Closed manifolds admitting no special generic maps whose codimensions are negative and their cohomology rings
aa r X i v : . [ m a t h . A T ] S e p CLOSED MANIFOLDS ADMITTING NO SPECIAL GENERICMAP WHOSE CODIMENSION IS NEGATIVE AND THEIRCOHOMOLOGY RINGS
NAOKI KITAZAWA
Abstract.
Special generic maps are higher dimensional versions of Morsefunctions with exactly two singular points, characterizing spheres topologicallyexcept 4-dimensional cases: in these cases standard spheres are characterized.Canonical projections of unit spheres are special generic. In suitable cases, it iseasy to construct special generic maps on manifolds represented as connectedsums of products of spheres for example. It is an interesting fact that thesemaps restrict the topologies and the differentiable structures admitting themstrictly in various cases. For example, exotic spheres, which are not diffeomor-phic to standard spheres, admit no special generic map into some Euclideanspaces in considerable cases.In general, it is difficult to find (families of) manifolds admitting no suchmaps of suitable classes. The present paper concerns a new result on this workwhere key objects are products of cohomology classes of the manifolds. We cansee that closed symplectic maifolds, real projective spaces, and so on, admitno special generic map into a connected open manifold in considerable casesfor example. Introduction.
Special generic maps are higher dimensional versions of Morse functions with ex-actly two singular points, characterizing spheres topologically except 4-dimensionalcases: in these cases standard spheres are characterized as smooth manifolds. Theyhave been attractive objects in understanding topologies and differentiable struc-tures of manifolds in geometric ways, which is a fundamental and important studyin geometry of manifolds.1.1.
Notation on differentiable maps and bundles.
Throughout the presentpaper, manifolds and maps between manifolds are smooth (of class C ∞ ). Diffeo-morphisms on manifolds are always assumed to be smooth. The diffeomorphismgroup of a manifold is the group of all diffeomorphisms on the manifold. The struc-ture groups of bundles whose fibers are manifolds are assumed to be subgroups ofthe diffeomorphism 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 aunit disc in a Euclidean space and whose structure group acts linearly in a canonicalway on the fiber.A singular point p ∈ X of a smooth map c : X → Y is a point at which therank of the differential dc is smaller than both the dimensions dim X and dim Y . Key words and phrases.
Singularities of differentiable maps; fold maps and special genericmaps. Cohomology classes. Higher dimensional closed and simply-connected manifolds.2020
Mathematics Subject Classification : Primary 57R45. Secondary 57R19.
We call the set S ( c ) of all singular points the singular set of c . We call c ( S ( c )) the singular value 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 the map.1.2. The definition of a special generic map, the topologies and the differ-entiable structures of manifolds admitting special generic maps, mean-ings in algebraic topology and differential topology of manifolds and themain theorem.
Let m > n ≥ m -dimensionalsmooth manifold with no boundary into an n -dimensional smooth manifold withno boundary is said to be a special generic map if at each singular point p , the mapis represented as ( x , · · · , x m ) ( x , · · · , x n − , m X k = n x k )for suitable coordinates. The restriction map to the singular set is an immersion.We can define for the cases m = n . However, we concentrate on cases where m > n hold.We can define fold maps as higher dimensional versions of Morse functions sim-ilarly. See [7] for introductory and systematic explanations for example.An exotic homotopy sphere is a homotopy sphere which is not diffeomorphic toany standard sphere. [19] is the first article on exotic homotopy spheres and [10]and so on are also on these homotopy spheres.Canonical projections of unit spheres are special generic. Every homotopy sphereof dimension > m > n ≥
1, on an m -dimensional manifold represented as aconnected sum of manifolds represented as products of two standard spheres suchthat the dimension of either of the two sphere for each manifold is smaller than n ,we can obtain a special generic map into R n (we will present in Example 1). Weintroduce interesting facts from the viewpoint of differential topology in [22], [23],[24], [25] and [28] for example. As studies before these recent ones, [2], [3], [6] andso on, and [26], [27], and so on, are also related important studies. Theorem . Exotic homotopy spheres of dimension m > R m − , R m − and R m − . Theorem . R .[5] is, for example, on classical theory of 7-dimensional homotopy spheres otherthan [19]. Theorem . Closed and connected manifolds represented as connected sumsof the following manifolds are characterized as manifolds admitting special genericmaps into the plane.(1) A homotopy sphere of dimension > ANIFOLDS ADMITTING NO SPECIAL GENERIC MAP AND THEIR COHOMOLOGIES 3
Theorem . For an integer 3 ≤ m ≤
5, closed manifolds representedas connected sums of the following manifolds are characterized as closed manifoldshaving free fundamental groups and admitting special generic maps into R .(1) The standard sphere of dimension m .(2) The total space of a bundle whose fiber is diffeomorphic to S m − over S .(3) The total space of a bundle whose fiber is diffeomorphic to S m − over S .For an integer m = 6 ,
7, closed and simply-connected manifolds admitting specialgeneric maps into R are represented as connected sums of the following manifolds.(1) Homotopy spheres admitting special generic maps into R .(2) The total space of a bundle whose fiber is diffeomorphic to a homotopysphere of dimension m − S .For this, see also [25]. As another study, in [20], Nishioka completely solvedthe so-called existence problem of special generic maps into Euclidean spaces on 5-dimensional closed and simply-connected manifolds, which are completely classifiedin [1]. He has shown that such manifolds admitting special generic maps into R n for some integer 1 ≤ n ≤ S over S . Notealso that in [11] and [12], it is shown that round fold maps of suitable classes restrictthe topologies and the differentiable structures of manifolds admitting these maps:a round fold map is a fold map such that the restriction to the singular set is anembedding and that the singular value set is concentric spheres, introduced andsystematically studied by the author in [11], [12] and [13] for example.In general, it is difficult to find (families of) manifolds admitting no specialgeneric maps of suitable classes in general. As a related small proposition, theauthor has proved in [14], [15] and so on first, for example, that the product ofgiven suitable two cohomology classes of a manifold admitting a special genericmap into the Euclidean space of a fixed dimension vanishes.The present paper concerns a kind of advanced and systematic studies on coho-mology rings of manifolds admitting special generic maps and ones admitting nospecial generic map. Main Theorem . Let M be a closed manifold of dimension m > ≤ n < m and k > A be a principal ideal domain (having a unique identityelement which is not the zero element). If there exists a sequence { a j } kj =1 ⊂ H ∗ ( M ; A ) of cohomology classes such that the product Q kj =1 a k does not vanish,that the degree of each class is smaller than or equal to m − n and that the sum ofthe degrees are greater than or equal to n , then M admits no special generic mapinto a connected open manifold N of dimension n .For example we have the following corollary. Main Corollary . Let m > m -dimensioal manifold whosecohomology ring is isomorphic to that of the m -dimensional real projective spaceadmits no special generic map into a connected open manifold N of dimension n for 1 ≤ n ≤ m − Z / Z .The definition of a c-symplectic manifold is explained in the last section and wepresent another corollary. NAOKI KITAZAWA
Main Corollary . A closed c-symplectic manifold M of dimension m = 2 k > N of dimension n < k − triple Massey product for a triplet of cohomology classes of a manifold admittinga special generic map of a suitable class for which we can define the triple Masseyproduct. However, we do not discuss this in the present paper. See [16] for Masseyproducts for example.Another main theorem is as the following. We present undefined several termi-nologies and notions in the last section. This is a theorem on Euclidean spaces intowhich given manifolds admit speciaql generic maps.
Main Theorem . Let A be a principal ideal domain (having a unique identityelement which is not the zero element).(1) Let M and M be closed and connected manifolds of dimension m >
2. Let n be an integer larger than 1 and smaller than m . If M and M satisfythe conditions Sp ≥ n and CohP A,m,n − , then a manifold represented as aconnected sum of these manifolds also does.(2) Let M ′ be a closed and connected manifold of dimension m ′ > n ′ larger than 1 and smaller than m ′ , let M ′ satisfy the conditionsSp ≥ n ′ and CohP A,m ′ ,n ′ − . We also assume that the homology group of M ′ whose coefficient is A is free. Let n be a positive integer larger than n ′ and F be a closed and connected manifold of dimension n − n ′ satisfyingthe following properties.(a) There exist a positive integer k and a sequence { a j } kj =1 ⊂ H ∗ ( F ; A )of cohomology classes such that the product Q kj =1 a k does not vanish,that the degree of each class is smaller than or equal to m ′ − n ′ + 1and that the sum of the degrees is equal to n − n ′ .(b) The homology group of F whose coefficient is A is free.(c) F can be immersed into R n so that the normal bundle is trivial.In this situation, M ′ × F satisfies the conditions CohP A,m ′ + n − n ′ ,n − and Sp ≥ n .1.3. The content of the present paper.
The organization of the paper is asthe following. We review structures of special generic maps. In explaining them,we use the
Reeb space of a special generic map, which is defined as the space of allconnected components of preimages. This is regarded as a compact manifold whosedimension is equal to that of the target space and which we can immerse into thetarget space. We also present Example 1 as simplest examples of special genericmaps. The last section is devoted to the main ingredient including Main Theorems.2.
Structures of special generic maps and a simplest example.
For a continuous map c : X → Y between topological spaces, we can define anequivalence relation ∼ c on X by the following rule: p ∼ c p if and only if p and p are in a same connected component of a preimage c − ( q ) ( q ∈ Y ). We call thequotient space W c := X/ ∼ c the Reeb space of c . We denote the quotient map by q c : X → W c . We can define in a unique way a map ¯ c satisfying f = ¯ c ◦ q c . ANIFOLDS ADMITTING NO SPECIAL GENERIC MAP AND THEIR COHOMOLOGIES 5
See also [21] (as a classical and important study) for example. In general Reebspaces are fundamental and strong tools in investigating manifolds via Morse func-tions, fold maps and more general good smooth maps.
Proposition . (1) The Reeb space of a special generic map f : M → N from a closed and connected manifold M of dimension m into an n -dimensional manifold N with no boundary satisfying m > n isa smooth manifold of dimension n immersed into N via ¯ f : W f → N and q f ( S ( f )) = ∂W f holds. Furthermore, around the boundary ∂W f , thecomposition of the restriction of q f to the preimage of a small collar neigh-borhood with a canonical projection to the boundary ∂W f gives a linearbundle whose fiber is diffeomorphic to D m − n +1 . Moreover, the composi-tion of the restriction of q f to the preimage of the complementary set of theinterior of the small collar neighborhood gives a smooth bundle whose fiberis diffeomorphic to S m − n : the bundle is linear in the case m − n = 1 , , W bounded by M and collapsing to W f and inthe case m − n = 1 , , W as a smooth manifold.(2) For any smooth immersion ¯ f N of a compact and connected manifold ¯ N of dimension n > n -dimensional manifold N with no boundaryand any integer m > n , there exists a closed and connected manifold M ofdimension m and a special generic map f : M → N satisfying W f = ¯ N and¯ f = ¯ f N . If N is orientable, then M can be taken as an orientable manifold.Example 1 . Let M be a closed manifold of dimension m > { S k j × S m − k j } of finitely many manifoldssatisfying 1 ≤ k j < n where 1 < n < m holds. This admits a special generic mapinto R n such that the restriction to the singular set is an embedding and that theimage is represented as a boundary connected sum of all manifolds in the family { S k j × D n − k j } .3. The main theorems, their proofs and applications.
The following proposition or essentially (almost) equivalent ones are shown in[11], [12], [13], [14], [15], [22] and so on.
Proposition . In the situation of Proposition 1 (1) (and (2)), assume that N is connected open and let us denote the inclusion map by i : M → W . For asuitable PL or piecewise smooth map giving a collapsing r : W → W f , q f = r ◦ i holds. Let A be a commutative group. Here, the induced morphisms i ∗ : H j ( M ; A ) → H j ( W f ; A ), i ∗ : H j ( W f ; A ) → H j ( M ; A ) and i ∗ : π j ( M ) → π j ( W f )are isomorphisms for 0 ≤ j ≤ m − n .We explain only the main ingredient of the proof of the statement on the iso-morphisms. The main ingredient of the proof of the statement on the isomorphisms. W f collapsesto an ( n − W is obtained by attaching handles whoseindices are larger than m − m + 1 to M × { } ⊂ M × [0 , M and M × { } via the map i M ( x ) := ( x,
0) and M × [0 ,
1] is regarded as a small collarneighborhood (in the category where we discuss). (cid:3)
NAOKI KITAZAWA
Theorem . Let M be a closed manifold of dimension m > ≤ n < m and k > A be a principal ideal domain (having a unique identity elementwhich is not the zero element). If there exists a sequence { a j } kj =1 ⊂ H ∗ ( M ; A ) ofcohomology classes such that the product Q kj =1 a k does not vanish, that the degreeof each class is smaller than or equal to m − n and that the sum of the degrees isgreater than or equal to n , then M admits no special generic map into a connectedopen manifold N of dimension n . Proof.
Suppose that M admits a special generic map f : M → N . We abusenotation in Proposition 2 and apply this proposition. There exists a unique se-quence { b j } kj =1 ⊂ H ∗ ( W ; A ) of cohomology classes satisfying a j = i ∗ ( b j ). Wehave Q kj =1 a k = Q kj =1 i ∗ ( b k ) = i ∗ ( Q kj =1 b k ) and this is zero since W collapses to an( n − (cid:3) We end the present paper by presenting important examples related to thepresent problem.Example 2 . Let m > m -dimensional torus adnits no specialgeneric map into a connected open manifold N of dimension n for 1 ≤ n ≤ m − . By virtue of known algebraic topological and differential topologicaltheory of special generic maps in the introduction and the previous sections, Propo-sition 2, and so on, S × S × S × S admits no special generic map into R . If itadmits one, then the fundamental group of the Reeb space, which is diffeomorphicto a 3-dimensional compact, connected and orientable manifold by Proposition 1,and that of S × S × S × S agree: however, this group is known to be neverisomorphic to any fundamental group of any 3-dimensional compact, connected andorientable manifold. Theorem 5 produces another explanation on this.A closed manifold X of dimension 2 k where k > c-symplectic if there exists a cohomology class c s ∈ H ( X ; R ) such that c sk is notzero. See also [9], [17], and so on. Corollary . A closed c-symplectic manifold M of dimension m = 2 k > N of dimension n < k − S × S isregarded as a symplectic manifold and admits a special generic map into R . Onthe other hand, the 2 k -dimensional torus is regarded as a symplectic manifold andadmits no special generic map into a connected open manifold N of dimension n for 1 ≤ n ≤ k − Corollary . Let m > m -dimensioal manifold M whose co-homology ring is isomorphic to that of the m -dimensional real projective spaceadmits no special generic map into a connected open manifold N of dimension n for 1 ≤ n ≤ m − Z / Z . ANIFOLDS ADMITTING NO SPECIAL GENERIC MAP AND THEIR COHOMOLOGIES 7
Proof.
There exists a cohomology class u ∈ H ( M ; Z / Z ) such that u m is notzero. Theorem 5 completes the proof. (cid:3) A weaker result with its proof in a different way is announced in [29]. This is onmanifolds homotopy equivalent to the 7-dimensional real projectiive plane.Definition 1 . Let M be a closed manifold of dimension m >
1. Let n be a positiveinteger smaller than m . M is said to satisfy the condition Sp ≥ n if M admits aspecial generic map into R n for any integer n o ≤ n < m .Example 3 . M in Example 1 satisfies the condition Sp ≥ min { k j +1 } j ∈ J where J de-notes a finite set and is not empty in the situation of Example 1.The following notion is introduced motivated by Theorem 5.Definition 2 . Let X be a closed and connected manifold of dimension dim X > n be a positive integer smaller than dim X . Let A be a principal ideal domain(having a unique identity element which is not the zero element). If there exist apositive integer k > { a j } kj =1 ⊂ H ∗ ( X ; A ) of cohomology classessuch that the product Q kj =1 a k does not vanish, that the degree of each class issmaller than or equal to dim X − n and that the sum of the degrees are greaterthan or equal to n , then X is said to satisfy the condition CohP A, dim X,n . Theorem . Let A be a principal ideal domain (having a unique identity elementwhich is not the zero element).(1) Let M and M be closed and connected manifolds of dimension m >
2. Let n be an integer larger than 1 and smaller than m . If M and M satisfythe conditions Sp ≥ n and CohP A,m,n − , then a manifold represented as aconnected sum of these manifolds also does.(2) Let M ′ be a closed and connected manifold of dimension m ′ > n ′ larger than 1 and smaller than m ′ , let M ′ satisfy the conditionsSp ≥ n ′ and CohP A,m ′ ,n ′ − . We also assume that the homology group of M ′ whose coefficient is A is free. Let n be a positive integer larger than n ′ and F be a closed and connected manifold of dimension n − n ′ satisfyingthe following properties.(a) There exist a positive integer k and a sequence { a j } kj =1 ⊂ H ∗ ( F ; A )of cohomology classes such that the product Q kj =1 a k does not vanish,that the degree of each class is smaller than or equal to m ′ − n ′ + 1and that the sum of the degrees is equal to n − n ′ .(b) The homology group of F whose coefficient is A is free.(c) F can be immersed into R n so that the normal bundle is trivial.In this situation, M ′ × F satisfies the conditions CohP A,m ′ + n − n ′ ,n − and Sp ≥ n . Proof.
We prove the the first statement. The condition CohP
A,m,n − follows im-mediately. We can construct a special generic map on a closed manifold representedas a connected sum of closed and connected manifolds admitting special genericmaps into R n easily (for any n satisfying 1 ≤ n ≤ m ). This is a fundamentalargument in [22] and so on. This completes the proof of the first statement.We show the second statement. By choosing suitable sequences of cohomologyclasses of M ′ and F , we can obtain a sequence of cohomology classes of M ′ × F of NAOKI KITAZAWA a finite length where the coefficient ring is A such that the degree of each class issmaller than or equal to m ′ − n ′ + 1 and that the sum of the degrees are greaterthan or equal to n − n ′ + n ′ − n −
1. This completes the proof on thecondition CohP
A,m ′ + n − n ′ ,n − . D n ′ + a × F can be immersed into R n + a by virtue of the last condition for anynon-negative integer a . M ′ satisfies the condition Sp ≥ n ′ . So we can take a specialgeneric map f a on M ′ into R n ′ + a for 0 ≤ a < m ′ − n ′ . By considering a productmap of such a special generic map (composed with an embedding into the interiorof a copy of the disc D n ′ + a ) and the identity map on F and compose the immersionof D n ′ + a × F gives a special generic map into R n ′ + a + n − n ′ = R n + a .This completes the proof. (cid:3) For example, we can apply Theorem 6 to connected sums and products of closedand connected manifolds inductively starting from standard spheres respecting con-ditions on dimensions of the closed and connected manifolds and the Euclideanspaces, and so on. 4.
Acknowledgement.
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