Standard Special Generic Maps of Homotopy Spheres into Euclidean Spaces
aa r X i v : . [ m a t h . G T ] A ug STANDARD SPECIAL GENERIC MAPS OF HOMOTOPYSPHERES INTO EUCLIDEAN SPACES
DOMINIK J. WRAZIDLO
Abstract.
A so-called special generic map is by definition a map of smoothmanifolds all of whose singularities are definite fold points. It is in general anopen problem posed by Saeki in 1993 to determine the set of integers p forwhich a given homotopy sphere admits a special generic map into R p .By means of the technique of Stein factorization we introduce and studycertain special generic maps of homotopy spheres into Euclidean spaces called standard . Modifying a construction due to Weiss, we show that standardspecial generic maps give naturally rise to a filtration of the group of homotopyspheres by subgroups that is strongly related to the Gromoll filtration. Finally,we apply our result to concrete homotopy spheres, which particularly answersSaeki’s problem for the Milnor 7-sphere. Introduction
A smooth map f between smooth manifolds is traditionally called a specialgeneric map if every singular point x of f is a definite fold point, i.e., f looksin suitable charts around x and f ( x ) like the multiple suspension of a positivedefinite quadratic form (see Section 2.1).For a closed smooth manifold M n of dimension n , let S ( M n ) denote the set of allintegers p ∈ { , . . . , n } for which there exists a special generic map M n → R p . Notethat S is a diffeomorphism invariant of smooth manifolds. In 1993, the followingproblem was posed by Saeki in [11, Problem 5.3, p. 177] (see also [12]). Problem 1.1.
Study the set S ( M n ).For orientable M n , Eliashberg [4] showed that n ∈ S ( M n ) if and only if M n isstably parallelizable. We are concerned with the case that 1 ∈ S ( M n ), i.e., the casethat M n admits a special generic map into R . Such a map is usually referred to as special generic function , and is nothing but a Morse function all of whose criticalpoints are extrema. If M n admits a special generic function, then every componentof M n is homeomorphic to S n by a well-known theorem of Reeb [9], and is for n ≤ S n . If Σ n denotes an exotic sphere of dimension n ≥
7, then according to [11, (5.3.4), p. 177] (compare Remark 5.5) we have { , , n } ⊂ S (Σ n ) ⊂ { , , . . . , n − , n } . (1.1) Date : September 20, 2018.2010
Mathematics Subject Classification.
Primary 57R45; Secondary 57R60, 58K15.
Key words and phrases.
Special generic map; Stein factorization; homotopy sphere; Gromollfiltration.This work was supported by a scholarship for doctoral research awarded by the German Na-tional Merit Foundation.
Special generic functions can also prove useful in the study of individual homo-topy spheres. Indeed, a homotopy sphere is said to have
Morse perfection ≥ p (seeSection 2.3) if it admits a family of special generic functions smoothly parametrizedby points of the unit p -sphere that is subject to an additional symmetry condition.In [13] it is shown that the notion of Morse perfection is related as follows to thecelebrated Gromoll filtration (see Section 2.5) of a homotopy sphere.
Theorem 1.2. If Σ n is a homotopy sphere of dimension n ≥ , then ( Gromoll filtration of Σ n ) − ≤ Morse perfection of Σ n . As pointed out in [13], it is possible by means of algebraic K -theory to deriveupper bounds for the Morse perfection of certain homotopy spheres in terms of thesignature of parallelizable null-cobordisms. Consequently, Theorem 1.2 allows todraw conclusions on the Gromoll filtration of concrete homotopy spheres such asMilnor spheres (compare Proposition 5.1( ii )). From the point of view of differen-tial geometry the upper bounds for the Morse perfection imply (see [13, p. 388])that certain homotopy spheres do not admit a Riemannian metric with sectionalcurvature “pinched” in the interval (1 / , standard special generic maps (see Definition 3.3)by which we roughly mean special generic maps from homotopy spheres into Eu-clidean spaces that factorize nicely over the closed unit ball (the required techniqueof Stein factorization is recapitulated in Section 2.2). According to Corollary 3.8a homotopy sphere that admits a standard special generic map into R p will alsoadmit such maps into R , . . . , R p − . The fold perfection of a homotopy sphere Σ n (see Definition 3.9) is defined to be the greatest integer p for which a standardspecial generic map Σ n → R p exists.In refinement of Theorem 1.2 our main result is the following Theorem 1.3. If Σ n is a homotopy sphere of dimension n ≥ , then ( i ) Gromoll filtration of Σ n ≤ fold perfection of Σ n , and ( ii ) ( fold perfection of Σ n ) − ≤ Morse perfection of Σ n . Compared to Morse perfection the notion of fold perfection has the naturalalgebraic advantage that it gives rise to a filtration of the group of homotopy spheresby sub groups (see Remark 3.12).The paper is organized as follows. In Section 2 we explain in detail all relevantnotions and techniques. Section 3 introduces standard special generic maps andstudies some of their properties. The proof of Theorem 1.3, which will be givenin Section 4, is based on a modification of the original proof of Theorem 1.2. Fi-nally, in Section 5, we discuss various applications of Theorem 1.3. Proposition 5.1exploits known results about the Gromoll filtration and the Morse perfection ofconcrete exotic spheres such as Milnor spheres to extract information about theirfold perfection. Furthermore, Corollary 5.2 and Proposition 5.4 focus on the impactto Problem 1.1, which includes an answer to Problem 1.1 for the Milnor 7-sphere.
Notation.
In the following M n will always denote a connected closed smoothmanifold of dimension n ≥
1. The symbol ∼ = will either mean diffeomorphism ofsmooth manifolds or isomorphism of groups. The singular locus of a smooth map f between smooth manifolds will be denoted by S ( f ). Euclidean p -space R p is TANDARD SPECIAL GENERIC MAPS OF HOMOTOPY SPHERES 3 always equipped with the Euclidean inner product ( u, v ) u · v , and || u || := √ u · u denotes the corresponding Euclidean norm. Let D p = { x ∈ R p ; || x || ≤ } denotethe closed unit ball in R p , and S p − := ∂D p the standard ( p − m × m unit matrix will be denoted by I m . For p > p let π p p : R p → R p denotethe projection to the last p coordinates. Acknowledgement.
The author is grateful to Professor Osamu Saeki for pointingout the important application to the Milnor 7-sphere.2.
Preliminaries
The purpose of the present section is to recall the concepts that play a role inthe presentation of Theorem 1.3.2.1.
Special generic maps.
Consider a smooth map f : M n → N p betweensmooth manifolds of dimensions n ≥ p ≥
1. A point x ∈ M is called fold point ifthere exist local coordinates x , . . . , x n around x and y , . . . , y p around y := f ( x )in which f takes the form y i ◦ f = x i , i = 1 , . . . , p − ,y p ◦ f = x p + · · · + x p + λ − − x p + λ − · · · − x n , for a suitable integer λ ∈ { , . . . , n − p + 1 } . The integer max { λ, n − p + 1 − λ } turnsout to be independent of the choice of coordinates, and is called absolute index of f at x . The map f is called fold map if every singular point of f is a fold point. Itis easy to see that the singular locus S ( f ) of a fold map f is a ( p − M n , and that f restricts to an immersion S ( f ) → N p . The absoluteindex is known to be a locally constant function on S ( f ). Definition 2.1.
The smooth map f : M n → N p is called special generic map ifits singular locus S ( f ) consists of definite fold points, i.e., fold points of absoluteindex n − p + 1. Special generic maps into N p = R are referred to as special genericfunctions .If M n admits a special generic function, then M n is homeomorphic to S n by awell-known result of Reeb [9], and for n ≤ S n according tothe proof of [11, (5.3.3), p. 177].We mention the following folkloric method that allows to check in some casesthat a given smooth map is a fold map. Proposition 2.2.
Let n ≥ q ≥ be integers. Let f : U → R be a smooth functiondefined on an open subset U ⊂ R q − × R n − q +1 . Then the smooth map F : U → R q , z = ( x, y ) F ( z ) = ( x, f ( z )) , is a fold map if and only if the Hessian H y ( f x ) := H y ( f | U ∩ ( { x }× R n − q +1 ) ) is non-degenerate for every point z = ( x, y ) ∈ S ( f ) = { z ∈ U | ∂ y f ( z ) = · · · = ∂ y n − q +1 f ( z ) = 0 } . In this case, the absolute index of F at z = ( x, y ) ∈ S ( f ) isgiven by max { λ, n − p + 1 − λ } , where λ denotes the number of negative diagonalentries of H y ( f x ) after diagonalization.Proof. The result can be obtained by combining [14, Proposition 3.3.4, p. 58] and[14, Proposition 4.5.3, p. 103]. (cid:3)
D.J. WRAZIDLO
Example . The standard sphere S n admits a special generic map into R p forall p ∈ { , . . . , n } . In fact, the composition of the inclusion S n ⊂ R n +1 with anyorthogonal projection R n +1 → R p is easily seen to be a special generic map.2.2. Stein factorization.
Special generic maps have first been studied by Burletand de Rham in [1] by means of the technique of Stein factorization, which has eversince played an essential role in the study of special generic maps, see e.g. [10].Let us recall the notion of Stein factorization of an arbitrary continuous map f : X → Y between topological spaces. Define an equivalence relation ∼ f on X asfollows. Two points x , x ∈ X are called equivalent, x ∼ f x , if they are mappedby f to the same point y := f ( x ) = f ( x ) ∈ Y , and lie in the same connectedcomponent of f − ( y ). The quotient map q f : X → X/ ∼ f gives rise to a uniqueset-theoretic factorization of f of the form X YX/ ∼ f fq f f .If we equip W f := X/ ∼ f with the quotient topology induced by the surjectivemap q f : X → W f , then it follows that the maps q f and f are continuous. Then,the above diagram (and sometimes W f itself) is called the Stein factorization of f .For our purposes, the relevant properties of the Stein factorization of a specialgeneric map can be summarized according to [10, p. 267] in the following Proposition 2.4.
Let f : M n → R p be a special generic map where p < n . Then, ( i ) the quotient space W f = M n / ∼ f can be equipped with the structure of asmooth p -dimensional manifold with boundary in such a way that q f : M n → W f is a smooth map satisfying q − f ( ∂W f ) = S ( f ) , and f : W f → R p is animmersion. ( ii ) the quotient map q f restricts to a diffeomorphism S ( f ) ∼ = −→ ∂W f . Morse perfection.
The concept of Morse perfection has been introduced in[13, Definition 0.1, p. 387].
Definition 2.5.
The
Morse perfection of M n is the greatest integer k ≥ − η : S k × M n → R with the following properties:(1) η restricts for every s ∈ S k to a special generic function (see Definition 2.1) η s : M n → R , η s ( x ) = η ( s, x ) . (2) η − s = − η s for all s ∈ S k .Note that every M n has Morse perfection ≥ −
1. (In fact, (1) and (2) are emptyconditions for k = − S − = ∅ by convention.)As pointed out in [13], the Morse perfection of M n is always ≤ n , and thestandard sphere S n has Morse perfection n .2.4. Homotopy spheres.
Let Θ n denote the group of h -cobordism classes of ori-ented homotopy n -spheres as introduced in [7]. It is known that Θ n is trivial for n ≤
6. (For n = 3 this follows from the classical Poincar´e conjecture proven by TANDARD SPECIAL GENERIC MAPS OF HOMOTOPY SPHERES 5
Perelman.) For n ≥ n -sphere is homeomorphic to S n , and theequivalence relation of h -cobordism coincides with that of orientation preservingdiffeomorphism. This interpretation of Θ n will be understood in the following.Consider the group Γ n = π (Diff( D n − , ∂ )) of isotopy classes of orientationpreserving diffeomorphisms of D n − that are the identity near the boundary. For n ≥ n ∼ = −→ Θ n due to Smale and Cerf (see [3]) can explicitly be described as follows. Let ι ± : D n − → S n − denote the embedding x ι ± ( x ) = ( ± p − || x || , x ). A representative g : D n − ∼ = −→ D n − of a given γ ∈ Γ n induces a diffeomorphism ϕ g : S n − ∼ = −→ S n − uniquely determined by requiring that ϕ g ◦ ι − = ι − ◦ g and ϕ g ◦ ι + = ι + . (Inparticular, ϕ g agrees with the identity map in a neighborhood of the hemisphereof S n − with non-negative first component.) Then, Σ( γ ) ∈ Θ n is represented bythe homotopy n -sphere that is obtained by identifying two copies of D n along theirboundaries via ϕ g .2.5. The Gromoll filtration.
Let n ≥ n = π (Diff( D n − , ∂ )) due to Gromoll [5] by subgroups of the form0 = Γ nn − ⊂ · · · ⊂ Γ n = Γ n . By definition, γ ∈ Γ np +1 if γ can be represented by a diffeomorphism g ∈ Diff( D n − , ∂ )such that the following diagram commutes. D n − D n − R p R p g id R p π n − p | D n − π n − p | D n − One says that γ ∈ Γ n (or Σ( γ ) ∈ Θ n ) has Gromoll filtration p if γ ∈ Γ np \ Γ np +1 .3. Fold perfection of homotopy spheres
The following result is due to Saeki [10, Proposition 4.1, p. 274].
Proposition 3.1.
Let f : M n → R p ( p < n ) be a special generic map. Then M n is a homotopy sphere if and only if the Stein factorization W f is contractible. Corollary 3.2. If f : Σ n → R p ( p < n ) is a special generic map on a homotopysphere Σ n , then the singular locus S ( f ) is a homology ( p − -sphere.Proof. Note that q f restricts by Proposition 2.4( ii ) to a diffeomorphism S ( f ) ∼ = ∂W f . Moreover, Proposition 3.1 implies that W f is contractible. Hence, the claimfollows from Poincar´e duality, H ∗ ( W f , ∂W f ; Z ) ∼ = H n −∗ ( W f ; Z ) as well as the longexact sequence for reduced integral homology of the pair ( W f , ∂W f ). (cid:3) However, in the situation of Corollary 3.2, W f is in general not diffeomorphic to D p because S ( f ) ∼ = ∂W f is in general not even a homotopy sphere as remarked in[10, Remark 4.4, p. 275]. Definition 3.3.
A special generic map f : Σ n → R p ( p < n ) on a homotopy sphereΣ n is called standard if the Stein factorization W f is diffeomorphic to D p .It is convenient to clarify in dependence of the dimension p what it means for aspecial generic map to be standard. D.J. WRAZIDLO
Proposition 3.4.
Suppose that f : Σ n → R p ( p < n ) is a special generic map ona homotopy sphere Σ n . Then the following statements hold: ( a ) If p ∈ { , , } , then f is a standard special generic map. ( b ) For ≤ p < n the following statements are equivalent: ( i ) The map f is a standard special generic map. ( ii ) The singular locus S ( f ) is diffeomorphic to S p − . ( iii ) The singular locus S ( f ) is simply connected. ( c ) For p = 5 the equivalence ( i ) ⇔ ( ii ) from part ( b ) is valid.Proof. The claim of part ( a ) follows for p ∈ { , } from the well-known classificationof compact smooth manifolds of dimension p . For p = 3 the contractible compactsmooth 3-manifold W f (see Proposition 3.1) is (according to Corollary 3.2) boundedby a homology 2-sphere ∂W f ∼ = S ( f ), which is necessarily diffeomorphic to S .Hence the claim follows from a standard application of the 3-dimensional version ofthe h -cobordism theorem. (The latter is by [8, p. 113] a consequence of the classicalPoincar´e conjecture proven by Perelman.) Concerning part ( b ), the implication( i ) ⇒ ( ii ) holds because q f restricts by Proposition 2.4( ii ) to a diffeomorphism S ( f ) ∼ = ∂W f , and ∂W f ∼ = S p − . Moreover, ( ii ) ⇒ ( iii ) as p >
2. In order toshow ( iii ) ⇒ ( i ), first note that W f is by Proposition 3.1 a contractible compactsmooth manifold with simply connected boundary S ( f ) ∼ = ∂W f by ( iii ). As W f isof dimension p ≥
6, the claim is a direct consequence of the h -cobordism theorem(see [8, Proposition A, p. 108]). Finally, part ( c ) is covered by [8, Proposition C1), p. 110f]. (cid:3) The following is a sufficient criterion for a special generic map to be standard.
Lemma 3.5.
Let f : M n → R p be a special generic map. If f ( M n ) = D p and f : W f → R p is injective, then M n is a homotopy sphere and f is standard.Proof. Since f ( W f ) = f ( M n ) = D p , it suffices by Proposition 3.1 and Definition 3.3to note that the immersion f (see Proposition 2.4( i )) is an embedding. This is truebecause f is injective by assumption and the domain W f = q f ( M n ) is compact, sothat the immersion f restricts to a homeomorphism onto its image. (cid:3) Example . Example 2.3 and Lemma 3.5 imply that any orthogonal projection R n +1 → R p restricts to a standard special generic map on the standard sphere S n for all p ∈ { , . . . , n − } . Proposition 3.7.
Let f : Σ n → R p be a standard special generic map. Then, forevery u ∈ S p − , the composition of the standard special generic map h : Σ n q f −→ W f ∼ = −→ D p ֒ → R p (where a diffeomorphism W f ∼ = −→ D p has been fixed) with the orthogonal projection π u : R p → R , v u · v, yields a special generic function π u ◦ h : Σ n → R with S ( π ◦ h ) = h − ( {± u } ) ∼ = S .Proof. Proposition 2.4 and Lemma 3.5 imply that h is a standard special genericmap with image h (Σ n ) = D p ⊂ R p and singular locus S ( h ) = h − ( S p − ). Alsonote that h restricts to a diffeomorphism S ( h ) ∼ = −→ S p − . Moreover, π := π u restricts by Example 2.3 to a special generic function S p − → R with singular locus TANDARD SPECIAL GENERIC MAPS OF HOMOTOPY SPHERES 7 S ( π | S p − ) = π − ( S ) = {± u } . Altogether, S ( π ◦ h ) = h − ( {± u } ) ∼ = S . It sufficesto show that both critical points of π ◦ h are non-degenerate.As every c ∈ S ( π ◦ h ) is also critical point of h , there exist by Definition 2.1charts ϕ : U → U ′ ⊂ R n = R p − × R n − p +1 and ψ : V → V ′ ⊂ R p = R p − × R with h ( U ) ⊂ V , c ∈ U , ϕ ( c ) = (0 , h ′ := ψ ◦ h ◦ ϕ − : U ′ → V ′ , h ′ ( x, y ) = ( x, || y || ) . Set π ′ := π ◦ ψ − : V ′ → R . A short computation shows that the Hessian of π ′ ◦ h ′ = π ◦ h ◦ ϕ − at ( x, y ) = (0 ,
0) = ϕ ( c ) ∈ U ′ ⊂ R p − × R n − p +1 is given by ablock matrix of the form H (0 , ( π ′ ◦ h ′ ) = (cid:18) H (0 , ( π ′ ◦ h ′ | U ′ ∩ ( R p − × ) 00 2 · ( ∂ p π ′ )(0 , · I n − p +1 (cid:19) . To establish that the Hessian H (0 , ( π ′ ◦ h ′ | U ′ ∩ ( R p − × ) is non-singular note that π ′ ◦ h ′ | U ′ ∩ ( R p − × = π ◦ ( h ◦ ϕ − ) | U ′ ∩ ( R p − × is the composition of the embedding h ◦ ϕ − | : U ′ ∩ ( R p − ×
0) = S ( h ′ ) = S ( h ◦ ϕ − ) → S p − with the Morse function π | S p − : S p − → R , and that ( h ◦ ϕ − )(0 ,
0) = h ( c ) ∈ {± u } = S ( π | S p − ).It remains to show that 2 · ( ∂ p π ′ )(0 , · I n − p +1 is non-singular, i.e., ( ∂ p π ′ )(0 , =0. For this purpose, note that ϕ ( c ) = (0 ,
0) is a critical point of π ′ ◦ h ′ = ( π ◦ h ) ◦ ϕ − because c is a critical point of π ◦ h and ϕ is a diffeomorphism. Hence, by the chainrule,0 = J ( π ′ ◦ h ′ , (0 , J ( π ′ , h ′ (0 , · J ( h ′ , (0 , (cid:0) ( ∂ π ′ )(0 , . . . ( ∂ p − π ′ )(0 ,
0) ( ∂ p π ′ )(0 , (cid:1) · (cid:18) I p − . . .
00 0 . . . (cid:19) = (cid:0) ( ∂ π ′ )(0 , . . . ( ∂ p − π ′ )(0 ,
0) 0 . . . (cid:1) . Thus, ( ∂ π ′ )(0 ,
0) = · · · = ( ∂ p − π ′ )(0 ,
0) = 0. Consequently, ( ∂ p π ′ )(0 , = 0 as π ′ : V ′ → R is a submersion. (cid:3) Corollary 3.8.
Let f : Σ n → R p be a standard special generic map. Then, forevery q ∈ { , . . . , p − } , the composition of the standard special generic map h : Σ n q f −→ W f ∼ = −→ D p ֒ → R p (where a diffeomorphism W f ∼ = −→ D p has been fixed) with the projection π : R p → R q to the first q components yields a standard special generic map π ◦ h : Σ n → R q .Proof. It suffices to show that g := π ◦ h : Σ n → R q is a special generic map. Infact, Lemma 3.5 will then imply that g is standard because g (Σ n ) = π ( D p ) = D q ,and the fibers of g are all connected. (To show that g has connected fibers, one usesthat q f and π | : D p → D q are surjections with connected fibers between compactHausdorff spaces.)Analogously to the proof of Proposition 3.7 one shows that h is a standardspecial generic map, and that S ( g ) = h − ( S ( π | S p − )) = h − ( S q − × ∼ = S q − .Furthermore, observe that g restricts to a diffeomorphism S ( g ) ∼ = −→ S q − .Let c ∈ S ( g ) be a singular point of g , and let u := g ( c ) ∈ S q − . Let π u : R q → R denote the orthogonal projection given by v u · v . Let { u , . . . , u q − , u } be anextension to an orthonormal basis of R q . Let π ⊥ u : R q → R q − denote the orthogonalprojection given by λ u + · · · + λ q − u q − + λu ( λ , . . . , λ q − ). One obtains alinear isomorphism ψ := ( π ⊥ u , π u ) : R q → R q . D.J. WRAZIDLO
The composition α := π ⊥ u ◦ g | S ( g ) is a local diffeomorphism at c such that α ( c ) =0 ∈ R q − . Thus, α restricts to a chart U S ∼ = −→ U ′ S of S ( g ) ∼ = S q − between suitableopen neighborhoods c ∈ U S ⊂ S ( g ) and 0 ∈ U ′ S ⊂ R q − . Chosen appropriately, U S has a trivial tubular neighborhood ν : U S × R n − q +1 ∼ = −→ U in Σ n . As π ⊥ u ◦ g | U S = α | U S , the Jacobian of ϕ := ( π ⊥ u ◦ g, pr R n − q +1 ◦ ν − ) : U → R q − × R n − q +1 is invertible at points in S ( g ) ∩ U = ν ( U S ×
0) = U S . In particular, ϕ restrictsto a chart U → U ′ of Σ n between suitable open neighborhoods c ∈ U ⊂ U and ϕ ( c ) = (0 , ∈ U ′ ⊂ R q − × R n − q +1 .Using that π ⊥ u ( g ( ϕ − ( x, y ))) = x by construction of ϕ , consider the composition ψ ◦ g ◦ ϕ − : U ′ → R q , ( x, y ) ( x, ( π u ◦ g ◦ ϕ − )( x, y )) . Observe that π u ◦ g = π ˜ u ◦ h , where ˜ u := ( u, ∈ S p − ⊂ R q × R p − q . NowProposition 3.7 implies that π ˜ u ◦ h is a special generic function whose singular locus S ( π ˜ u ◦ h ) = h − ( {± ˜ u } ) contains c = h − (˜ u ). Consequently, (0 ,
0) = ϕ ( c ) is acritical point of π u ◦ g ◦ ϕ − such that the Hessian H (0 , ( π u ◦ g ◦ ϕ − ) is definite.Therefore, (0 ,
0) is also a critical point of π u ◦ g ◦ ϕ − | U ′ ∩ (0 × R n − q +1 ) such that theHessian H (0 , ( π u ◦ g ◦ ϕ − | U ′ ∩ (0 × R n − q +1 ) ) is definite. In conclusion, Proposition 2.2implies that c is a definite fold point of g . (cid:3) In view of Corollary 3.8 we define the notion of fold perfection of homotopyspheres.
Definition 3.9.
The fold perfection of a homotopy sphere Σ n ( n ≥
7) is the greatestinteger p ≥ n → R p .Note that by (1.1) and Example 3.6 the standard sphere S n is the only homotopy n -sphere of fold perfection n −
1. By (1.1) and Proposition 3.4( a ) any homotopysphere Σ n has fold perfection ≥
2. In Proposition 5.1( ii ) we will see that the foldperfection of Milnor spheres equals 2. Furthermore, Proposition 5.1( i ) shows thatthere exist exotic spheres with fold perfection greater than 2. Remark . The notion of fold perfection has only been defined for homotopyspheres of dimension ≥
7. As discussed in Section 2, this is due to the fact thatin lower dimensions the standard sphere turns out to be the only homotopy spherethat admits a special generic function. In fact, it is well-known that there are nohomotopy n -spheres except for the standard sphere S n in dimension n ≤ n = 4.The following fact will notably be applied to the Milnor 7-sphere in Corollary 5.2. Proposition 3.11. If Σ n ( n ≥ ) is a homotopy sphere of fold perfection , then / ∈ S (Σ n ) .Proof. Any special generic map Σ n → R is standard by Proposition 3.4( a ). Hence3 ∈ S (Σ n ) would imply that Σ n has fold perfection ≥ (cid:3) Remark . By Corollary 3.8 the notion of fold perfection gives rise to a filtrationof Θ n ( n ≥
7) by subsets F nn − ⊂ · · · ⊂ F n ⊂ Θ n , where, by definition, [Σ n ] ∈ F np ifthe fold perfection of Σ n is ≥ p . The proof of [10, Lemma 5.4, p. 278] implies that F np is in fact a filtration of Θ n by sub groups , which contains the Gromoll filtrationΓ np by Theorem 1.3( i ). We do not know whether the groups F np and Γ np do ingeneral coincide or not. TANDARD SPECIAL GENERIC MAPS OF HOMOTOPY SPHERES 9 Proof of Theorem 1.3
Let Σ n be a homotopy sphere of dimension n ≥ i ) is a modification of the original proof due to Weiss (see [13, §
4, pp. 403 ff]) of Theorem 1.2.According to Section 2.4 there exists a diffeomorphism g ∈ Diff( D n − , ∂ ) suchthat Σ n is diffeomorphic to the twisted sphere obtained by gluing the open balls V − := S n \ { (1 , , . . . , } ,V + := S n \ { ( − , , . . . , } , along the open cylinder V − ∩ V + = S n \ { ( ± , , . . . , } ∼ = S n − × ( − ,
1) =: C .The result of the gluing procedure is described by a pushout diagram of the form C V + V − Σ n j − j + ◦ ( ϕ g × id ( − , ) k + k − ,where the embeddings j − : C → V − and j + : C → V + are both given by the formula( z, t ) ( p − t · z, t ) ∈ V − ∩ V + ⊂ R n × R . Supposing that Σ n has Gromoll filtration > p , we may assume by Section 2.5 that g is chosen such that π n − p ( g ( x )) = π n − p ( x ) for all x ∈ D n − . For the associateddiffeomorphism ϕ g : S n − → S n − this implies π np ( ϕ g ( z )) = π np ( z ) for all z ∈ S n − . (In fact, in terms of the embeddings ι ± : D n − → S n − of Section 2.4 thediffeomorphism ϕ g satisfies for all x ∈ D n − the equations π np ( ϕ g ( ι − ( x ))) = π np ( ι − ( g ( x ))) = π np ( − p − || g ( x ) || , g ( x )) = π n − p ( g ( x )) = π np ( x )as well as π np ( ϕ g ( ι + ( x ))) = π np ( ι + ( x )) = π np ( p − || x || , x ) = π n − p ( x ).)Consequently, the following diagram commutes. C V + R n +1 V − R n +1 R p +1 j − j + ◦ ( ϕ g × id ( − , ) incl π n +1 p +1 incl π n +1 p +1 (In fact, for all ( z, t ) ∈ C we have( π n +1 p +1 ◦ j + ◦ ( ϕ g × id ( − , ))( z, t )= π n +1 p +1 ( j + ( ϕ g ( z ) , t )) = π n +1 p +1 ( p − t · ϕ g ( z ) , t )= ( p − t · π np ( ϕ g ( z )) , t ) = ( p − t · π np ( z ) , t )= π n +1 p +1 ( p − t · z, t ) = ( π n +1 p +1 ◦ j − )( z, t ) . )Thus, the universal property of the above pushout diagram gives rise to a map f : Σ n → R p +1 such that f ◦ k ± = π n +1 p +1 | V ± . With k − ( V − ) ∪ k + ( V + ) being an open cover of Σ n , Example 2.3 and Lemma 3.5 imply that f is a standard special genericmap. In conclusion, Σ n has fold perfection > p .The proof of part ( ii ) follows immediately from Proposition 3.7. In fact, supposethat Σ n has fold perfection ≥ p . If f : Σ n → R p is a standard special generic mapand ι : W f ∼ = D p is a diffeomorphism, then the smooth map η : S p − × Σ n → R , η ( u, x ) = u · ι ( q f ( x ))satisfies properties (1) and (2) of Definition 2.5, which shows that Σ n has Morseperfection ≥ p − Applications
We start with an application of Theorem 1.3 to some concrete exotic spheres.
Proposition 5.1. ( i ) In certain dimensions n ≥ the group Θ n is known tocontain exotic spheres of great depth in the Gromoll filtration. For instance, Γ = 0 and Γ = 0 by [3, Appendix A] . Hence, these exotic spheres haveat least an accordingly great fold perfection by Theorem 1.3 ( i ) . ( ii ) Let n = 4 k − for some integer k ≥ , and let Σ nM denote the Milnor n -sphere , i.e., Σ nM = ∂W n +1 for some parallelizable cobordism W n withsignature . By [13, p. 390] one has ( Gromoll filtration of Σ nM ) − Morse perfection of Σ nM . Consequently, the fold perfection of Σ nM isprecisely by Theorem 1.3. In view of (1.1), Proposition 5.1( ii ) implies the following answer of Problem 1.1for the Milnor 7-sphere by invoking Proposition 3.11. Corollary 5.2.
The Milnor -sphere Σ M of Proposition 5.1 ( ii ) satisfies S (Σ M ) = { , , } . Remark . Since the subgroup F ⊂ Θ (see Remark 3.12) has at least index 2 dueto [Σ M ] / ∈ F , there are in fact at least 14 exotic 7-spheres Σ with S (Σ ) = { , , } .We do not know the actual size of the groups Γ ⊂ F .With regard to Problem 1.1 we obtain the following immediate consequence ofTheorem 1.3( i ) and Corollary 3.8. Proposition 5.4. If Σ n is a homotopy sphere of dimension n ≥ whose Gromollfiltration is ≥ k , then { , . . . , k, n } ⊂ S (Σ n ) . Remark . Let us compare Proposition 5.4 with the inclusions of (1.1).( i ) Proposition 5.4 implies that a homotopy sphere Σ n ( n ≥
7) with Gromollfiltration ≥ n − S n because the second inclusionof (1.1) shows that n − / ∈ S (Σ n ) whenever Σ n is an exotic sphere. Indeed,both of these facts are known to follow from Hatcher’s proof [6] of the Smaleconjecture according to the introduction of [3] and [11, Remark 2.4, p. 165].( ii ) Furthermore, it is known that every homotopy sphere Σ n of dimension n ≥ ≥
2, and this fact implies via Proposition 5.4the inclusion { , , n } ⊂ S (Σ n ) of (1.1). Both of these facts follow fromCerf’s work [2] as pointed out in the introduction of [3] and in [10, p. 279]. TANDARD SPECIAL GENERIC MAPS OF HOMOTOPY SPHERES 11
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Mathematisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Im NeuenheimerFeld 205, 69120 Heidelberg, Germany
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