Surgery operations to fold maps to increase connected components of singular sets by two
aa r X i v : . [ m a t h . K T ] M a y SURGERY OPERATIONS TO FOLD MAPS TO INCREASECONNECTED COMPONENTS OF SINGULAR SETS BY TWO
NAOKI KITAZAWA
Abstract.
In geometry, understanding the topologies and the differentiablestructures of manifolds in constructive ways is fundamental and important. Itis in general difficult, especially for higher dimensional manifolds.The author is interested in this and trying to understand manifolds viaconstruction of explicit fold maps: differentiable maps locally represented asproduct maps of Morse functions and identity maps on open balls. Fold mapshave been fundamental and useful in investigating the manifolds by observing(the sets of) singular points and values and preimages as Thom and Whitney’spioneering studies and recent studies of Kobayashi, Saeki, Sakuma, and so on,show. Here, construction of explicit fold maps on explicit manifolds is difficult.The author constructed several explicit families of fold maps and inves-tigated the manifolds admitting the maps. Main fundamental methods aresurgery operations ( bubbling operations ), the author recently introduced mo-tivated by Kobayashi and Saeki’s studies such as operations to deform genericdifferentiable maps whose codimensions are negative into the plane preservingthe differentiable structure of the manifold in 1996 and so on. We removea neighborhood of a (an immersed) submanifold consisting of regular valuesin the target space, attach a new map and obtain a new fold map such thatthe number of connected components of the set consisting of singular pointsincreases. In this paper, we investigate cases where the numbers increase bytwo and obtain cases of a new type. Introduction and fundamental notation and terminologies.
In geometry, understanding the topologies and the differentiable structures ofmanifolds in geometric and constructive ways is fundamental and important. Itis in general difficult, especially for higher dimensional manifolds, although theywere considerably understood in the latter half of the last century via sophisticatedalgebraic topological theory and abstract differential topological theory such ashomotopy theory, fundamental theory of Morse functions, surgery theory, and soon. See also [12], [13] and [22] for example to know related tools more precisely.This paper concerns this understanding via fold maps, regarded as higher di-mensional versions of Morse functions.We first review fold maps. Before this, we explain several fundamental termi-nologies on differentiable manifolds and maps.Throughout this paper, for a differentiable map, a singular point is a point atwhich the rank of the differential of the map drops. a singular value is a point inthe target space realized as a value at a singular point, and a regular value is a
Mathematics Subject Classification.
Primary 57R45. Secondary 57N15.
Key words and phrases.
Singularities of differentiable maps; generic maps. Differential topol-ogy. Reeb spaces. point in the target space which is not a singular value: they are defined as this inmost of introductory books on differentiable manifolds and maps.Moreover, the singular set of the map is the set of all singular points, the singularvalue set is the image of the singular set, and the regular value set is the complementof the singular value set.Throughout the present paper, manifolds are differentiable and smooth (of class C ∞ ), maps are differentiable and smooth (of class C ∞ ) unless otherwise stated. Adiffeomorphism is always assumed to be smooth and the diffeomorphism group ofa manifold means the group consisting of all diffeomorphisms on the manifold.For a smooth manifold X , we denote the tangent bundle by T X and the tangentvector at p ∈ X by T p X ⊂ T X .1.1.
Fold maps.
Fold maps have been fundamental and useful in investigating themanifolds by observing the singular sets and the singular value sets and preimagesas Thom and Whitney’s pioneering studies ([21] and [23]) and recent studies ofKobayashi, Saeki, Sakuma, and so on, which we will introduce later, show.Definition 1 . Let m > n ≥ m -dimensionalsmooth manifold with no boundary into an n -dimensional smooth manifold with noboundary is said to be a fold map if at each singular point p , the map is representedas ( x , · · · , x m ) ( x , · · · , x n − , m − i X k = n x k − m X k = m − i +1 x k )for some coordinates and an integer 0 ≤ i ( p ) ≤ m − n +12 .For a fold map, the following properties hold. • For any singular point p , the i ( p ) in Definition 1 is unique : i ( p ) is the index of p . • The set consisting of all singular points of a fixed index of the map is asmooth and closed submanifold of the domain with no boundary of dimen-sion n − • The restriction to the singular set of the original map is a smooth immer-sion.1.2. (Normal) crossings of a family of smooth immersions.
We define a crossing and a normal crossing of a family of smooth immersions.Let a > { c j : X j → Y } aj =1 be a family of a smooth im-mersions from m j -dimensional smooth manifolds X j without boundaries into an n -dimensional smooth manifold Y with no boundary. A crossing of the family ofthe smooth immersions is a point y ∈ Y such that S aj =1 c j − ( y ) has at least twopoints. A crossing is said to be normal if the following properties hold.(1) The disjoint union S aj =1 c j − ( y ) of these preimages is a finite set consistingof exactly b y > S aj =1 c j − ( y ) by { p y,j } b y j =1 . Let y ( j ) be thenumber satisfying p y,j ∈ X y ( j ) , which we can determine uniquely. Wedenote the dimension of the intersection T b y j =1 dc y ( j ) p y,j ( T p y,j X y ( j ) ) of theimages of the differentials at all points p y,j by y ( c ). In this situation, y ( c ) + Σ b y j =1 ( n − m y ( j ) ) = n . NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 3
We can also consider the case for a single immersion (the case a = 1) similarly.Definition 2 . A stable fold map is a fold map whose restriction to the singular setis a smooth immersion such that the crossings of the restriction to the singular setof the original fold map are always normal.Remark 1 . Stable fold maps are actually defined as a fold map which is stable .However, we can also define as Definition 2. For more precise and systematicexplanations on stable (fold) maps, see [1] for example.In the present paper, we consider crossings which are normal and preimages ofwhich consist of at most two points.1.3.
Reeb spaces.
Reeb spaces are also fundamental and important tools in in-vestigating the topologies of the domains of smooth maps whose codimensions arenegative.Let X and Y be topological spaces. For p , p ∈ X and for a continuous map c : X → Y , we define a relation ∼ c on X in the following way: p ∼ c p if and onlyif p and p are in a same connected component of c − ( p ) for some p ∈ Y . Thus ∼ c is an equivalence relation on X . We denote the quotient space X/ ∼ c by W c Definition 3 . We call W c the Reeb space of c .We denote the induced quotient map from X into W c by q c . We can define¯ c : W c → Y in a unique way so that the relation c = ¯ c ◦ q c holds. Proposition . For (stable) fold maps, the Reeb spaces are polyhedra and thedimensions are equal to the dimensions of the target manifolds.For Reeb spaces, see also [14] for example.1.4.
Explicit fold maps and their Reeb spaces.
We present fundamental andimportant examples of fold maps here. We explain terminologies on bundles.For a topological space X , an X-bundle is a bundle whose fiber is X .Hereafter, the structure groups of bundles such that their base spaces and fibersare manifolds are assumed to be (subgroups of) diffeomorphism groups except casessuch as situations in a sketch of the proof of Proposition 3: the bundles are smooth ina word except several cases. For an integer k >
0, a linear bundle is a smooth bundlewhose fiber is a k -dimensional unit disc (standard closed disc of a fixed diameter) orthe ( k − R k and whose structure group is a subgroupof the k -dimensional orthogonal group O ( k ) acting linearly and canonically.Example 1 . (1) A special generic map is defined as a fold map such that theindex of each singular point is 0. Canonical projections of unit spheres aresimplest stable special generic maps. According to studies [15], [16], [18],[24] and so on, homotopy spheres which are not diffeomorphic to standardspheres do not admit special generic maps into sufficiently high dimensionalEuclidean spaces whose dimensions are smaller than those of the homotopyspheres: on the other hand, homotopy spheres except 4-dimensional oneswhich are not diffeomorphic to S admit special generic maps into the planesuch that the restriction to the singular set is an embedding and the singularset is a circle. Moreover, 4-dimensional homotopy spheres which are notdiffeomorphic to S admit no special generic maps into the Euclidean spaces R , R and R and such homotopy spheres are still undiscovered. NAOKI KITAZAWA
The Reeb space of a (stable) special generic map f from a closed andconnected manifold of dimension m into R n satisfying the relation m >n ≥ n -dimensional compact and connected manifoldwe can immerse into R n . The image is regarded as the image of a suitableimmersion of the n -dimensional manifold. The boundary of the Reeb spaceand the image of the singular set by q f agree.Conversely, for arbitrary integers m > n ≥ n -dimensional com-pact manifold we can immerse into R n , we can construct a (stable) specialgeneric map from a suitable closed and connected manifold of dimension m into R n whose Reeb space is diffeomorphic to the n -dimensional manifold.Moreover, we have the following bundles for a general special genericmap f from a closed and connected manifold of dimension m into R n .(a) If we restrict the map q f to the preimage of the interior of the Reebspace, then it gives a smooth S m − n -bundle over the interior of theReeb space.(b) If we restrict the map q f to the preimage of a small collar neighborhoodof the boundary of the Reeb space and consider the composition of thiswith a canonical projection onto the boundary, then it gives a linear S m − n +1 -bundle over the boundary.Moreover, for an arbitrary n -dimensional compact manifold we can immerseinto R n , we can construct a stable special generic map on a suitable closedand connected manifold of dimension m into R n so that the Reeb spaceis diffeomorphic to the n -dimensional manifold and that these bundles aretrivial. We can replace R n by an arbitrary manifold N of dimension n withno boundary. See [15], the articles [6] and [7] by the author, and so on.(2) ([2], [3] and [5]) Let l > m > n ≥ l − S m − n × S n into R n (this is a standardsphere for l = 1) satisfying the following properties.(a) The singular value set is ⊔ lj =1 {|| x || = j | x ∈ R n } .(b) Preimages of regular values are disjoint unions of standard spheres.(c) In the target space R n , the number of connected components of apreimage increases as we go straight to the origin 0 ∈ R n of the targetEuclidean space starting from a point in the complement of the image.The Reeb space is simple homotopy equivalent to a bouquet of l − n -dimensional spheres for l > l = 1. We can constructa map satisfying these properties for manifolds obtained by changing theproducts to total spaces of general smooth S m − n -bundles over S n .(3) In [6], [7], [8] and so on present construction of stable fold maps satisfyingthe assumption of Proposition 3 in the last and information of the topologies(the homology groups and the cohomology rings) of the Reeb spaces (and asa result those of the manifolds). The maps are obtained by finite iterationsof surgery operations ( bubbling operations ) starting from fundamental foldmaps: this operation is introduced by the author in [6] respecting ideas of[9], [10] and [11] for example. More explicitly, we start from stable specialgeneric maps of suitable classes mainly, by changing maps and manifoldsby bearing new connected components of singular (value) sets of stable fold NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 5 maps one after another, we obtain desired maps. The previous examplealso shows simplest examples of these maps.1.5.
Construction of explicit fold maps by new surgery operations (bub-bling operations) and the organization of this paper.
In this paper, wepresent further studies on construction in Example 1 (3). We apply bubblingoperations first defined in [8], improved versions of original versions in [6]. Theorganization of the paper is as the following. In the next section, we introduce a bubbling operation first introduced based on [8]. The last section is devoted to apresentation of new results: Proposition 2 and Theorem 1. We present constructionof new families of explicit fold maps and investigate the cohomology rings of theReeb spaces. We see that the cohomology rings of newly obtained Reeb spaces arealso obtained first in the present paper. Last we present Proposition 3. This givesexplicit situations where Reeb spaces of fold maps on manifolds know much topo-logical information of the manifolds. We can apply this to new maps obtained byapplying Proposition 2 and Theorem 1 explicitly. For Proposition 3, an essentiallysame explanation is in [8].1.6.
Acknowledgement.
The author is a member of the project Grant-in-Aidfor Scientific Research (S) (17H06128 Principal Investigator: Osamu Saeki) ”In-novative research of geometric topology and singularities of differentiable map-pings” (https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17H06128/ ) andsupported by this.2.
Bubbling operations and fold maps such that preimages ofregular values are disjoint unions of spheres.
We introduce bubbling operations , first introduced in [6], referring to [8]. Moreprecisely, in the present paper, we essentially consider only the operations satisfyingseveral good properties:
ATSS operations .Hereafter, m > n ≥ M is a smooth, closed and connected manifoldof dimension m , N is a smooth manifold of dimension n with no boundary, and f : M → N is a smooth map.For a smooth map c , we denote the singular set S ( c ).Definition 4 . For a stable 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 an open manifold diffeomorphicto ¯ 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) Crossings of the family { c j | ∂N ( S j ) : ∂N ( S j ) → P } lj =1 are normal and eachpreimage consists of exactly two points. The number of crossings of thisfamily is finite. NAOKI KITAZAWA (3) Crossings of { c j | S j : S j → P } lj =1 are normal and each preimage consists ofexactly two points. The number of crossings of this family is finite.(4) We denote the set of all the crossings of the family { c j | S j : S j → P } lj =1 of the immersions by { p j ′ } l ′ j ′ =1 . For each p j ′ , there exist two integers 1 ≤ a ( j ′ ) , b ( j ′ ) ≤ l and small standard closed discs D j ′ − ⊂ S a ( j ′ ) and D j ′ ⊂ S b ( j ′ ) satisfying the following 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.(d) If we restrict the base space of the bundle N ( S a ( j ′ ) ) to D j ′ − and thatof the bundle N ( S b ( j ′ ) ) to D j ′ , then the images of the total spacesof these resulting bundles by the immersions c a ( j ′ ) and c b ( j ′ ) agree assubsets in R n .(e) The restrictions to the total spaces of the two immersions just beforeare embeddings.(f) The set of all the crossings of the family of { 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 4, 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. For a suitable r ,same properties as presented in Definition 4 hold. In other words, we can obtainanother family { ( S j , N ′ ( S j ) , c j | N ′ ( S j ) : N ′ ( S j ) → P ) } lj =1 and this is also regardedas a normal system of submanifolds compatible with f : we can identify each fiber,which is a standard closed disc of diameter r with a unit disc via the diffeomorphismdefined by the correspondence t r t .Definition 5 . (1) The familiy { ( S j , N ( S j ) , c j : N ( S j ) → P ) } lj =1 is said tobe a wider normal 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 .(2) For a stable fold map f : M → N and an integer l >
0, let P be a connectedcomponent 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 systemsupporting this as before. Assume that we can construct a stable fold map f ′ on an m -dimensional closed manifold M ′ into R n satisfying the followingproperties.(a) Q is the preimage f − ( S lj =1 c j ′ ( N ′ ( S j ))).(b) M − Int Q is realized as a compact submanifold of M ′ of dimension m by considering a suitable smooth embedding e : M − Int Q → M ′ .(c) f | M − Int Q = f ′ ◦ e | M − Int Q holds.(d) f ′ ( S ( f ′ )) is the disjoint union of f ( S ( f )) and S nj =1 c j ( ∂N ( S j )).(e) The indices of points in the preimages of new connected componentsin the resulting singular value set are all 1. NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 7 (f) For each regular value p of the resulting map sufficiently close tothe union S lj =1 c j ( S j ), the preimages are disjoint unions of standardspheres.This yields a procedure of constructing f ′ from f . We call it an ATSS op-eration to f . The union S lj =1 c j ( S j ) is called the generating normal system of the operation.We show a local fold map around p j ′ in Definition 4. This is also presented in[8] with FIGURE 2. S is the two point set with the discrete topology.First in the situation of Example 1 (2), we explain a restriction of a fold mapfor l = 1 to the preimage of the set of all points x ∈ R n satisfying || x || ≤ . Weconstruct a product bundle D n × S m − n over D n . We also set D n as the set ofall points x ∈ R n satisfying || x || ≤ . We also set a Morse function ˜ f m − n, on amanifold obtained by removing the interior of a standard closed disc of dimension m − n + 1 embedded smoothly in the interior of S m − n × [ − ,
1] onto [ , ] ⊂ (0 , + ∞ ) ⊂ R such that the following four hold.(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.(4) The value at the singular point is 1.We glue the projection of the product bundle and the map ˜ f m − n, × id S n − :[ , + ∞ ) × S n − where we identify the base space D n of the product bundle witha standard closed disc of dimension n whose diameter is and whose center is theorigin 0 ∈ R n . By gluing suitably, we have a desired smooth map onto the standardclosed disc of dimension n whose center is the origin and whose diameter is in R n . This is a desired map, obtained by restricting the original fold map for l = 1.See also [2] and [5] for the fold map for l = 1.We denote the resulting map onto the standard closed disc of dimension n by˜ f m,n, .We consider the composition of ˜ f m − n +dim D j ′− , dim D j ′− , with a suitable dif-feomorphism, we have a smooth map onto a sufficiently small standard closeddisc D ′ j ′ − ⊃ D j ′ − of dimension dim D j ′ − satisfying S a ( j ′ ) ⊃ D ′ j ′ − ⊃ Int D ′ j ′ − ⊃ D j ′ − . We can take a sufficiently small standard closed disc D ′ j ′ ⊃ D j ′ of dimension dim D j ′ satisfying similar properties. We can consider the prod-uct map of the previous smooth map and the identity map id D ′ j ′ . We composethe resulting map with a suitable diffeomorphism onto a manifold, regarded as c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ ) and containing p j ′ ∈ R n in its interior.We can restrict the map to a total space of a trivial D m − n -bundle over the targetspace, regarded as c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ ) and containing p j ′ in the interior.For the composition of ˜ f m − n +dim D j ′ , dim D j ′ , with a suitable diffeomorphism,we can restrict the map to a total space of a trivial D m − n -bundle over the targetspace, diffeomorphic to a standard closed disc of dimension dim D j ′ . This gives atrivial D m − n -bundle. The total space is regarded as a submanifold of the domain ofthe original map and we can also restrict the composition of ˜ f m − n +dim D j ′ , dim D j ′ , with the suitable diffeomorphism to the closure of its complement in the domain. NAOKI KITAZAWA
The closure of the complement is also a compact submanifold of dimension m − n + dim D j ′ . We consider the composition of the product map of the restrictionto this complement and the identity map id D ′ j ′− with a suitable diffeomorphismonto a manifold, regarded as c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ )In a suitable way, we replace the original projection of the trivial D m − n -bundleover the target space, identified with c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ ), containing p j ′ in the interior, by the map just before: the composition of the product map of therestriction to the complement of a total space of a trivial D m − n -bundle over thetarget space, diffeomorphic to a standard closed disc of dimension dim D j ′ , and theidentity map id D ′ j ′− with a suitable diffeomorphism onto a manifold, regarded as c a ( j ′ ) ( D ′ j ′ − ) × c b ( j ′ ) ( D ′ j ′ ). The resulting map is a desired local map and said tobe a local canonical fold map around a crossing for an ATSS operation.By the definition, the following corollary immediately follows. Corollary . Let f : M → N be a stable fold map on an m -dimensional closedand connected manifold M into an n -dimensional manifold N with no boundarysatisfying m − n >
1. If an ATSS operation is performed to f and a new map f ′ isobtained as a result, then W f is a proper subset of W f ′ and ¯ f ′ | W f = ¯ f : W f → N .3. New families of stable fold maps and their Reeb spaces obtainedby ATSS operations increasing the numbers of connectedcomponents of singular sets by two.
In this section, we need fundamental theory of graded commutative algebras overcommutative rings (PIDs) and cohomology rings.Definition 6 . Let A be a module over a commutative ring R having a unique identityelement 1 = 0 ∈ R . Let a ∈ A be a non-zero element such that the following twohold. • a is not represented as a = ra ′ where ( r, a ′ ) ∈ R × A and r is not a unit. • For r ∈ R , ra = 0 if and only if r = 0.Let A be represented as an inner direct sum of the submodule generated by theone element set { a } and another submodule B . We can define a homomorphism a ∗ : A → R between the modules over R such that a ∗ ( a ) = 1 and that a ∗ ( B ) = 0for any such B . We call a ′ the dual of a ∈ A .For a graded commutative algebra A over a graded commutative ring R and anon-negative integer i ≥
0, we call the module of all elements of degree i the i -thmodule of A . Hereafter, we assume that the 0-th module is R and equipped with acanonical action by R .Definition 7 . Let A and A be graded commutative algebras over a commutativering R having a unique identity element 1 = 0 ∈ R . Let A be a graded commutativealgebra over R satisfying the following properties.(1) The i -th module is the direct sum of the i -th modules of A and A .(2) Let i , i > a i , , a i , ) , ( a i , , a i , ) ∈ A ⊕ A , which areelements of i -th and i -th modules, respectively, the product is ( a i , a i , , a i , a i , ) ∈ A ⊕ A .(3) For r ∈ R , where we define this as an element of the 0-th module and takethe product of this and an element ( a i, , a i, ) ∈ A ⊕ A of the i -th modulewhere i >
0. The product is ( ra i, , ra i, ) ∈ A ⊕ A . NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 9 A is called a graded commutative algebra obtained canonically from the directsum A ⊕ A .We first obtain examples as Proposition 2. Proposition . Let R be a PID having an identity element 1 ∈ R satisfying 1 = 0 ∈ R . Let l i ∈ R be an element represented as l i, ∈ Z times the identity element 1for i = 1 ,
2. Let f : M → N be a stable fold map on an m -dimensional closed andconnected manifold into an n -dimensional manifold with no boundary satisfying m − n >
1. Let N be not closed. We also assume at least one of the followingconditions.Let U be an open set in N − f ( S ( f )) such that f | f − ( U ) : f − ( U ) → U gives atrivial S m − n -bundle.In this situation, by an ATSS operation to f , we have a new fold map f ′ satisfyingthe following properties if n is even.(1) H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = n and H i ( W f ; R ) ⊕ R ⊕ R for i = n , n .(2) The cohomology group H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = n and H i ( W f ; R ) ⊕ R ⊕ R for i = n , n : we can set isomorphisms betweenmodules over R for identifications respecting Definition 7 and we will abusethese identifications. The cohomology group H i ( W f ; R ) is isomorphic to H i ( W f ′ ; R ) for i = n , n and identified with H i ( W f ; R ) ⊕ { } ⊕ { } in H i ( W f ; R ) ⊕ R ⊕ R before for i = n , n via a map i f,f ′ ( x ) := ( x, ,
0) for x ∈ H i ( W f ; R ). Furthermore this gives a monomorphism over R from H i ( W f ; R ) into H i ( W f ′ ; R ) for all i : for i = n , n , i f,f ′ ( x ) = x for x ∈ H i ( W f ; R ), which we abbreviated just before.(3) We can define a graded commutative algebra A R over R such that the i -thmodule is isomorphic to { } for i = 0 , n , n and isomorphic to R ⊕ R andidentified with { } ⊕ R ⊕ R in H i ( W f ; R ) ⊕ R ⊕ R before for i = n , n andthat the following rules are satisfied.(a) Under the explained identifications, the product of (0 , , ∈ { }⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R and (0 , , ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R is (0 , l , , l , ) ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R .(b) Under the explained identifications, the product of (0 , , ∈ { }⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R and (0 , , ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R is (0 , , ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R .(c) Under the explained identifications, the product of (0 , , ∈ { }⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R and (0 , , ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R is (0 , , ∈ { } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R We also have a new fold map f ′ satisfying the following properties if n is an arbitrarypositive integer and k is a positive integer 2 k < n .(1) H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = k, n − k, n , H i ( W f ; R ) ⊕ R for i = k, n − k and H n ( W f ; R ) ⊕ R ⊕ R for i = n .(2) The cohomology group H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = k, n − k, n , H i ( W f ; R ) ⊕ R for i = k, n − k and H i ( W f ; R ) ⊕ R ⊕ R for i = n : we can set isomorphisms between modules over R for identifica-tions respecting Definition 7. The cohomology group H i ( W f ; R ) is isomor-phic to H i ( W f ′ ; R ) for i = k, n − k, n , identified with H i ( W f ; R ) ⊕ { } in H i ( W f ; R ) ⊕ R before for i = k, n − k via a map i f,f ′ ( x ) := ( x,
0) for x ∈ H i ( W f ; R ) and identified with H i ( W f ; R ) ⊕ { } ⊕ { } in H i ( W f ; R ) ⊕ R ⊕ R before for i = n via a map i f,f ′ ( x ) := ( x, ,
0) for x ∈ H i ( W f ; R ). Fur-thermore the maps give a monomorphism over R from H i ( W f ; R ) into H i ( W f ′ ; R ) for all i : for i = k, n − k, n , i f,f ′ ( x ) = x for x ∈ H i ( W f ; R ),which we abbreviated just before.(3) We can define a graded commutative algebra A R over R such that the i -th module is isomorphic to { } for i = 0 , k, n − k, n , isomorphic to R and identified with { } ⊕ R in H i ( W f ; R ) ⊕ R before for i = k, n − k andisomorphic to R ⊕ R and identified with { } ⊕ R ⊕ R in H i ( W f ; R ) ⊕ R ⊕ R before for n and that the following rules are satisfied.(a) Under the explained identifications, the product of (0 , ∈ { } ⊕ R ⊂ H k ( W f ; R ) ⊕ R and (0 , ∈ { } ⊕ R ⊂ H n − k ( W f ; R ) ⊕ R is (0 , l , l ) ∈{ } ⊕ R ⊕ R ⊂ H n ( W f ; R ) ⊕ R ⊕ R .(b) Under the explained identifications, the square of (0 , ∈ { } ⊕ R ⊂ H k ( W f ; R ) ⊕ R vanishes.(c) Under the explained identifications, the square of (0 , ∈ { } ⊕ R ⊂ H n − k ( W f ; R ) ⊕ R vanishes.Last the resulting cohomology ring H ∗ ( W f ; R ) is regarded as a subalgebra of H ∗ ( W f ′ ; R ) via the following rules.(1) For two elements c ∈ H i ( W f ; R ) and c ∈ H i ( W f ; R ) where i , i > c j , ∈ H i j ( W f ; R ) ⊕ { } ⊂ H i j ( W f ′ ; R ) for j = 1 , c c , ∈ H i + i ( W f ; R ) ⊕ { } ⊂ H i + i ( W f ′ ; R ): we can identify thiswith c c ∈ H i + i ( W f ; R ).(2) For elements r ∈ H ( W f ; R ) and c ∈ H i ( W f ; R ) where i >
0, we considerthe natural identifications so that they are regarded as r ∈ H ( W f ; R )where the group is identified with H ( W f ′ ; R ) and ( c, ∈ H i ( W f ; R ) ⊕{ } ⊂ H i ( W f ′ ; R ), respectively, and the product is ( rc, ∈ H i ( W f ; R ) ⊕{ } ⊂ H i ( W f ′ ; R ): we can identify this with rc ∈ H i ( W f ; R ).We prove this using tools and ideas used for the proof of Proposition 3 of [8]. Proof.
In the proof, notation in Definition 4 and around this will be used. Letus find a suitable normal system of submanifolds compatible with f . As this, wewill find { ( S , N ( S ) , c : N ( S ) → P ) , ( S , N ( S ) , c : N ( S ) → P ) } such that S is a standard sphere of dimension 0 < k ≤ n and that S is standard sphere ofdimension n − k .We can take { ( S , N ( S ) , c : N ( S ) → P ) , ( S , N ( S ) , c : N ( S ) → P ) } sothat the family of immersions has exactly | l , | + | l , | ≥ | l , | + | l , | ) crossings), that the normal bundle of the immersion is trivialand that c ( N ( S )) S c ( N ( S )) ⊂ U . We can perform an ATSS operation whosegenerating normal system is c ( S ) S c ( S ) to obtain a new fold map f ′ . Weuse local canonical fold maps around crossings in section 2 around crossings in c ( S ) S c ( S ). Around the remaining singular values and regular values, we con-struct products of Morse functions with exactly one singular point, which is of index1, and identity maps on ( n − S m − n -bundles over n -dimensional manifolds, respectively. We can glue all the local maps together. Wecan perform the construction thanks to the assumption that f | f − ( U ) : f − ( U ) → U gives a trivial S m − n -bundle. NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 11
By the definition of a normal bubbling operation, W f ′ is regarded as a spaceobtained by attaching a polyhedron A we can obtain by identifying exactly ( | l , | + | l , | )-pairs of disjointly embedded PL discs of dimension n , represented as a productof of a k -dimensional disc and an ( n − k )-dimensional dicsc, two smooth (PL)manifolds S × S n − k and S × S k to B := ¯ f − ( c ( N ( S ))) S ¯ f − ( c ( N ( S ))) ⊂ W f .Moreover, S × D n − k and S × D k are attached: D n − k ⊂ S n − k and D k ⊂ S k arehemispheres.We explain the topologies of the Reeb spaces and the polyhedra without usingsophisticated terminologies on Mayer-Vietoris sequences and other notions of alge-braic topology. Rigorous understandings via these terminologies are left to readersand see also Proposition 3 of [8]. H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = k , H i ( W f ; R ) ⊕ R for i = k = n , H i ( W f ; R ) ⊕ R ⊕ R for i = k = n and H i ( W f ; R ) ⊕ R for i = n − k = n . We canidentify the modules for these cases.We explain about the summands R . For i = k, n − k , the summands R areseen to be generated by the class represented by {∗} × S n − k ⊂ S × S n − k or {∗} × S k ⊂ S × S k in the original manifolds to obtain A where ∗ is a suitable pointin S or S . In the case i = n , the summands R are seen to be generated by theclasses represented by suitable subpolyhedra of A obtained after the deformationof the original manifolds to obtain A , which are n -dimensional polyhedra obtainedfrom the original n -dimensional closed, connected and orientable manifolds.We discuss the cohomology rings. By the construction, the resulting cohomol-ogy ring is isomorphic to and can be identified with a graded commutative algebraobtained canonically from the direct sum of H ∗ ( W f ; R ) and a new graded commu-tative algebra A R . We denote the j -th module of A R by A R,j . This is zero unless j = k, n − k, n . Both in the cases where k = n − k and k = n , we consider the prod-ucts in H n ( W f ′ ; R ) of the two classes in H k ( W f ′ ; R ) and H n − k ( W f ′ ; R ) generatingthe summands R . A R,i is isomorphic to and identified with R for i = k, n − k and k = n . A R,i is isomorphic to and identified with R ⊕ R for an even n and i = n . A R,n is isomorphic to and identified with R ⊕ R .We note that H i ( W f ′ ; R ) is isomorphic to H i ( W f ; R ) for i = k , H i ( W f ; R ) ⊕ R for i = k = n , H i ( W f ; R ) ⊕ R ⊕ R for i = k = n and H i ( W f ; R ) ⊕ R for i = n − k = n and that we can identify the modules for these cases respecting Definition 7 asexplained.For k = n , consider the product of 1 ∈ R , identified with A R,k , and 1 ∈ R ,identified with A R,n − k and for k = n , consider the product of (1 , ∈ R ⊕ R and(0 , ∈ R ⊕ R , where the modules are identified with A R,k
In the case where the numbers l , and l , are 0, the product vanishes (see alsothe proofs of some propositions and theorems of [7]). In the case where the number l , is 1 and the number l , is 0, the product can be (1 , ∈ R ⊕ R , identified with A R,n . Let us explain this more precisely.We can define cohomology classes regarded as the duals of the homology classesrepresented by {∗ } × S k ⊂ S × S k and {∗ } × S n − k ⊂ S × S n − k before: thevalue of the dual at the original homology class is the identity element 1 ∈ R , thevalue at the remaining class of the two classes is zero, and the values at classesin H i ( W f ; R ) ⊕ { } are zero where natural identifications with H i ( W f ′ ; R ) beforeare considered ( i = k, n − k ). We evaluate the value of the product at the classesrepresented by the n -dimensional polyhedra obtained from the products of two standard spheres in A : more precisely, the classes obtained after the manifolds aredeformed and attached to W f . For the original manifolds S × S n − k and S × S k , theclass represented by S × {∗ ′ } ⊂ S × S n − k can be mapped to the class representedby {∗ ′′ } × S k in the other original manifold deformed and attached to and regardedas a subspace in W f ′ . It also can be mapped to zero if we perform an ATSSoperation in a suitable way. The class represented by S × {∗ ′′′ } ⊂ S × S k can bemapped to zero.We give an additional explanation on this using FIGURE 3 of [8].We consider one point in the pair ( p , p ) of the crossing in c ( S ): take p .We can take D , D , D and D as in Definition 4. Dots denote S × {∗ ′ } ⊂ S × S n − k and S × {∗ ′ } ⊂ S × S k deformed and attached to the Reeb space. Werepresent the original manifold S × S n − k by S × ( D n − k S D n − k ) where D n − k and D n − k are hemispheres and the boundaries are an equator of S n − k . We canperform an ATSS operation so that D and { q } × D n − k ⊂ D × S n − k agree in W f ′ for a suitable point q . We can perform an ATSS operation so that D and { q } × D n − k ⊂ D × S n − k agree and also perform an ATSS operation so that D and { q } × D n − k ⊂ D × S n − k agree in W f ′ for a suitable point q .The argument yields the fact that the pair of the values of the product before atthe classes represented by the two n -dimensional polyhedra or the classes obtainedafter the original manifolds, which are diffeomorphic to products of two spheres, aredeformed and attached to W f , can be (1 , ∈ R ⊕ R . The product is representedas (1 , ∈ R ⊕ R , identified with { } ⊕ R ⊕ R and { } ⊕ A R,n ⊂ H n ( W f ′ ; R ) (underthe suitable identification of H n ( W f ; R ) ⊕ A R,n with H n ( W f ′ ; R )). The productcan also vanish if we perform the operation in another suitable way.For the pair ( l , , l , ) = (0 , k and n − k exchange.For a pair ( l , , l , ) of non-negative integers and arbitrary integers, we can arguesimilarly. Moreover, on the square of each of the k -th and ( n − k )-th cohomologyclasses, we can see the vanishing easily observing the topologies of the Reeb spaces.By the construction via the ATSS operation, we can easily see that last statementholds: H ∗ ( W f ; R ) is regarded as a subalgebra of H ∗ ( W f ′ ; R ).This completes the proof. (cid:3) We introduce a connected sum of two stable fold maps.For integers m > n ≥
1, let M i ( i = 1 ,
2) be a closed and connected manifold ofdimension m and f : M → N be a stable fold map such N − f ( M ) is not emptyand f . , : M → R n be a stable fold map.Let e ∈ R n be the point such that the first component is 1 and that theremaining components are all 0. Let D n , := { x = ( x , · · · , x n ) ∈ R n | || x − e || ≤ } . We consider the canonical projection of a unit sphere S m ⊂ R m +1 to R n defined as the composition of the canonical inclusion with the projection π m +1 ,n (( x , · · · , x n , · · · , x m +1 )) = ( x , · · · , x n ) and the restriction to the preimageof D n , : we denote the restriction by π m,n,SB .We consider a composition of f . with an embedding e : R n → S n and wedenote the resulting map by f . We can take a standard closed disc P i of dimension n such that there exists a pair (Φ i , φ i ) of diffeomorphisms satisfying the relation φ i ◦ f i | f i − ( P i ) = π m,n,SB ◦ Φ i for i = 1 , P i and thatof the right map is D n , ). NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 13
We set N := N and N := S n . We can glue the maps f i | f i − ( N i − Int P i ) : f i − ( N i − Int P i ) → N i − Int P i ( i = 1 ,
2) on the boundaries to obtain a new mapand by composing a diffeomorphism from the new target space to N = N , weobtain a smooth map into N = N so that the resulting domain is represented asa connected sum of the original manifolds M and M . The resulting fold mapis a connected sum of f and f . . For N := R n , a connected sum is essentiallyequivalent to one presented in [6], [7], and so on.Proposition 2 yields the following main theorem. Theorem . Let R be a PID having an identity element 1 ∈ R satisfying 1 = 0 ∈ R .Let f : M → N be a stable fold map on an m -dimensional closed and connectedmanifold into an n -dimensional manifold with no boundary satisfying m − n > N be not closed.Let k > k ≤ n . Let l i ∈ R be an element representedas l i, ∈ Z times the identity element 1 for i = 1 ,
2. Let A R be a graded commutativealgebra over R isomorphic to the cohomology ring whose coefficient ring is R of abouquet of a finite number of closed and connected manifolds whose dimensions aresmaller than n and which can be embedded into R n .By a connected sum of f and a special generic map into R n such that therestriction to the singular set is an embedding, we can construct a stable fold map f . By an ATSS operation to f , we have a new fold map f ′ : M ′ → N satisfyingthe following properties.(1) The cohomology ring H ∗ ( W f ′ ; R ) is isomorphic to and identified with agraded commutative algebra obtained canonically from the direct sum of H ∗ ( W f ; R ) and a graded commutative algebra B R over R : we denote the i -th module of B R by B R,i and we abuse this identification in explainingthe products and so on.(2) A R is regarded as a subalgebra of B R . We denote the i -th module of A R by A R,i and as a module over R , B R,i is represented as a direct sum of A R,i and a suitable module C R,i over R : we identify B R,i and A R,i ⊕ C R,i via asuitable isomorphism between modules over R in the remaining properties.(3) C R,i is zero for i = k, n − k , R ⊕ R for i = k = n and i = n , R for i = k, n − k where k = n .(4) For k = n , the product of (0 , , ∈ A R,k ⊕ C R,k = A R,k ⊕ R ⊕ R and(0 , , ∈ A R,k ⊕ C R,k = A R,k ⊕ R ⊕ R is (0 , l , l ) ∈ A R,n ⊕ C R,n = A R,n ⊕ R ⊕ R and the squares of (0 , , ∈ A R,k ⊕ C R,k and (0 , , ∈ A R,k ⊕ C R,k vanish. For k = n , the product of (0 , ∈ A R,k ⊕ C R,k = A R,k ⊕ R and (0 , ∈ A R,n − k ⊕ C R,n − k = A R,n − k ⊕ R is (0 , l , l ) ∈ A R,n ⊕ C R,n = A R,n ⊕ R ⊕ R . Here the square of each of these two elements (0 ,
1) vanishes.(5) For k = n , the product of (0 , , ∈ A R,k ⊕ C R,k = A R,k ⊕ R ⊕ R and anyelement ( a, ∈ A R,i ⊕ C R,i for any i vanishes. For k = n , the product of(0 , ∈ A R,k ⊕ C R,k = A R,k ⊕ R and any element ( a, ∈ A R,i ⊕ C R,i forany i vanishes.Furthermore, in addition, let A R,k be not zero and let a k ∈ A R,k be a non-zeroelement such that for any element r ∈ R which is not a unit, we cannot representas a k = ra k ′ for an element a k ′ ∈ A R,k , that ra k = 0 if and only if r ∈ R iszero, and that the homology class whose dual is a k is represented by a standardsphere of dimension k embedded in the interior of the smooth manifold represented as a regular neighborhood of bouquet of a finite number of closed and connectedmanifolds in the beginning.Let l ∈ R be represented as l ∈ Z times the identity element 1 ∈ R . We canconstruct the map satisfying either of the following property in addition to the fiveproperties before.(1) For k = n , the product of ( a k , , ∈ A R,k ⊕ C R,k = A R,k ⊕ R ⊕ R and (0 , , ∈ A R,k ⊕ C R,k = A R,k ⊕ R ⊕ R is (0 , , l ) ∈ A R,n ⊕ C R,n = A R,n ⊕ R ⊕ R .(2) For k = n , the product of ( a k , ∈ A R,k ⊕ C R,k = A R,k ⊕ R and (0 , ∈ A R,n − k ⊕ C R,n − k = A R,n − k ⊕ R is (0 , , l ) ∈ A R,n ⊕ C R,n = A R,n ⊕ R ⊕ R . Proof.
We have an n -dimensional compact manifold embedded into R n whose coho-mology ring is isomorphic to A R by the assumption. Example 1 (1) yields a specialgeneric map into R n whose Reeb space is diffeomorphic to the n -dimensional mani-fold. We can obtain the special generic map such that the restriction to the singularset is an embedding and that the two kinds of the presented bundles in Example 1(1) are trivial.We consider a connected sum of f and this special generic map. The resultingmap is f . H ∗ ( W f ; R ) is a graded commutative algebra obtained canonically fromthe direct sum of H ∗ ( W f ; R ) and a graded algebra A R over R :The key ingredient is the proof of Proposition 2. This yields the first five prop-erties. We explain the proof of the remaining property. a k is, by the definition andconstruction of special generic map f , regarded as the dual of a homology classrepresented by a standard k -dimensional sphere S k embedded in the regular valueset of f and f | f − ( S k ) : f − ( S k ) → S k gives a trivial S m − n -bundle. We canrepresent l times the class a k by a standard k -dimensional sphere. In the proof ofProposition 2, we take { ( S , N ( S ) , c : N ( S ) → P ) , ( S , N ( S ) , c : N ( S ) → P ) } so that c ( S ) is the k -dimensional standard sphere. We can see that we can obtainthe map satisfying the last two properties on products of the cohomology ring byobserving the topologies of the resulting Reeb spaces: for more precise discussions,see the proofs of several propositions and theorems in [7]. (cid:3) Example 2 . The class of
CPS manifolds are characterized as the minimal classof manifolds satisfying the following conditions (see also [7], in which the authorintroduced the class of these manifolds first).(1) A standard sphere whose dimension is positive is CPS.(2) A product of two CPS manifolds is CPS.(3) A manifold represented as a connected sum of two CPS manifolds is CPS.One of good properties of CPS manifolds is that we can embed these manifoldsinto Euclidean spaces as codimension 1 closed submanifolds. This is introducedas a reasonable class for obtaining special generic maps as in the beginning of theproof of Theorem 1 and Example 1 (1). This class also contributes to producingvarious explicit situations explaining Theorem 1 and several theorems previouslyobtained by the author well.In [7], in situations similar to that of Theorem 1, ATSS operations are studiedunder the condition that graded commutative algebras playing roles as A R playsare isomorphic to some integral cohomology rings of bouquets of finite numbersof CPS manifolds and additional conditions that the immersions of the standardspheres are embeddings and that the images are disjoint: the numbers of embedded NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 15 standard spheres are general. Let f be a canonical projection of a unit sphere ofdimension m > R and A R be an algebra over R := Z isomorphic to theintegral cohomology ring of S × S . Consider a case where ( S , S ) = ( S , S ) and( l , l , , l , ) is a general triplet of integers. The fundamental group of the resultingReeb space vanishes. In general, an integral cohomology ring isomorphic to thatof this resulting Reeb space cannot be obtained as the integral cohomology ring ofanother resulting Reeb space if we consider cases such that the immersions of thestandard spheres are embeddings and that the images are disjoint only in Theorem1. We see about this.First consider a case where the bouquet of CPS manifolds is represented as S × S satisfying the conditions that the immersions of standard spheres areembeddings, that the number of the immersions is two, that the two immersedspheres are S and S , respectively, and that the images are disjoint. For the re-sulting Reeb space in this case, the 2nd integral cohomology group is isomorphic to R ⊕ R ⊕ R = Z ⊕ Z ⊕ Z . The rank of the submodule consisting of all 2nd cohomol-ogy classes such that the products with 2nd cohomology classes are always zero isexactly 1. For such a class, for any 3rd integral cohomology class, the product mustvanish. Note that the integral homology group of the resulting Reeb space W f ′ inthis case is isomorphic to that of the Reeb space before, and that the fundamentalgroup of the space vanishes. The rank of the image of the homomorphism definedby taking the product of an element of H ( W f ′ ; Q ) and an element of H ( W f ′ ; Q )is at most 1. In general, in the situation of Theorem 1 as before, the rank may be2. Let the target space be R . To obtain Reeb spaces whose integral homologygroups are isomorphic to the integral homology groups of the Reeb spaces before,and whose fundamental groups vanish, such that the integral cohomology ring of S × S is regarded as subalgebras of the integral cohomology rings of the Reebspaces, A R must be a graded commutative algebra over R := Z isomorphic to theintegral cohomology ring of a bouquet of S × S and S or that of S × S and S : we do not consider the previous case. In the former case, for the family ofthe embeddings of standard spheres, the number of the embeddings are two andthe spheres are a point and a standard sphere of dimension 3, respectively. In thelatter case, for the family of embeddings of standard spheres, the number of theembeddings are two and the spheres are a point and a standard sphere of dimension2, respectively. For each case, the rank of the image of the homomorphism definedby taking the product of an element of H ( W f ′ ; Q ) and an element of H ( W f ′ ; Q )is at most 1. In general, in the situation of Theorem 1 as before, the rank may be2. We have the following general fact respecting some of Example 2. Theorem . In the situation of Theorem 1, let R := Z and let 2 k < n . Assumethat A R,n − k is zero and that the immersions of S k and S n − k , respectively, areembeddings. In this situation, consider about the graded commutative algebra B R .The rank of the image of the homomorphism defined by taking the product of anelement of B R,k ⊗ Q and an element of B R,n − k ⊗ Q is at most 1.Last we introduce a proposition stating that the Reeb space of a stable fold mapsatisfying several properties on indices of singular points and preimages know muchabout homology groups and the cohomology ring of the manifold of the domain. Proposition . For a stable fold map f : M → N from an m -dimensional closed, connected and orientable manifold M into an n -dimensional manifold N with no boundary such that the following properties hold.(1) m − n > f to the singular set S ( f ), whichis an immersion, the preimages consist of exactly two points.In this situation, we have the following two properties for a commutative group A .(1) Three induced homomorphisms q f ∗ : π j ( M ) → π j ( W f ), q f ∗ : H j ( M ; A ) → H j ( W f ; A ) and q f ∗ : H j ( W f ; A ) → H j ( M ; A ) are isomorphisms for 0 ≤ j ≤ m − n − A is a commutative ring. Let J be the set of all integersgreater than or equal to 0 and smaller than or equal to m − n − ⊕ j ∈ J H j ( W f ; A ) and ⊕ j ∈ J H j ( M ; A ) are algebras such that the sums andthe products are canonically induced from the cohomology rings H ∗ ( W f ; A )and H ∗ ( M ; A ) respectively and that the maximal degrees are m − n − m − n − q f inducesan isomorphism between the algebras ⊕ j ∈ J H j ( W f ; A ) and ⊕ j ∈ J H j ( M ; A )and the isomorphism is given by the restriction of q f ∗ to ⊕ j ∈ J H j ( W f ; A ).(3) Suppose that A is a principal ideal domain and that m = 2 n holds. Underthese assumptions, the rank of H n ( M ; A ) is twice the rank of H n ( W f ; A )and in addition if H n − ( W f ; A ) is free, then the H n − ( M ; R ), H n ( M ; A )and H n ( W f ; A ) are also free modules over A .Hereafter, the description is essentially same as that of the last part of [8].The piecewise smooth category is the category such that objects are smooth man-ifolds having canonically defined PL structures and that morphisms are piecewisesmooth maps between these manifolds with these PL structures. This category isknown to be equivalent to the PL category.We give a sketch of the proof of this. We can give more rigorous proofs thisreferring to the discussions in the first three referred articles and [17]. A sketch of the proof.
For each point in the image and an n -dimensional small stan-dard closed disc containing the point as its neighborhood, each connected compo-nent of the preimage is either of the following types as smooth manifolds whichmay have corners, • A product of an ( m − n )-dimensional standard sphere and an n -dimensionalstandard closed disc. • A product of an manifold obtained by removing the interior of the disjointunion of three disjointly and smoothly embedded standard closed discs in S m − n +1 and an ( n − • A product of an manifold obtained by removing the interior of the disjointunion of four disjointly and smoothly embedded standard closed discs in S m − n +1 , a closed interval I and an ( n − NCREASING TWO CONNECTED COMPONENTS OF SINGULAR SETS OF FOLD MAPS 17 points in the Reeb space of each of the three types form manifolds whose dimensionsare n , n −
1, and n −
2, respectively.For each case, we can construct bundles whose fibers are as above in the piecewiselinear category and we can construct bundles whose fibers are D m − n +1 , D m − n +2 , or D m − n +2 × I in the category whose subbundles obtained by restricting the fibers tosuitable compact submanifolds of the boundaries of the discs D m − n +1 or D m − n +2 are the original bundles: note that the dimensions of the suitable compact subman-ifolds are same as those of the boundaries and that in the last case first we restrict D m − n +2 × I to ∂D m − n +2 × I = S m − n +1 × I . We can locally construct these bundlesand glue them in the piecewise smooth category or PL category. As a result, wehave a desired ( m + 1)-dimensional compact PL manifold collapsing to W f . W f isan n -dimensional polyhedron. In the PL category, the resulting ( m +1)-dimensionalmanifold is a PL manifold obtained by attaching handles whose indices are largerthan or equal to m − n to M × { } ⊂ M × [0 , (cid:3) We can apply this to explicit fold maps obtained in Theorem 1 (Proposition 2)in suitable situations.
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