aa r X i v : . [ m a t h . DG ] J a n DUAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES
FUQUAN FANG
Dedicated to Professor Boju Jiang on the occasion of his 80th birthday A bstract . Let Σ be a simply connected rational homology sphere. A pair of disjoint closedsubmanifolds M + , M − ⊂ Σ are called dual to each other if the complement Σ − M + stronglyhomotopy retracts onto M − or vice-versa. In this paper we are concerned with the basic problemof which integral triples ( n ; m + , m − ) ∈ N can appear, where n = dim Σ − m ± = codim M ± − ff erential geometry:(i) the theory of isoparametric hypersurfaces and Dupin hypersurfaces in the unit sphere S n + initiated by ´Elie Cartan, where M ± are the focal manifolds of the hypersurface M ⊂ S n + , and m ± coincide with the multiplicities of principal curvatures of M .(ii) the Grove-Ziller construction of non-negatively curved riemannian metrics on exoticspheres, where M ± are the singular orbits of a cohomogeneity one action on Σ .Based on important result of Grove-Halperin [16], we provide a surprisingly simple answer,namely, if and only if one of the following holds true: • m + = m − = n , • m + = m − = n ∈ { , , , } , • m + = m − = n ∈ { , } , • m + = m − = n ∈ { , } , • nm + + m − = m + + m − is odd if min( m + , m − ) ≥ Σ is a homotopy sphere and the ratio nm + + m − = ≤ m − < m + ), we observe that, the proof in Stolz [23] applies almost identically to concludethat, the pair can be realized if and only if, either ( m + , m − ) = (5 ,
4) or m + + m − + δ ( m − −
1) (cf. the table on page 2), which is equivalent to the existence of ( m − − S m + + m − by Adams’ celebrated work. In contrast,infinitely many counterexamples are given if Σ is a rational homology sphere.
1. I ntroduction
Let Σ be a simply connected rational homology sphere of dimension n +
1, and let M + ⊔ M − ⊂ Σ be an embedded closed submanifold with two connected components. We call M + is dual to M − if the complement Σ − M + is strongly homotopy retracts onto M − ⊂ Σ . It is clear that thedual relation is reflective, i.e., if M + is dual to M − , then M − is dual to M + . Moreover, note that Σ = D ( ν + ) ∪ ∂ D ( ν − )where ν ± is the normal disk bundles of M ± in Σ .Let m ± denote the dimensions of the normal spheres to M ± ⊂ Σ . In this paper we are con-cerned with the following The author is supported by a NSFC key grant, the Ministry of Education in China, and the municipal adminis-tration of Beijing.
Problem.
Which integral triples ( n ; m + , m − ) ∈ N can be realized as the dimension of M,codimensions of M + and M − ? Besides the interests in its own rights from algebraic topology, the problem has roots in atleast two important themes in di ff erential geometry: • the theory of isoparametric hypersurfaces or more generally of Dupin hypersurfaces in theunit sphere S n + initiated by ´Elie Cartan in [3][4][5][6], where M ± are the focal manifolds ofthe hypersurface M ⊂ S n + , and m ± are referred as the multiplicities of principal curvatures of M . • cohomogeneity one isometric actions on a homology sphere Σ where M ± are the singularorbits, which produce many important examples in riemannian geometry including Einsteinmetrics, minimal submanifolds (cf. Hsiang-Lawson [18]), and very recently, new examples ofnon-negatively curved riemannian metrics on exotic 7-spheres by the Grove-Ziller [17].It is clear that, given a trivial k -knot S k ⊂ S k + ℓ + , there is a dual trivial ℓ -knot S ℓ ⊂ S k + ℓ + .A nontrivial dual pair are the embedded real projective planes RP ± in S , for which the nor-mal S -bundles of RP ± have the same total space S / Q where Q = {± , ± i , ± j , ± k } ⊂ S thequaternion subgroup. Both examples arise in the theory of isoparametric hypersurfaces in theunit spheres, i.e., the principal curvatures are constant. From the classification of ´Elie Car-tan [3][4][5][6], the focal manifold of any isoparametric hypersurface in the unit sphere withtwo distinct principal curvatures is the union of a dual pair of trivial knots, moreover, isopara-metric hypersurface with three distinct principal curvatures only occurs in the unit spheres ofdimensions 4 , ,
13 and respectively 25, whose focal manifold is FP ± with F = R , C , H andrespectively F / Spin (9) ± where the multiplicities m + = m − = , , ,
8. In general, a cele-brated theorem of M ¨unzner [22] shows that the number g of distinct principal curvatures of anisoparametric hypersurface M ⊂ S n + must be 1, 2, 3, 4 or 6. Furthermore, when g is odd, themultiplicities of the g principal curvatures are all the same, m : = m = m = · · · = m g , and when g is even, we have equalities m = m = · · · = m g − : = m + and m = m = · · · = m g : = m − .From definition the dimension of M and its multiplicities satisfy the formula 2 n = g ( m − + m + ).In very recent two decades, breakthroughs have been made for the classification of isopara-metric hypersurfaces in the spheres in the series [9][10][11][21] answered in positive an openproblem in the list of S.T.Yau [27], it turns out that, an isoparametric hypersurface in the unitsphere is either homogeneous or one of the Ferus-Karcher-M ¨unzner examples from the repre-sentations of Cli ff ord algebra. More precisely, the homogeneous isoparametric hypersurfacesare the principal orbits of a cohomogeneity one isometric action on S n + where the focal man-ifolds are the two singular orbits. The Ferus-Karcher-M ¨unzner [15] examples are constructedusing the orthogonal representations of Cli ff ord algebra as follows: Let Cl , m + denote the Clif-ford algebra spanned by 1 , e , · · · , e m satisfying e i = e i e j = − e j e i for i , j . For anynontrivial ( n + Cl , m + , e , · · · , e m give rise matri-ces P , · · · , P m satisfying that P i = I and P i P j = − P j P i for i , j . Let f ( x ) = h x , x i − P mi = h P i ( x ) , x i , x ∈ R n + . The function f maps the unit sphere to [ − ,
1] and satisfies the Cartan-M ¨unzner equations, i.e, k d f k and the Laplacian ∆ f are both functions of f . By [22], for any regular value c ∈ [ − , M = f − ( c ) is an isoparametric hypersurface with four distinct principal cur-vatures and multiplicities m , n − m , m , n − m . Its scalar curvature is constant and equal to UAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES 3 n − n . Notice that all irreducible representations of Cl , m + have the same dimension 2 δ ( m )where δ ( m ) , m ≥ δ ( m + k ) = k δ ( m ) and m δ ( m ) 1 2 4 4 8 8 8 8From the construction we know that ( n +
2) must be a multiple of 2 δ ( m ). We refer to [15] formore details about those examples.On the other hand, given a cohomogeneity one action of a connected compact Lie group G ona simply connected rational homology sphere Σ , the orbit space Σ / G is isometric to an interval[ − , M ± corresponding to ± ∈ [ − , M + and M − are dual pair in Σ . For a linear (representation) cohomogeneity one action onthe sphere S n + , the principal orbits are isoparametric hypersurfaces, and the singular orbits arethe focal manifolds. However, there are infinitely many nonlinear cohomogeneous one actionson spheres. Recently, Grove-Ziller [17] constructed non-negatively curved riemannian metricson the Milnor spheres Σ (homotopy 7 spheres which are S bundles over S ) via cohomogene-ity one actions whose the singular orbits are of codimensions 2, i.e., m ± =
1. Very recently,Goette-Kerin-Shankar has announced a generalized result along the line of Grove-Ziller that all7-dimensional exotic spheres admit riemannian metrics with non-negative sectional curvature,once again there is a pair of dual submanifolds in Σ of codimension 2. So far there is no anyexample of exotic spheres of dimension > > m + = m − = heorem A. Let Σ be a simply connected rational homology sphere of dimension n + andlet M ± ⊂ Σ be a pair of dual submanifolds of codimensions m ± + . Then one of the followingholds: • m + = m − = n, • m + = m − = n ∈ { , , , } , • m + = m − = n ∈ { , } , • m + = m − = n ∈ { , } , • nm + + m − = or , and for the latter case, m + + m − is odd if min ( m + , m − ) ≥ . We remark that Theorem A implies immediately the celebrated theorem of M ¨unzner, that thenumber g ∈ { , , , , } of distinct principal curvatures for an isoparametric hypersurface, bythe formula 2 n = g ( m + + m − ) when g is even, and n = gm + = gm − when g is odd.It is also clear to read from Theorem A thatC orollary Let Σ be a simply connected rational homology sphere of dimension n + with dual submanifolds M ± ⊂ Σ . If m + = m − = , then n ∈ { , , , , } . C orollary There is no riemannian submersion π : M → Σ from a simply connectedcohomogeneity one manifold M whose singular orbits are of codimension onto a rationalhomology sphere Σ of dimension at least . FUQUAN FANG
Therefore, it seems that the Grove-Ziller construction does not produce riemannian metricson rational homology spheres, in particular exotic spheres of dimension at least 8 with non-negative sectional curvature.Note that, when n = m + + m − , any pair ( m + , m − ) can be realized as the trivial dual knots in thesphere. When nm + + m − = m + , m − , it is very di ffi cult to determine which pair of integers canbe realized. Without loss of generality, we may assume that m − < m + , in the special case that Σ is in addition a homotopy sphere, though the Stolz’s theorem was formulated in riemanniangeometry (cf. Theorem 4.1), we observe that the proof of Stolz’s theorem [23] applies almostidentically, to prove the following pure topological theorem, we tribute to Stephan Stolz.T heorem Let Σ be a homotopy sphere of dimension n + , and let M ± ⊂ Σ be apair of dual submanifolds codimensions m ± + where m − < m + . If n = m + + m − ) , then either ( m + , m − ) = (5 , , or m + + m − + is divisible by δ ( m − − .Remark . It is natural to ask whether a similar result as above holds true for rationalhomology sphere. We will provide infinitely many examples of dual submanifolds in rationalhomology spheres of dimension 4 m −
1, such that any pair of integers ( m + , m − ) so that m + + m − = m − M ± . Remark . Though Theorem 1.3 does not hold for rational homology spheres in general, itis still interesting to wonder, under what constraints on Σ , Theorem 1.3 holds true. The proof ofthe Stolz’s theorem [23] depends heavily the cell structure of the hypersurfaces M and the focalmanifolds M ± which seems to be impossible to generalize. In [12], ahead the work of Stolz[23], the K -theory of isoparametric hypersurfaces was developed which solved in half cases ofthe multiplicities problem. It might be useful to apply K -theory for the problem.In view of the above results it follows that, for a pair of dual submanifolds M + ⊔ M − ⊂ Σ with the dimension data ( n ; m + , m − ) where Σ is a homotopy sphere, there is an isoparametrichypersurface N ⊂ S n + with focal submanifold N + ⊔ N − ⊂ S n + with the same dimensions data.Moreover, by Tables 2.2 and 2.3 the fundamental groups and homology groups of N ± coincidewith that of M ± . We conclude this section with the following natural but di ffi cult problem whichis already highly nontrivial and interesting when M is a Dupin hypersurface in the sphere (cf.[13]). Problem.
Is every pair of dual submanifolds M + ⊔ M − ⊂ Σ in a homotopy sphere topo-logically homeomorphic to the pair of focal manifolds of an isoparametric hypersurface in asphere? Acknowledgement . The author would like to thank Karsten Grove for useful discussionsmotivated the corollaries in the paper. 2. P reliminaries
In this section we present some basic preliminary results of Grove-Halperin [16].2.1.
Double mapping cylinder and rational homotopy theory.
In [16] the authors considerthe so called double mapping cylinder DE of maps φ ± : E → B ± with homotopy fibers of S m ± up to weak homotopy equivalence, i.e., the gluing of the mapping cylinders of φ ± along E . Let UAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES 5 F denote a path connected component of a homotopy fiber of the inclusion j : E → DE . Forsimplicity let us assume that • E , B ± are connected CW complexes, DE is simply connected, and the homotopy fibres S m ± satisfy m ± ≥ m + ≥ m − ≥
1. If m − =
1, then [ φ + ( S m − )] ∈ π ( B + ) actson the homology group H m + ( S m + ; Z ) of the homotopy fibers of φ + : E → B + . By [16], φ + iscalled twisted if this action is non trivial, hence by −
1. Similarly, φ − is twisted if m + = φ − ( S m + )] ∈ π ( B − ) acts by − H m − ( S m − ; Z ) of the homotopy fibers of φ − : E → B − .Under this assumption, we have the following important result in [16].T heorem The fundamental group π ( F ) and the homology group H ∗ ( F ; Z ) are given in the following tables where Q = {± , ± i , ± j , ± k } ⊂ S is the order quaterniongroup. ( m + , m − ) m ± > m + > m − = m + = m − = m + = m − = m + = m − = φ ± twisted one of φ ± twisted both φ ± twisted π ( F ) 1 Z Z ⊕ Z Z ⊕ Z Q T able π ( F ) FUQUAN FANG ( m + , m − ) and twists H i ( F ; Z ) im + , m − Z i = i = m + , m − mod ( m + + m − )no twists Z ⊕ Z i > i = m + + m − ) m + = m − Z i = Z ⊕ Z i > i = m + ) m + > m − = Z i = i = ± m + + φ + twisted Z ⊕ Z i > i = m + + Z i = m + , m + + m + + m + = m − = Z i = i = φ + twisted Z ⊕ Z i = φ − not twisted Z i = Z ⊕ Z i > i = m + = m − = Z i = φ ± both twisted Z ⊕ Z i > i = Z ⊕ Z i = able H ∗ ( F ; Z )T heorem The rational homotopy type of F is given in the followingtable. Moreover, the exceptional cases A (4) × Ω S , A (4) × Ω S do not occur if DE is ahomotopy sphere and φ ± are normal sphere bundles of B ± . UAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES 7 ( m + , m − ) and twists Q homotopy type of Fm + = m − =
1; [ SO (3) / ( Z ⊕ Z )] × Ω S φ ± both twisted ≃ Q [ SO (4) / ( Z ⊕ Z )] × Ω S m + = m − = φ + twisted, not φ − [( SO (2) × SO (3)) / Z ] × Ω S m + = m − = S × S × Ω S φ ± both not twisted S × Ω S m + > m − = m + odd, φ + twisted S × S m + + × Ω S m + + m + > m − = S × S m + × Ω S m + + φ + not twisted ≃ Q S × S m + × S m + + × Ω S m + + if m + = m + > m − ≥ S m + × S m − × Ω S m + + m − + m + = m − odd S m + × S m + × Ω S m + + ≃ Q S m + × Ω S m + + m + = m − even S m + × S m + × Ω S m + + S m + × Ω S m + + m + = m − = SU (3) / T × Ω S ; Sp (2) / T × Ω S G / T × Ω S m + = m − = Sp (3) / Sp (1) × Ω S A (4) × Ω S A (4) × Ω S m + = m − = F / Spin (8) × Ω S T able F Note that the cohomology ring H ∗ ( Ω S k − ; Q ) = Q [ x ] FUQUAN FANG the free polynomial algebra, where x is a degree 2 k − Ω S k ≃ Q S k − × Ω S k − . Let A m ( k ) be the simply connected space ( k even, m = , , , ,
6) unique up to rational homotopy type, with cohomology algebra H ∗ ( A m ( k ); Q ) (cid:27) Q [ x , y ] where x , y are of degree k subject to relations x m = x + y = m = , , . x m = x + y = m = , . Note that A ( k ) ≃ Q S k ; A ( k ) ≃ Q S k × S k and when m = , , k = , ,
8. Moreover, SU (3) / T ≃ Q A (2) ; Sp (2) / T ≃ Q A (2) ; G / T ≃ Q A (2) Sp (3) / Sp (1) ≃ Q A (4) ; F / Spin (8) ≃ Q A (8)3. P roof of T heorem AIf either of m ± equals n , the corresponding manifold M ± is a point, hence the dual one is alsoa point and so m + = m − = n . In this case Σ is forced to be a homotopy sphere. In the followingwe assume m ± < n .By definition H ∗ ( Σ ; Q ) (cid:27) H ∗ ( S n + ; Q ) since Σ is a rational homology sphere Σ . The degreeone map f : Σ → S n + is a rational homotopy equivalence. Let i : M ⊂ Σ be the hypersurfaceand let F denote the homotopy fiber of the inclusion. Consider the homotopy fibrations ΩΣ → F → M → Σ where ΩΣ is the based loop space of Σ .For simplicity we will often use ≃ Q to denote rational homotopy equivalence. Since f ◦ i : M → Σ → S n + is contractible, it follows that F ≃ Q ΩΣ × M ≃ Q Ω S n + × M . Therefore, thecohomology rings H ∗ ( F ; Q ) (cid:27) H ∗ ( Ω S n + ; Q ) ⊗ H ∗ ( M ; Q ) . Note that, if n is even, then H ∗ ( Ω S n + ; Q ) (cid:27) Q [ e n ] is a polynomial ring on a variable e n of degree n ; and if n is odd, then H ∗ ( Ω S n + ; Q ) (cid:27) Q [ e n + ] ⊗ E ( e n ) where E ( e n ) is the exterior algebra on e n . Indeed, it is well known that Ω S n + ≃ Q S n × Ω S n + when n is odd.On the other hand, by Theorem 2.4 it follows that the rational homotopy type of F is eitherof the form S k × S l × Ω S k + l + or of the form A m ( k ) × Ω S mk + with m = , , , , k even,where A ( k ) ≃ Q S k ; A ( k ) ≃ Q S k × S k . Moreover, if m = , ,
6, then k = , , or 8. Now wecompare this with the previous rational equivalence F ≃ Q Ω S n + × M . The point of departure isL emma If Ω S n + × X ≃ Q Ω S ℓ + × Y, where X and Y are finite CW complexes, then m = nif m , n have the same pairities. Moreover, if n is odd but m is even, then m = n.Proof. Recall that the loop space Ω S n + (resp. Ω S ℓ + ) contains a factor of the form Ω S i + nomatter n + H ∗ ( Ω S i + ; Q ) is a free polynomial algebra ona generator e i of degree 2 i . The cohomology ring H ∗ ( Ω S n + × X ; Q ) = H ∗ ( Ω S n + ; Q ) ⊗ H ∗ ( X ; Q )where H ∗ ( X ; Q ) is of finite dimensional. Therefore, the factors of free polynomial algebras ofthe isomorphism H ∗ ( Ω S n + × X , Q ) (cid:27) H ∗ ( Ω S ℓ + × Y , Q ) must be of the same degree. The desiredresult follows. (cid:3) UAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES 9 If F ≃ Q S k × S l × Ω S k + l + , it is clear to note that k + l ≤ n . By the above lemma we getthat, either n = k + l or n = k + l ). By Table 2.5 it follows that, if m + > m − ≥
2, then( k , l ) = ( m + , m − ), and therefore nm + + m − ∈ { , } . Moreover, if m + > m − ≥ m + + m − is even, then nm + + m − =
1. It is straightforward to check that, if either m + > m − = m + = m − =
1, then either k + l = ( m + + m − ), 2( m + + m − ) or 3( m + + m − ). In particular, if m + = m − =
1, then n = , , F ≃ Q A m ( k ) × Ω S mk + with m ∈ { , , , , } and k even. Since A ( k ) ≃ Q S k and A ( k ) ≃ Q S k × S k and so F is of the form in the previous case, we may assume that m ∈ { , , } , then m + = m − = k ∈ { , , } . By using Lemma 3.1 again it follows that, n = mk if n and mk havethe same pairity, otherwise, either n = mk or 2 n = mk , the latter can not occur for dimensionalreasoning. For the former, since mk is always even, hence n = mk implies that n is also evenand has the same pairity as mk , a contradiction. Therefore, n = mk with k ∈ { , , } and m ∈ { , , } . By Table 2.5 it su ffi ces to exclude the cases A (4) × Ω S and A (4) × Ω S , where Σ is a simply connected rational homology sphere of dimension 17 and respectively 25.The following lemma is probably well-known to experts.L emma Let γ denote the universal S -bundle p : B SO (4) → B SO (5) . Then the Eulerclass e ( γ ) = .Proof. Note that in general the Euler class of an odd dimensional vector bundle is an order 2element in the cohomology group with integer coe ffi cient. It is well-known (from the definition)that e ( γ ) can be calculated from the Gysin exact sequence with integer coe ffi cents H ( B SO (5)) p ∗ −→ H ( B SO (4)) → H ( B SO (5)) ∪ e ( γ ) −→ H ( B SO (5)) p ∗ −→ H ( B SO (4)) → H ( B SO (4)) and H ( B SO (5)) can be calculated from the Serre spec-tral sequences with integer coe ffi cients of the homotopy fibration B Spin (4) → B SO (4) → K ( Z ,
2) and the homotopy fibration B Spin (5) → B SO (5) → K ( Z , B Spin (4) = B S × B S and B Spin (5) = B Sp (2). It is routine to see that only the E -term E , == H ( K ( Z , H ( B Spin (4))) (cid:27) Z and respectively E , = H ( K ( Z , H ( B Sp (2))) (cid:27) Z survives contributing to H ( B SO (4)) and respectively H ( B SO (5)). Therefore, the last homo-morphism p ∗ in the above Gysin sequence is an isomorphism, and it follows that e ( γ ) = (cid:3) By the above Lemma it follows that the Euler classes the oriented S bundles π ± : M → M ± are zero, hence the long exact sequences (up to degree 5) of the S -bundles implies that thereare cohomology classes α ± ∈ H ( M ) such that the restrictions of α ± on the fibers of π ± aregenerators of the cohomology groups H ( S ) (di ff erent fibers). From the Leray-Hirsch Lemmait follows that H ∗ ( M ) (cid:27) H ∗ ( M + )[1 , α + ] (cid:27) H ∗ ( M − )[1 , α − ] ⋆ as free modules.On the other hand, since F is 3-connected, Σ is simply connected, there is an exact sequenceup to degree 5 from the homotopy fibration F → M → Σ H ( Σ ) → H ( M ) → H ( F ) (cid:27) Z ⊕ Z = h α + , α − i where H ( Σ ) is a torsion group since Σ is a rational homology sphere, it follows that the torsionfree part of H ( M ), i.e., modulo torsion, H ( M ) / T = h α + , α − i . In particular, α + , α − are linearly independent. This together with ⋆ implies that α + lies in the image of π ∗− : H ( M − ) → H ( M )and α − lies in the image of π ∗ + : H ( M + ) → H ( M ).Now we need to derive the multiplicative structure of H ∗ ( M ) / T , where T is the torsion.Let α + i , α − i ∈ H i ( M ) denote the generator of free part (isomorphic to Z ) of the image of π ∗± : H i ( M ± ) → H i ( M ) (in particular, α ± = α ± ).For the boundary homomorphism ∂ ∗ : H ∗ ( M ) → H ∗ + ( Σ ) in the Mayer-Vietories exact se-quence of ( Σ ; D ( γ + ) , D ( γ − )), note that ∂ ∗ ( αβ ) = ∂ ∗ ( α ) β + α∂ ∗ ( β ) when α, β are both of evendegree. Therefore, ∂ ∗ ( α + α − ) =
0, since ∂ ∗ ( α + ) = ∂ ∗ ( α − ) =
0. It follows that α + α − can beexpressed as a combination of α + , α − ∈ H ( M ). By ⋆ we know that both { α + α − , α + } and { α + α − , α − } are basis of H ( M ) / T (cid:27) Z , hence { α + , α − } is also a basis of H ( M ) / T . Re-call that H ∗ ( M ) / T ⊗ Q (cid:27) A (4) or respectively A (4), generated by Q [ x , y ] modulo relations x = x + y = x = x + y =
0. In particular, for any β , β m − ,
0, where m = α + i , α − i of H ∗ ( M ) / T .To finish the proof, by the same argument of [16] on page 456 it follows that, the Stiefel-Whitney class of the normal bundle γ + , w ( γ + ) = α − (mod 2)but the first Pontryagin class p ( γ + ) = M ± ) = j − : S → M ofthe bundle π − : M → M − and π + : M → M + and the pullback bundle ( π + ◦ j − ) ∗ ( γ + ) : = η onthe 4-sphere. Note that w ( η ) = α − (mod 2) , p ( η ) =
0. Since π ( SO ) (cid:27) Z , p ( η ) = η is stably trivial and so w ( η ) =
0. A contradiction. Thisexcludes the cases of A (4) × Ω S and A (4) × Ω S , and the desired result follows. Example . Let π : Σ → S m be an S m − -bundle with nontrivial Euler class, e.g., the unittangent bundle of the sphere S m . Note that Σ is a rational homology sphere. If M ± ⊂ S m isa pair of dual submanifolds, then π − ( M ± ) ⊂ Σ is a pair of dual submanifolds in Σ of the samecodimensions, but the ratio dim Mm + + m − gets doubled. For any pair ( m + , m − ) where m + + m − + = m and the trivial m ± -knots S m ± ⊂ S m , the preimages π − ( S m ± ) ⊂ Σ is a pair of dual submanifoldswith the same codimensions of the knots in S m . Therefore, the ratio dim Mm + + m − = Example . For the Hopf fibration π : S → S , by pullback the isoparametric hypersur-faces and their focal manifolds in S we indeed get isoparametric hypersurfaces in S , e.g, thepullback of Cartan’s example S / Q )(with g =
3) is an isoparametric hypersurface in S with g =
6, clearly di ff eomorphic to S × S / Q . In a similar vein, for the boundary S × S ⊂ S ofthe trivial knots, its preimage S × S × S ⊂ S is an isoparametric hypersurface with g = S → S also produces isoparametric hypersurface in the total space with g =
4, but none of g = Example . The Milnor spheres Σ are 7-dimensional homotopy spheres which are S -bundles over S . Up to orientation reversing, there are exactly 10 of them are exotic spheres(cf. [17]). For the bundle projections π : Σ → S , the pullback of the dual submanifolds RP ± , π − ( RP ± ) = S × RP ± ⊂ Σ is a pair of 5-dimensional submanifolds in the Milnor spheres. UAL SUBMANIFOLDS IN RATIONAL HOMOLOGY SPHERES 11
4. P roof of T heorem /after S tephan S tolz As a natural generalization of isoparametric hypersurface, a Dupin hypersurface M ⊂ S n + is a compact hypersurface where the number of distinct principal curvatures are constant andthe principal curvatures λ ( x ) ≤ λ ( x ) · · · ≤ λ g ( x ) are constant along the leaves of the foliationsdefined by the eigenspaces of λ ( x ) , · · · , λ g ( x ). According to M ¨unzner and Thorgbersson, g ∈{ , , , , } . The multiplicities of λ , · · · , λ g satisfy that, if g is even, then m = m = · · · = m g − : = m + and m = m = · · · = m g : = m − ; and if g is odd, m = m = · · · = m g . Alongstanding problem was which pair of integers ( m + , m − ) can be realized as the multiplicitiesof a Dupin hypersurface of g =
4. Partial results were obtained in [1][16][24][12][14]. Theproblem was completely solved by Stolz in [23].T heorem
Let M n ⊂ S n + be a Dupin hypersurface with distinct principal curva-tures and multiplicities ( m − , m + ) . For simplicity let us assume m − ≤ m + . Then ( m − , m + ) = (2 , or (4 , , or m + + m − + is a multiple of δ ( m − − . The proof of Stolz’s theorem used heavily stable homotopy theory, which applies identicallyto the situation of Theorem 1.3 where Σ is a homotopy sphere. For reader’s convenience wewill give a very brief review on his beautiful proof.The point of departure is the Thom-Pontryagin construction which gives a stable map c : Σ → S ∧ M since the normal line bundle of M in Σ is trivial. In case Σ is a homotopy sphere, c gives a stablehomotopy class in π sn ( M ). Note that c is a degree one map, i.e., induces an isomorphism on thehomology group H n ( − , Z ). By composition with the bundle projections p ± : M → M ± thereis a map ( p + ∧ p − ) ◦ c : S n → M → M + ∧ M − . For dimension reason the image of the maplies in the n -skeleton ( M + ∧ M − ) ( n ) of the wedge. Therefore, it gives a stable homotopy class in π sn (( M + ∧ M − ) ( n ) ).An important observation in [23] (cf. Proposition 2.7 therein) is that, if ( m + , m − ) , (5 , M + ∧ M − ) ( n ) desuspends ℓ times for any ℓ ≤ m − −
1, i.e, homotopy equivalentto S ℓ ∧ X , the ℓ -th suspension of some CW complex X . This follows completely elementarydepending on the cell structure of M ± , M in the homotopy sphere Σ forced by the homologygroups of M ± and M .The geometric construction of May-Milgram-Segal plays a key role (cf. [20]). For anyconnected topological space X and any given integer k ≥
1, let D k , ( X ) = ( S k − + ∧ X ∧ X ) / Z ,where S k − + is the ( k −
1) sphere equipped with a disjoint base point and Z acts on S k − by theantipodal map and on X ∧ X by flipping the factors. One may take k = ∞ , and let D ( X ) denote D ∞ , ( X ) for simplicity. It is well-known that, D ( S ℓ ) = S ℓ ∧ RP ∞ ℓ , the ℓ -times suspension of thestunted real projective space RP ∞ ℓ = RP ∞ / RP ℓ − , by collapsing the subspace RP ℓ − ⊂ RP ∞ . By[19][20], for any q < r −
1, there is a generalized EHP exact sequence π q ( X ) → π sq ( X ) → π q ( D ( X )) → π q − ( X ) → π sq − ( X )whenever X is ( r − r ≥
2. The homomorphism from π sq ( X ) to π q ( D ( X )) in thesequence is called the Hopf invariant H .Now let X be the CW complex such that ( M + ∧ M − ) ( n ) = S ℓ ∧ X for any ℓ < m − . Let m = m + + m − . Note that n = m . By the previous section we know that m is odd. Moreover, by the homology groups calculation of M ± it follows that, M + ∧ M − is ( m − H m ( M + ∧ M − ; Z ) (cid:27) Z . Therefore, π m ( M + ∧ M − ) (cid:27) Z by the Hurewicz theorem. Let i : S m ⊂ ( M + ∧ M − ) (2 m ) be the inclusion of the bottom dimensional cell. Let f : S m − ℓ → X bethe stable homotopy class such that S ℓ ∧ f = ( p + ∧ p − ) ◦ c : S m → ( M + ∧ M − ) (2 m ) = S ℓ ∧ X .An important step in [23], based on the framed bordism theory and Brown-Kervaire invariant,is to prove that the Hopf invariant H ( f ) is not zero. Applying the D functor to i : S m − ℓ ⊂ X weget a map D ( i ) : S m − ℓ ∧ RP ∞ m − ℓ = D ( S m − ℓ ) → D ( X )From the cell structure of X it is easy to see that D ( i ) induces an epimorphism on the stablehomotopy groups π m − ℓ ( − ).On the other hand, there is a diagonal map S ℓ ∧ D ( X ) → D ( S ℓ ∧ X ) which induces ahomomorphism ∆ ∗ : π m ( S ℓ ∧ D ( X )) → π m ( D ( S ℓ ∧ X )). It is a technical result that ∆ ∗ commutes with Hopf invariants under the suspension isomorphism (cf. Lemma 4.8 in [23])on the stable homotopy groups, therefore, there is an element α ∈ π m ( S ℓ ∧ D ( X )) such that ∆ ∗ ( α ) = H ( f ) ,
0, the Hopf invariant of S ℓ ∧ f .Since D ( i ) induces an epimorphism on the stable homotopy groups (here we are often shift-ing the degree in the category of stable homotopy), we may write α = ( S ℓ ∧ D ( i )) ∗ ( s ) for some s ∈ π m ( S ℓ ∧ D ( S m − ℓ ) = π sm ( RP ∞ m − ℓ ). Note that D ( S ℓ ∧ X ) is (2 m − H ( f ) induces a nonzero homomorphism on the 2 m -th homology groups by the Hurewicztheorem. Therefore, the stable map s : S m → RP ∞ m − ℓ also induces a nonzero homomorphismon the m -th integral homology groups, note here there is a degree shifting due to the suspen-sion isomorphism. From homotopy theory it is not di ffi cult to see that s is homotopy to a mapinto the m -skeleton RP mm − ℓ such that, its composition with the collapsing map onto the sphere S m = RP mm − ℓ / RP mm − S m → RP mm − ℓ → S m has degree one (cf. page 264 in [23]). In other words, the top cell of RP mm − ℓ splits o ff . It is nowa classical result of Adams [2] in solving the vector fields problem on the spheres, that m + δ ( ℓ ). R eferences [1] Abresch, U., Isoparametric hypersurfaces with four or six distinct principal curvatures , Math. Ann., (1983), 283-302[2] Adams, J. F.,
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