New explicit construction of fold maps on general 7-dimensional closed and simply-connected spin manifolds
aa r X i v : . [ m a t h . A T ] O c t NEW EXPLICIT CONSTRUCTION OF FOLD MAPS ONGENERAL 7-DIMENSIONAL CLOSED ANDSIMPLY-CONNECTED SPIN MANIFOLDS
NAOKI KITAZAWA
Abstract. exotic spheres by Milnor. It has influenced on theunderstanding of higher dimensional closed and simply-connected manifoldsvia algebraic and abstract objects. Recently this class is still being activelystudied via more concrete notions from algebraic topology such as concretebordism theory by Crowley, Kreck, and so on.As a new kind of fundamental and important studies, the author has beenchallenging understanding the class in constructive ways via construction of fold maps, which are higher dimensional versions of Morse functions. Thepresent paper presents a new general method to construct ones on spin mani-folds of the class. Introduction and fold maps.
Closed (and simply-connected) manifolds whose dimensions are m ≥ exotic spheres [15] and a related work [4]show and the class has been attractive as [2], [3], [14], and so on, show: an exotic (homotopy) sphere is a homotopy sphere which is not diffeomorphic to any standardsphere.1.1. Fold maps. A singular point p ∈ X of a differentiable map c : X → Y between two differentiable manifolds is a point at which the rank of the differential dc p of the map is smaller than both the dimensions dim X and dim Y . S ( c ) denotesthe set of all singular points of c and this is the singular set of c . c ( S ( c )) is che singular value set of c . and Y − c ( S ( c )) is the regular value set of c . A singular ( regular ) value is a point in the singular (resp. regular) value set of c .Throughout the present paper, manifolds and maps between manifolds are smooth(of class C ∞ ) unless otherwise stated.Definition 1 . Let m > n ≥ f from an m -dimensionalsmooth manifold with no boundary into an n -dimensional smooth manifold with no 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. boundary is said to be a fold map if at each singular point p , the map has the form( x , · · · , x m ) ( x , · · · , x n − , P m − ik = n x k − P mk = m − i +1 x k ) for suitable coordinatesand an integer 0 ≤ i ( p ) ≤ m − n +12 . Proposition . In the situation of Definition 1, the following properties hold.(1) For any singular point p , i ( p ) is unique.(2) The set consisting of all singular points of a fixed index of the map is a closedsubmanifold of dimension n − m -dimensionalmanifold.(3) The restriction map to the singular set is an immersion.We define i ( p ) in Proposition 1 as the index of p . A special generic map is a foldmap such that the index of each singular point is 0. The class of special genericmaps contains all Morse functions on closed manifolds with exactly two singularpoints, which are central objects in so-called Reeb’s theorem, characterizing spherestopologically except the case where the manifold is 4-dimensional. A standard 4-dimensional sphere is characterized as this in the case. The class of special genericmaps also contains all canonical projections of unit spheres. Fold maps have beenfundamental and strong tools in studying algebraic topological, differential topolog-ical and more general algebraic or geometric properties of manifolds in the branchof the singularity theory of differentiable maps as Morse functions have been in so-called Morse theory. Studies related to fold maps were essentially started in 1950sby Thom and Whitney ([24] and [25]). These studies are on smooth maps on man-ifolds whose dimensions are equal to or larger than 2 into the plane. After variousstudies, recently, Saeki, Sakuma and so on, have been found interesting facts onfold maps satisfying appropriate conditions, especially, special generic maps, andmanifolds admitting them in [17], [18], [19], [20], [21], [22], and so on. [5], [6], [7],[8] and so on, of the author, are motivated by these studies.1.2. Explicit fold maps on -dimensional closed and simply-connectedmanifolds of several classes and the main theorem. We assume that dif-feomorphisms on manifolds are smooth. We define the diffeomorphism group of amanifold as the group of all diffeomorphisms on the manifold. Unless otherwisestated, the structure groups of bundles whose fibers are manifolds are subgroups ofthe diffeomorphism groups. In other words the bundles are smooth bundles.Definition 2 . For a fold map f : M → R n on a closed and connected manifold M ,we also assume that f | S ( f ) is an embedding and that for each connected component C of the singular value set and its small closed tubular neighborhood N ( C ), thecomposition of f | f − ( N ( C )) : f − ( N ( C )) → N ( C ) with a canonical projection to C gives a trivial bundle over C . In this situation we say that f is S-trivial . Theorem . Let A , B and C be free commutative groups of rank a , b and c . Let { a i,j } aj =1 be a sequence of integers where 1 ≤ i ≤ b is an integer.Let p ∈ B ⊕ C . Let ( h i,j ) be a symmetric b × b matrix such that the ( i, j )-thcomponent is an integer and that the diagonal elements are 0. In this situation,there exist a 7-dimensional closed and simply-connected spin manifold and a foldmap f : M → R such that the following properties hold.(1) H ∗ ( M ; Z ) is free. Isomorphisms H ( M ; Z ) ∼ = A ⊕ B and H ( M ; Z ) ∼ = B ⊕ C hold and by fixing suitable identifications we have the following properties. EW FOLD MAPS ON 7-DIMENSIONAL CLOSED AND SIMPLY-CONNECTED MANIFOLDS3 (a) Products of elements in A ⊕ { } ⊂ H ( M ; Z ) vanish.(b) Consider a suitable basis { ( a j ∗ , } aj =1 of A ⊕ { } ⊂ H ( M ; Z ) anda suitable basis { (0 , b j ∗ ) } bj =1 of { } ⊕ B ⊂ H ( M ; Z ). The prod-uct of ( a j ∗ ,
0) and (0 , b j ∗ ) is regarded as ( a j ,j b j ∗ , ∈ B ⊕ { } ⊂ H ( M ; Z ). The product of (0 , b j ∗ ) and (0 , b j ∗ ) is regarded as ( h j ,j b j ∗ + h j ,j b j ∗ , ∈ H ( M ; Z ).(2) The 1st Pontryagin class of M is 4 p ∈ H ( M ; Z ) where the identificationbefore is considered.(3) The index of each singular point of f is always at most 1 and preimages ofregular values are disjoint unions of at most 3 copies of S . Furthermore, if( h i,j ) is the zero matrix, then we can construct this map f as an S-trivialmap such that preimages of regular values are disjoint unions of at most 2copies of S .We explain several facts implying explicitly that for understanding classes of7-dimensional closed and simply-connected spin manifolds in more geometric andconstructive ways, fold maps into R are interesting.It is known that there exist exactly 28 types of 7-dimensional oriented homotopyspheres (see [15] and see also [4]).Hereafter, for p in the Euclidean space R k , || p || denotes the distance between theorigin 0 and p where the underlying space is endowed with the Euclidean metric.For positive integers k and r , we denote the set { x ∈ R k | || x || ∈ N , ≤ || x || ≤ r } by D N ,rk . Theorem . Every 7-dimensional homotopy sphere admits an S-trivial fold map f into R satisfying the following properties.(1) f ( S ( f )) = D N , .(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.Moreover, we can show the following two facts.(1) M admits an S-trivial fold map f into R satisfying the last two propertiesof the previous three properties and the first property replaced by f ( S ( f )) = D N , if and only if M is a standard sphere.(2) M admits an S-trivial fold map f into R satisfying the last two propertiesof the previous three properties and the first property replaced by f ( S ( f )) = D N , if and only if the homotopy sphere M is oriented and one of 16 typesof the 28 types, where the standard sphere is one of the 16 types.We review known results on special generic maps on homotopy spheres. As aspecific case, if a homotopy sphere of dimension 7 admits a special generic map into R , then it is diffeomorphic to a standard sphere. Theorem . ([1], [17], [18], [26] and so on.) Every exotic homotopy sphere ofdimension m > R k for k = m − , m − , m − R . NAOKI KITAZAWA
Theorem . In the situation of Theorem 1, M admits no specialgeneric map into R when p ∈ B ⊕ C is not zero or there exists a non-zero elementin { a i,j } aj =1 or in { h i,j } .In the present paper, the main theorem is the following. Main Theorem.
Let A and B be free commutative groups of rank a and b . Let { k j } a + bj =1 be a sequence of integers such that integers in { k j + a } bj =1 are 0 ore 1. Let Y be a 4-dimensional closed and simply-connected spin manifold whose integralcohomology ring is isomorphic to that of a manifold represented as a connectedsum of finitely many copies of S × S and let us denote H j ( Y ; Z ) by H j . In thissituation, there exist a 7-dimensional closed and simply-connected spin manifold M and an S-trivial fold map f : M → R satisfying the following properties.(1) H ∗ ( M ; Z ) is free.(2) Isomorphisms H ( M ; Z ) ∼ = A ⊕ H , H ( M ; Z ) ∼ = B ⊕ H , H ( M ; Z ) ∼ = B ⊕ H and H ( M ; Z ) ∼ = A ⊕ H hold and by fixing suitable identificationswe have the following properties.(a) Products of elements in A ⊕ { } ⊂ H ( M ; Z ) vanish.(b) Products of elements in A ⊕ { } ⊂ H ( M ; Z ) and B ⊕ { } ⊂ H ( M ; Z )vanish.(c) Consider a suitable basis { ( a j ∗ , } aj =1 for A ⊕ { } ⊂ H ( M ; Z ) anda suitable basis for { ( b j ∗ , } bj =1 for B ⊕ { } ⊂ H ( M ; Z ). We alsotake a suitable basis { (0 , h j ∗ ) } rank H j =1 for { } ⊕ H ⊂ H ( M ; Z ). Theproduct of ( a j ∗ ,
0) and (0 , h j ∗ ) is regarded as ( k j h j ∗ , ∈ { } ⊕ H ⊂ H ( M ; Z ). The product of ( b j ∗ ,
0) and (0 , h j ∗ ) is regarded as( k a + j h j ∗ , ∈ { } ⊕ H ⊂ H ( M ; Z ).(d) For the suitable basis { (0 , h j ∗ ) } rank H j =1 for { } ⊕ H ⊂ H ( M ; Z ) justbefore, we have the following properties.(i) rank H is even.(ii) We can take the basis so that the dual PD X ( h j ∗ ) ∗ of PD X ( h j ∗ )is h rank H + j ∗ for 1 ≤ j ≤ rank H .(iii) For the suitable basis before, the product of (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) and (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) van-ishes unless | j − j | = rank H and the product of (0 , h j ∗ ) ∈{ } ⊕ H ⊂ H ( M ; Z ) and (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) isΣ bj =1 ( k a + j b j ∗ , ∈ B ⊕{ } ∼ = H ( M ; Z ) where | j − j | = rank H .For this class of manifolds, we can also show the following theorem. Main Theorem.
In Main Theorem 1.2, we can obtain manifolds which we cannotobtain in Theorem 1 under the constraint that the matrix ( h i,j ) is the zero matrix.1.3. The content of the present paper.
The organization of the paper is as thefollowing. In the next section, we demonstrate construction of new fold maps on 7-dimensional closed, simply-connected and spin manifolds and show Main Theorems(Theorems 5 and 6). Key methods resemble methods in the referred articles inTheorems 1 and 4 in a sense and are also mainly based on arguments in [9], [10]and so on. It is an important fact that these two articles of the author are mainlyfor studies of topological properties of
Reeb spaces of fold maps. The
Reeb space W c of a map c : X → Y is the quotient space X/ ∼ c of X from the following EW FOLD MAPS ON 7-DIMENSIONAL CLOSED AND SIMPLY-CONNECTED MANIFOLDS5 equivalence relation ∼ c on X : for x , x ∈ X , x ∼ c x if and only if they are ina same connected component of a same preimage. Reeb spaces already appearedin [16] for example. For a fold map, the Reeb space has been shown to be apolyhedron whose dimension is equal to that of the target space in [13], [23], and soon, and Reeb spaces have been fundamental tools in studying algebraic or geometricproperties, especially, (algebraic) topological properties of the manifolds. Note alsothat investigating the homology groups and the cohomology rings of the manifoldsare different from investigating those of the Reeb spaces and more difficult. Inconsiderable cases, Reeb spaces inherit informastion of topological invariants of themanifolds admitting the maps such as homology groups, cohomology rings, and soon.2. Construction of new family of fold maps on -dimensional closedand simply-connected manifolds of a new class. Hereafter, M denotes a closed and connected manifold of dimension m , let n < m be a positive integer and let f : M → R n denote a smooth map unless otherwisestated. For a topological space X such as a manifold, which is regarded as apolyhedron in a canonical way, and a general polyhedron, let c be a homologyclass. The class c is represented by a closed and compact submanifold Y with noboundary, if for a homology class ν Y of degree dim Y canonically obtained from Y , i ∗ ( ν Y ) = c where i : Y → X denotes the inclusion: in other words ν Y is the fundamental class if Y is connected, orientable and oriented and characterized asthe generator of the homology group of degree dim Y respecting the orientation.The following special generic maps play important roles. Hereafter, a linear bundle means a smooth bundle whose fiber is a unit sphere or disc and whosestructure group acts linearly in a natural way on the fiber.Example 1 . Let l ≥ m > n ≥ M of dimension m represented as a connected sum of l manifolds in a family { S l j × S m − l j } lj =1 of products of exactly two standard spheressatisfying 1 ≤ l j ≤ n − f : M → R n satisfying thefollowing properties.(1) f | S ( f ) is an embedding.(2) f ( M ) is a compact submanifold and represented as a boundary connectedsum of l manifolds in the family { S l j × D n − l j } lj =1 of product manifolds.(3) The following two submersions, or smooth maps with no singular point,give a trivial liner bundle whose fiber is D m − n +1 and a smooth bundlewhose fiber is S m − n , respectively.(a) The composition of the restriction of f to the preimage of a small collarneighborhood of ∂f ( M ) with the canonical projection to ∂f ( M ).(b) The restriction of f to the preimage of the complementary set of theinterior of the collar neighborhood before in f ( M ).(4) We denote the surjection obtained by restricting the target space of f to f ( M ) by f M . The homomorphism f M ∗ between the homology groups mapsa class represented by S l j × {∗ j, } ⊂ S l j × S m − l j in the connected sum toa class represented by S l j × {∗ j, } ⊂ S l j × Int D n − l j ⊂ S l j × D n − l j in theboundary connected sum f ( M ). NAOKI KITAZAWA
The following proposition is a fundamental fact and rigorous proofs are left toreaders.
Proposition . In the situation of Example 1, let ( m, n ) = (7 , l > l a , l b ≥ l a + l b = l and let l j = 2 for 1 ≤ j ≤ l a and l j = 3 for l a + 1 ≤ j ≤ l . We choose a suitable class represented by S l j × {∗ j, } ⊂ S l j × S m − l j in the connected sum by ν j . We have the following two.(1) For a copy X of S , put a generator ν X, of its 3rd integral homology group,isomorphic to Z , or a fundamental class. Let { a j + l a } l b j =1 be a sequence ofintegers of length l b such that the integers are 0 or 1. In this situation,there exists an embedding i X,f ( M ) of X into the interior of f ( M ) such that i X,f ( M ) ∗ ( ν X, ) = Σ l b j =1 a j + l a f M ∗ ( ν j + l a ).(2) For a copy X of S × S , put a generator ν X, of its 2nd integral ho-mology group, isomorphic to Z , and a generator ν X, of its 3rd integralhomology group, isomorphic to Z , or a fundamental class. Let { a j } lj =1 be a sequence of integers of length l such that the integers in { a j + l a } l b j =1 are 0 or 1. In this situation, there exists an embedding i X,f ( M ) of X intothe interior of f ( M ) such that i X,f ( M ) ∗ ( ν X, ) = Σ l a j =1 a j f M ∗ ( ν j ) and that i X,f ( M ) ∗ ( ν X, ) = Σ l b j =1 a j + l a f M ∗ ( ν j + l a ).For a compact manifold X , let there exist a closed and connected manifold X such that X is obtained by removing the interior of the union of two smoothly anddisjointly embedded unit discs of dim X and a Morse function c : X → R suchthat at distinct singular points the values are distinct, that there exist exactly twolocal extrema a < b , that their preimages are in the two embedded unit discs andthat on the distinctly embedded unit discs there exists no other singular point. Wedenote the restriction c | X by c X, ( X ,X ) .For a graded commutative algebra A over Z , we define the i -th module as themodule consisting of all elements of degree i of A . We also assume that the 0-thmodule is isomorphic to Z .For a non-negative integer i ≥
0, we define the ≤ i -part A ≤ i of A as a gradedcommutative algebra over Z as the following and as a graded module, this is re-garded as a submodule of the module A .(1) The j -th module is same as that of A for any j ≤ i .(2) The product of two elements such that the sum of the degrees is smallerthan or equal to i is same as that in the case of A .(3) The j -th module is the zero module for any j > i . Proposition . In the situation just before, let X be orientable satisfying dim X > i X : X → X denote the inclusion.(1) The restriction of i X ∗ : H ∗ ( X ; Z ) → H ∗ ( X ; Z ) to H ∗ ( X ; Z ) ≤ dim X − is anisomorphism between the graded commutative algebras H ∗ ( X ; Z ) ≤ dim X − and H ∗ ( X ; Z ) ≤ dim X − .(2) The restriction of i X ∗ : H ∗ ( X ; Z ) → H ∗ ( X ; Z ) to H ∗ ( X ; Z ) ≤ dim X − givesa monomorphism between the graded commutative algebras H ∗ ( X ; Z ) ≤ dim X − and H ∗ ( X ; Z ) ≤ dim X − and H ∗ ( X ; Z ) ≤ dim X − is represented as the internaldirect sum of the image of the monomorphism and a commutative subgroupG of H ∗ ( X ; Z ) ≤ dim X − isomorphic to Z . EW FOLD MAPS ON 7-DIMENSIONAL CLOSED AND SIMPLY-CONNECTED MANIFOLDS7
We need several notions and explain them. For a closed, connected and orientedmanifold X , we denote the so-called Poincar´e dual to an integral (co)homologyclass c by PD X ( c ). If for a (compact) topological space X whose integral homologygroup is free, then the dual c ∗ ∈ H j ( X ; Z ) of a homology class c ∈ H j ( X ; Z ) wecannot represent as a form kc ′ such that k = 0 , , − c ′ isnot zero is defined as a uniquely defined element such that c ∗ ( c ) = 1 and that forany subgroup G of H j ( X ; Z ) making H j ( X ; Z ) the internal direct sum of the groupgenerated by c and G and for any g ∈ G , c ∗ ( g ) = 0.Hereafter, ∼ = between groups means that the groups are isomorphic. Theorem . Let A and B be free commutative groups of rank a and b . Let { k j } a + bj =1 be a sequence of integers such that integers in { k j + a } bj =1 are 0 ore 1. Let Y be a 4-dimensional closed and simply-connected spin manifold whose integral cohomologyring is isomorphic to that of a manifold represented as a connected sum of finitelymany copies of S × S and let us denote H j ( Y ; Z ) by H j . In this situation, thereexist a 7-dimensional closed and simply-connected spin manifold M and an S-trivialfold map f : M → R satisfying the following properties.(1) H ∗ ( M ; Z ) is free.(2) Isomorphisms H ( M ; Z ) ∼ = A ⊕ H , H ( M ; Z ) ∼ = B ⊕ H , H ( M ; Z ) ∼ = B ⊕ H and H ( M ; Z ) ∼ = A ⊕ H hold and by fixing suitable identificationswe have the following properties.(a) Products of elements in A ⊕ { } ⊂ H ( M ; Z ) vanish.(b) Products of elements in A ⊕ { } ⊂ H ( M ; Z ) and B ⊕ { } ⊂ H ( M ; Z )vanish.(c) Consider a suitable basis { ( a j ∗ , } aj =1 for A ⊕ { } ⊂ H ( M ; Z ) and asuitable basis { ( b j ∗ , } bj =1 for B ⊕ { } ⊂ H ( M ; Z ). We also takea suitable basis { (0 , h j ∗ ) } rank H j =1 for { } ⊕ H ⊂ H ( M ; Z ). Theproduct of ( a j ∗ ,
0) and (0 , h j ∗ ) is regarded as (0 , k j h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ). The product of ( b j ∗ ,
0) and (0 , h j ∗ ) is regarded as(0 , k a + j h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ).(d) For the suitable basis { (0 , h j ∗ ) } rank H j =1 for { } ⊕ H ⊂ H ( M ; Z ) justbefore, we have the following properties.(i) rank H is even.(ii) We can take the basis so that the dual PD X ( h j ∗ ) ∗ of PD X ( h j ∗ )is h rank H + j ∗ for 1 ≤ j ≤ rank H .(iii) For the suitable basis before, the product of (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) and (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) van-ishes unless | j − j | = rank H and the product of (0 , h j ∗ ) ∈{ } ⊕ H ⊂ H ( M ; Z ) and (0 , h j ∗ ) ∈ { } ⊕ H ⊂ H ( M ; Z ) isΣ bj =1 ( k a + j b j ∗ , ∈ B ⊕{ } ∼ = H ( M ; Z ) where | j − j | = rank H . Proof.
First we construct a special generic map f on an 7-dimensional manifold M into R in Example 1 by setting ( l a , l b ) = ( a, b ) in Proposition 2. We replace a j by k j there. Take a copy X in Proposition 2 and remove the interior of a smallclosed tubular neighborhood N ( X ) and its preimage.Replace this removed map by a product map of c ( Y ,Y ) before where X and X are replaced by a suitable compact manifold Y and Y with a suitable Morsefunction c respectively and the identity map on a manifold diffeomorphic to X . NAOKI KITAZAWA
Note that this is regarded as a finite iteration of normal bubbling operations in [9],[10], and so on. Thus we have a desired fold map f .We observe the integral homology group and the integral cohomology ring ofthe manifold M to complete the proof. First we show the first statement (1) andthe second statement (2) on the cohomology group. H ( M ; Z ) and H ( M ; Z )are generated by classes represented by standard spheres in M and apart from f − ( N ( X )) and from these standard spheres and homology classes representedby them we have subgroups isomorphic to H ( M ; Z ) and H ( M ; Z ) respectively.We put A := H ( M ; Z ) and B = H ( M ; Z ): we take the duals of classes in thebases. Consider a generator of H ( Y ; Z ) ∼ = H ( Y ; Z ) such that we can define thedual of each of the elements in the basis. We go to (2d). Since Y is closed andsimply-connected, the homology groups and the cohomology groups of Y and Y are free and the first two statements hold (for H ). Furthermore, we can obtainthe generator(s) by first obtaining a generator of H ( Y ; Z ) ∼ = H ( Y ; Z ) for eachof which we can define the dual and taking the duals. By considering elements ofthe generators of A and H as integral cohomology classes of M in a canonicalway, we can observe the structure of H ( M ; Z ) and obtain a desired identification. H ( M ; Z ) is regarded as the internal direct sum of the following two subgroups A , and A , .(1) The classes represented by standard spheres in M and apart from f − ( N ( X ))and forming a basis for H ( M ; Z ). Consider the classes represented bythese spheres in M and their Poincar´e duals. We define A , as the sub-group generated by all the Poincar´e duals. This is isomorphic to B .(2) We take the image of a section of the trivial bundle define by the restrictionof f to f − ( X ). Each element of the suitable generator of H ( Y ; Z ) ∼ = H ( Y ; Z ) used to obtain the basis for H before is represented by a 2-dimensional closed submanifold with no boundary and we can obtain aproduct of the image of the section over S × {∗} ⊂ X = S × S and thissubmanifold in M in a canonical way. We can take the dual of the 4-thintegral homology class represented by the submanifold as a 4-th integralcohomology class of M . A , is defined as a subgroup generated by the setof all such classes. This is isomorphic to H .By considering elements of the generator of B and H , isomorphic to A , , as theimage of the map corresponding each element of A , to the Poincar´e dual, we canobserve the structure of H ( M ; Z ) and obtain a desired identification. H ( M ; Z ) isregarded as the internal direct sum of the following two subgroups A , and A , .(1) The classes represented by standard spheres in M and apart from f − ( N ( X ))and forming a basis for H ( M ; Z ). Consider the classes represented bythese spheres in M and their Poincar´e duals. We define A , as the sub-group generated by all the Poincar´e duals. This is isomorphic to A .(2) We take the image of a section of the trivial bundle define by the restrictionof f to f − ( X ). Each element of the suitable generator of H ( Y ; Z ) ∼ = H ( Y ; Z ) used to obtain the basis for H before is represented by a 2-dimensional closed submanifold with no boundary and we can obtain aproduct of the image of the section over X = S × S and this submanifoldin M in a canonical way. We can take the dual as a 5-th integral cohomologyclass of M . A , is defined as a subgroup generated by the set of all suchclasses. This is isomorphic to H . EW FOLD MAPS ON 7-DIMENSIONAL CLOSED AND SIMPLY-CONNECTED MANIFOLDS9
Note also that we can demonstrate this construction so that M is simply-connected and spin.We can show (2a) and (2b) easily by the structures of the cohomologygroups and (2c) follows by the structures of the cohomology groups andProposition 2. The third statement of (2d) follows by the structures of thecohomology groups and Propositions 2 and 3. This completes the proof. (cid:3) Theorem . In Theorem 5, we have manifolds M having integral cohomology ringswe cannot obtain as the integral cohomology rings of the manifolds in Theorem 1where the matrix ( h i,j ) is the zero matrix. Proof.
In Theorem 5, set a = 1, b = 1, k = 0, k = 0 and Y := S × S . Wecan easily see that H j ( M ; Z ) is free and of rank 3 for j = 2 , , ,
5. We cannot takea submodule of rank 2 of H ( M ; Z ) consisting of elements such that the squaresare zero. In Theorem 1 under the constraint that the matrix ( h i,j ) is the zeromatrix, we consider a 7-dimensional closed and simply-connected manifold suchthat H ( M ; Z ) is free and of rank 3. Products of elements in A ⊕ { } ⊂ H ( M ; Z )and products of elements in { } ⊕ B ⊂ H ( M ; Z ) vanish. This implies that wecan take a submodule of rank 2 of H ( M ; Z ) consisting of elements such that thesquares are zero.This completes the proof. (cid:3) Acknowledgement.
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