohomology classes satisfying the following: each cohomologyclass does not vanish at the corresponding cycle and the cycles do not intersect.We prove the fourth product in Theorem 1 (2b). We investigate several classes.The class c j ∗ ([ S j ]) coincides with Σ aj =1 a j ,j a j ∈ H ( W f ; Z ). φ , maps each( n − dim S j )-th homology class represented by E S × {∗ E S ′ } in the proof of Propo-sition 2 to the dual of the homology class represented by a product of the image ofan embedding of S j into f − ( c j ( S j )) such that the composition with f and theimmersion c j agree in the proof of the proposition and {∗ E S ′′ } × S m − n ⊂ E D ⊂ E there. We can see that the product of a j ∗ and φ , (( b j , a j ,j φ , ((0 , b j )) ∈ H ( M ; Z ) for any j and j .We prove the fifth case. The proof is a new ingredient. The coefficient of a j in the class Σ aj =1 a i,j a j ∈ H ( W f ; Z ), which is equal to c i ∗ ([ S i ]), is a i,j , By virtueof the discussion in the proof of Proposition 2 just before and by considering theclass to which the isomorphism φ , ◦ φ − maps a j ∈ H ( M ; Z ), for the productof b i, ∗ and φ , (( b i , φ , ◦ φ − ( a j ) ∈ H ( M ; Z ) for a j ∈ H ( M ; Z ) is a i,j where the identification of elements in A andthe homology group is considered. We must consider that the ATSS operationrespects Proposition 3 and this requires a new ingredient. Consider the value atthe class whose dual is φ , ◦ φ − ( b j ) ∈ H ( M ; Z ) for b j ∈ H ( M ; Z ) where theidentification as before is considered: it is h i,j .This completes the proof of the last case and the proof for cases where productsare in H ( M ; Z ).We explain about cases where products are in H ( M ; Z ). Proofs of the facts thatthe products vanish are done in the following way: for the pair of the cohomologyclasses, we can take corresponding cycles disjointly, representing the classes whoseduals are these cohomology classes so that each class does not vanish at thesecycles. Remaining cases can be shown by virtue of the definitions of φ , and φ , and Poincare’s duality theorem.For the third property, Proposition 2 and Remark 1 complete the proof.We show the fourth property. H ( M ; Z ) is generated by a finite set consistingof the classes represented by connected components of preimages of regular values.We exchange small tubular neighborhoods of the connected components, yieldingtrivial S -bundles over D , and change the map. More precisely, we change bundleisomorphisms for identifications between the boundaries: each connected compo-nent of the boundary is S × ∂D . Classical important facts on Pontryagin classesof linear S -bundles over S yield the fourth property. See also [16], [25] and so on.More precise explanations on this are in [8] for example.This completes the proof. (cid:3) It is important that this theorem explains the cohomology ring of the manifold M completely as Theorem 1 does. The following example explains an explicit relationbetween Theorems 1 and 5. EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 15 Example 3 . We can construct a fold map on a closed, simply-connected and spinmanifold whose integral cohomology ring is isomorphic to that of C P × S by virtueof Theorem 5. We set a = 0, b = 1, b ′ = 0 and H = ( h , ) = (1) to see this. Wecannot obtain a 7-dimensional closed and simply-connected manifold whose integralcohomology ring is isomorphic to this by applying Theorem 1. On a closed andsimply-connected manifold whose integral cohomology ring is isomorphic to that of C P × S , a fold map into R such that the restriction to the singular set is anembedding is also constructed in [8], which is not presented in the present paper asa theorem. However, the manifold is not spin.We can show the following as an extension of Theorem 3. We omit the proofand proofs are left to readers: see the original article [8]. Proposition . In the situation of Theorem 5, if at least one of the following hold,then M never admits a special generic map into R .(1) p ∈ B ⊕ C is not zero.(2) In { a i,j } aj =1 , at least one non-zero number exists.(3) There exists at least one entry of H which is not zero.3. A remark on fold maps on a -dimensional closed andsimply-connected manifold whose integral cohomology ring isisomorphic to that of C P × S . We remark on Example 3. Theorem . (1) Let X be a 7-dimensionalclosed, simply-connected and spin manifold whose integral cohomology ringis isomorphic to that of C P × S . Let φ be an isomorphism between in-tegral cohomology rings from the integral cohomology ring of C P × S onto that of X and p be a generator of H ( C P × S ; Z ). In this situation,the 1st Pontryagin class of X is represented as the form 4 kφ ( P ) for someinteger k .(2) The topology of a 7-dimensional closed, simply-connected and spin manifoldwhose integral cohomology ring is isomorphic to C P × S is determinedby its 1st Pontryagin class.(3) For any pair of mutually homeomorphic 7-dimensional closed, simply-connectedand spin manifolds whose integral cohomology rings are isomorphic to theintegral cohomology ring of C P × S , one of these two manifolds is rep-resented as a connected sum of the other manifold and a 7-dimensionalhomotopy sphere. Moreover, for a 7-dimensional closed, simply-connectedand spin manifold whose integral cohomology ring is isomorphic to the inte-gral cohomology ring of C P × S , consider two manifolds each of which isrepresented as a connected sum of the given manifold and some homotopysphere. They are diffeomorphic if and only if the homotopy spheres arediffeomorphic. Theorem . A closed, simply-connected and spin manifold M of dimension m = 7whose integral cohomology ring is isomorphic to that of C P × S admits a foldmap f : M → R satisfying the following properties.(1) The immersion f | S ( f ) is normal.(2) The index of each singular point is 0 or 1. (3) Preimages of regular values in the image of f are diffeomorphic to S , S ⊔ S or S ⊔ S ⊔ S . Proof. We consider a fold map obtained by Theorem 5 and presented in Example3. We also consider a fold map in Theorem 2. By virtue of Theorem 6, consideringa so-called connected sum of these two maps completes the proof. For connectedsums of two fold maps, see [7] for example. (cid:3) References [1] J. J. Eells and N. H. Kuiper, An invariant for certain smooth manifolds , Ann. Mat. PuraAppl. 60 (1962), 93–110.[2] N. Kitazawa, On round fold maps (in Japanese), RIMS Kokyuroku Bessatsu B38 (2013),45–59.[3] N. Kitazawa, On manifolds admitting fold maps with singular value sets of concentric spheres ,Doctoral Dissertation, Tokyo Institute of Technology (2014).[4] N. Kitazawa, Fold maps with singular value sets of concentric spheres , Hokkaido Mathemat-ical Journal Vol.43, No.3 (2014), 327–359.[5] N. Kitazawa, Round fold maps and the topologies and the differentiable structures of mani-folds admitting explicit ones , submitted to a refereed journal, arXiv:1304.0618 (the title haschanged).[6] N. Kitazawa, Constructing fold maps by surgery operations and homological informationof their Reeb spaces , submitted to a refereed journal, arxiv:1508.05630 (the title has beenchanged).[7] N. Kitazawa, Notes on fold maps obtained by surgery operations and algebraic informationof their Reeb spaces , submitted to a refereed journal, arxiv:1811.04080.[8] N. Kitazawa Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional Euclidean space , submitted to a refereed journal, arxiv:1911.11274.[9] N. Kitazawa Surgery operations to fold maps to construct fold maps whose singular valuesets may have crossings , arxiv:2003.04147.[10] N. Kitazawa Surgery operations to fold maps to increase connected components of singularsets by two , arxiv:2004.03583.[11] M. Kobayashi, Stable mappings with trivial monodromies and application to inactive log-transformations , RIMS Kokyuroku. 815 (1992), 47–53.[12] M. Kobayashi, Bubbling surgery on a smooth map , preprint.[13] M. Kobayashi and O. Saeki, Simplifying stable mappings into the plane from a global view-point , Trans. Amer. Math. Soc. 348 (1996), 2607–2636.[14] M. Kreck, On the classification of -connected -manifolds with torsion free second homology ,to appear in the Journal of Topology, arxiv:1805.02391.[15] J. Milnor, On manifolds homeomorphic to the -sphere , Ann. of Math. (2) 64 (1956), 399–405.[16] J. Milnor and J. Stasheff, Characteristic classes , Annals of Mathematics Studies, No. 76.Princeton, N. J; Princeton University Press (1974).[17] G. Reeb, Sur les points singuliers d`une forme de Pfaff completement integrable ou dunefonction numerique , -C. R. A. S. Paris 222 (1946), 847–849.[18] O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces , TopologyAppl. 49 (1993), 265–293.[19] O. Saeki, Topology of special generic maps into R , Workshop on Real and Complex Singu-larities (Sao Carlos, 1992), Mat. Contemp. 5 (1993), 161–186.[20] O. Saeki and K. Sakuma, On special generic maps into R , Pacific J. Math. 184 (1998),175–193.[21] O. Saeki and K. Suzuoka, Generic smooth maps with sphere fibers J. Math. Soc. JapanVolume 57, Number 3 (2005), 881–902.[22] K. Sakuma, On special generic maps of simply connected n -manifolds into R ,[23] K. Sakuma, On the topology of simple fold maps , Tokyo J. of Math. Volume 17, Number 1(1994), 21–32.[24] M. Shiota, Thom’s conjecture on triangulations of maps , Topology 39 (2000), 383–399.[25] N. Steenrod, The topology of fibre bundles , Princeton University Press (1951). EW EXPLICIT FOLD MAPS ON 7-DIMENSIONAL MANIFOLDS 17 [26] R. Thom, Les singularites des applications differentiables , Ann. Inst. Fourier (Grenoble) 6(1955-56), 43–87.[27] X. Wang On the classification of certain -connected -manifolds and related problems ,arXiv:1810.08474.[28] H. Whitney, On singularities of mappings of Euclidean spaces: I, mappings of the plane intothe plane , Ann. of Math. 62 (1955), 374–410.[29] D. J. Wrazidlo, Standard special generic maps of homotopy spheres into Eucidean spaces ,Topology Appl. 234 (2018), 348–358, arxiv:1707.08646. 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