Math. Z. 245, 711–724 (2003)

Mathematische Zeitschrift

DOI: 10.1007/s00209-003-0567-2

A problem of Kreck on Poincar´e manifolds Haibao Duan, Banghe Li∗ Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, P.R. China (e-mail: [email protected]; [email protected]) Received: 12 March 2002; in final form: 4 March 2003 / Published online: 4 September 2003 – © Springer-Verlag 2003

Abstract. An 1-connected closed manifold M is called a Poincar´e manifold if any other 1-connected closed manifold with the same homology as that of M is homeomorphic to M. In the metastable range 3 ≤ p ≤ q < 2p − 3 we answer the question raised by Kreck : “for which p and q is the product of two spheres S p × S q a Poincar´e manifold ?”. 1. Introduction Let T be the set of all homeomorphism types of 1-connected, closed topological manifolds. In [Kr] M. Kreck introduced the following Definition. An M ∈ T is called a Poincar´e manifold if #{N ∈ T | Hr (N ) ∼ = Hr (M), r ≥ 0} = 1. Clearly, with this concept, the generalized Poincar´e conjecture may be re-phrased as “the n-dimensional sphere S n is a Poincar´e manifold”. This notion is so inspiring as to promote various classification problems in topology. A direct but challenging one is the following problem inverse to the Poincar´e conjecture. “Which non-spherical manifolds are Poincar´e manifolds? [Kr]” Instead of merely asking the verification of a given manifold (e.g. S n ) being Poincar´e manifold, it calls also for a thorough understanding of the constructions and invariants for the set of 1-connected manifolds with a fixed homology. A number of interesting ideas, results and problems circled around Poincar´e manifolds are discussed in [Kr], where all Poincar´e manifolds in dimensions 4, 5 and 6 are determined. This paper concerns the question raised by Kreck [Kr] ∗

The second named author is partially supported by the 973 program of China.

712

H. Duan, B. Li

“for which p and q is the product of two spheres S p ×S q a Poincar´e manifold ?”. We will reduce this geometric problem to the problem of effective computation of homotopy groups, to which a great deal of efforts has been devoted during the past century. 2. Main results By a weakly smooth manifold, or simply a W-manifold, we mean a pair (M, DM ) in which 1) M is a closed, 1-connected topological manifold; 2) DM ⊂ M is an embedded disc, dimDM =dimM; and 3) M\ intDM is furnished with a fixed smooth structure. A W-homeomorphism between two W -manifolds (M, DM ) and (N, DN ) is a homeomorphism F : M → N that restricts to a diffeomorphism M\ intDM → N \ intDN . It is clear that W -homeomorphisms yield an equivalence relation among all W -manifolds. Denote by W the set of the equivalence classes. For a pair of integers 1 ≤ p < q we let Mp,q = {(M, DM ) ∈ W | Hr (M) ∼ = Hr (S p × S q ), r ≥ 0}. We call S p × S q a Poincar´e manifold with respect to W if #Mp,q = 1. The category W introduced above is also of classical interests. C.T.C. Wall classified all (n − 1) connected 2n- and (2n + 1)-manifolds, n ≥ 3, exactly in this category [W1 ], [W2 ]. Moreover, the slightly restricted version of Poincar´e manifolds is interesting by itself for its relevance to Poincar´e manifolds of the form S p × S q in the smooth category. Corollary 1. Assume that S p × S q a Poincar´e manifold with respect to W, and that p + q ≥ 5. Then for any 1-connected smooth manifold M with Hr (M) ∼ = Hr (S p × S q ), r ≥ 0, one has a diffeomorphism M ∼ = S p × S q #
for some homotopy sphere
of dimension p + q. While looking for Poincar´e manifolds with respect to W of the form S p × S q , we may exclude the case p = q from our consideration for, by Wall’s classification on (n − 1) connected 2n-manifolds, S 3 × S 3 ; S 7 × S 7 and S 8s+6 × S 8s+6 (s ≥ 1) are the only Poincar´e manifolds of the form S p × S p . Let SO(p) be the special orthogonal group of order p, and let i : SO(p) → SO(p + 1) be the natural inclusion. Our first result is Theorem 1. Assume that 3 ≤ p < q < 2p − 3. Then S p × S q is a Poincar´e manifold with respect to W if and only if 1) p ≡ 3, 5, 6, 7 mod 8; 2) πq−1 (SO(p + 1) = 0 for p + 1 < q; 3) the induced map i∗ : πp (SO(p)) → πp (SO(p + 1)) is trivial for p + 1 = q.

Poincar´e manifolds

713

Theorem 1 is ready to apply to provide concrete examples. Corollary 2. The only weakly smooth Poincar´e manifolds of the form S p × S p+r with p ≥ 3, 1 ≤ r < min{15, p − 3} are r = 1 : S 8s+5 × S 8s+6 ,

s ≥ 0;

r =6:S

8s−1

s ≥ 2;

r =7:S

8s−2

×S

8s+5

,

×S , r = 14 : S 8s−1 × S 8s+13 , 8s+5

s ≥ 2; s ≥ 3;

The only likely weakly smooth Poincar´e manifolds of the form S p × S p+r with p ≥ 3, 15 ≤ r < min{31, p − 3} are r = 15 : S 8s−2 × S 8s+13 ,

s ≥ 3;

r = 30 : S

s ≥ 5.

8s−1

×S

8s+29

,

Proof. Consider first the case r = 1. Following the first constraint of Theorem 1, we examine the sequence i∗

· · · → πp (SO(p)) → πp (SO(p + 1)) → Z → · · · . for p ≡ 3, 5, 6, 7 mod 8. These can be distinguished, by Kervaire [Ke], into three cases i∗

a) p ≡ 3, 7 mod 8:

··· → Z → Z ⊕ Z → Z → ···;

b) p ≡ 5 mod 8:

· · · → Z2 → Z → Z → · · · ;

c) p ≡ 6 mod 8:

· · · → Z4 → Z2 → Z → · · · .

i∗

i∗

The assertion for the case r = 1 is verified by 3) of Theorem 1. Assume next that r > 1. In view of the isomorphism [M, p.5] πq−1 (SO(p + 1)) = πq−1 (SO) ⊕ πq (Pp+1 ), 12 < p, q < 2p − 1, the conditions in Theorem 1 are equivalent to (1) p ≡ 3, 5, 6, 7 mod 8; (2) q ≡ 3, 5, 6, 7 mod 8 (by the Bott periodicity theorem); and (3) πq (Pp+1 ) = 0. The assertions for the case 1 ≤ r < min{15, p − 3} (resp. for the case 15 ≤ r < min{31, p − 3}) are verified by the table in [HM] (resp. in [M]) which lists the groups πq (Pp+1 ) for q = p + r with r ≤ 14 (resp. the 2-primary components of the groups πq (Pp+1 ) for q = p + r with r ≤ 30). 

714

H. Duan, B. Li

Let P be the set of all piecewise linear homeomorphism types of 1-connected, closed piecewise linear manifolds, and set Ep,q = {M ∈ P | Hr (M) ∼ = Hr (S p × S q ), r ≥ 0}, 1 ≤ p < q. S p × S q is called a Poincar´e manifold with respect to P if #Ep,q = 1.
For an n > 0 let P Ln be the structure group of n-block bundles 1 [BS1 ] (or equivalently, combinatorial n-prebundles [K1 ]). The operation of suspension

defines the natural inclusion i : P Ln → P Ln+1 . The analogue of Theorem 1 in the PL-category is Theorem 2. Assume that 3 ≤ p < q <2p − 3. Then S p × S q is a Poincar´e manifold with respect to P if and only if
Lq ) = 0; 1) πp−1 (P
2) πq−1 (P Lp+1 ) = 0 for p + 1 < q;

3) the induced map i∗ : πp (P Lp ) → πp (P Lp+1 ) is trivial for q = p + 1. The range 3 ≤ p < q < 2p − 3 in which Theorem 1 and 2 are valid is known as the metastable range. The following result is useful in eliminating the search of Poincar´e manifolds outside this range. Theorem 3. If 3 ≤ p < q, then a necessary condition for S p ×S q to be a Poincar´e manifold with respect to W (resp. P) is πp−1 (SO(q)) ⊕ Im{πq−1 (SO(p) → πq−1 (SO(p + 1))} = 0


(resp.πp−1 (P Lq ) ⊕ Im{πq−1 (P Lp ) → πq−1 (P Lp+1 )} = 0). The main body of the paper is arranged as follows. After preparations in Section 3, 4 and 5, the proof of Theorem 1 is done in Section 6. The proof of Theorem 2 and 3 will be devoted to section 7 and 8.

3. Embeddings of Sm in Sn × Sm Let N and M be two smooth compact manifolds of dimension m and m + n respectively, and let {N, M} be the set of diffeotopy classes of smooth embeddings N → M. The set of all homotopy classes of continuous maps N → M is denoted by [N, M]. Reduction from the diffeotopy classes to the homotopy classes yields the correspondence ϕ : {N, M} → [N, M], h → ϕ(h) = [h]. From Haefliger [H1 ], [H2 ] we have 1

In the PL-setting we choose to work with block bundles, rather than microbundles (with the structure groups of semi-simplicial group of germs of PL-automorphisms of Rn ), because of normal microbundles do not always exist in the PL-category [HW], [BS1 ].

Poincar´e manifolds

715

Lemma 1. If N and M are k-connected, then ϕ is a bijection whenever n > m − k and 2n − 3 > m. In particular, if 3 ≤ n ≤ m < 2n − 3, the correspondence ϕ :{S m , S n × S m } → πm (S n × S m ) = πm (S n ) ⊕ πm (S m ) is a bijection. Let lm ∈ πm (S m ) = Z be the class of identity, and set Kn,m = {h ∈ {S m , S n × S m } | [h] = θ ⊕ lm , θ ∈ πm (S n )}. Consider the map ψ : Kn,m → πm−1 (SO(n)) given by ψ(h) = the characteristic map of the normal bundle of h(S m ). In view of Lemma 1, if 2 ≤ n ≤ m < 2n − 3, the correspondence ψn,m : πm (S n ) → πm−1 (SO(n)), ψn,m (θ ) = ψϕ −1 (θ ⊕ lm ), is well defined. Lemma 2. In the homotopy sequence ∂n,m

i∗

· · · → πm (S n ) → πm−1 (SO(n)) → πm−1 (SO(n + 1)) → · · · of the fibration p : SO(n + 1) → S n we have 1) if 2 ≤ n ≤ m, Im ψ ⊆Ker{i∗ : πm−1 (SO(n)) → πm−1 (SO(n + 1))}; 2) if 2 ≤ n ≤ m < 2n − 3, ψn,m = ∂n,m . Proof. Denote by T M the tangent bundle of a smooth manifold M, and let p1 : S n × S m → S n (resp. p2 : S n × S m → S m ) be the projection onto the first (resp. the second) factor. Let h : S m → S n × S m be a smooth embedding with [h] = θ ⊕ lm for some θ ∈ πm (S n ) (i.e. h ∈ Kn,m ). It follows from p1 ◦ h
θ : S m → S n ; p2 ◦ h
lm : S m → S m that the induced bundle h∗ T (S n × S m ) has the ready made splitting θ ∗ T S n ⊕ T S m . That is, the normal bundle of h is θ ∗ T S n . We get 1) from θ ∗ T S n ⊕  = (n + 1)ε since T S n ⊕  = (n + 1)ε, where ε is the trivial line bundle over S m . If 2 ≤ n ≤ m < 2n − 3, the stability of homotopy groups of spheres implies that θ = Eθ  for some θ  ∈ πm−1 (S n−1 ), where E is the suspension operator. It follows that ψn,m (θ ) = ∂n,n (ln ) ◦ θ  : S m−1 → SO(n) since the characteristic map of T S n is ∂n,n (ln ). 2) follows from the commutativity of the diagram induced by θ  ∂n,n

πn (S n ) → πn−1 (SO(n))  θ∗ = (Eθ )∗ ↓ ↓ θ∗ ∂n,m

πm (S n ) → πm−1 (SO(n)). 

716

H. Duan, B. Li

4. Constructions in Mp,q : Sphere bundles over spheres Assume throughout this section that 3 ≤ p < q. Let S be the set of all diffeomorphism types of closed, 1-connected smooth manifolds of dimensions > 5. For an M ∈ S with a smooth embedding j : D n → M, n =dimM, the equivalence class (M, j (D n )) ∈ W is independent of the choice of j since, if j  is another embedding, there is a diffeomorphism F : M → M isotopic to the identity and carrying j (D n ) to j  (D n ) [S, Theorem 1.4]. Thus we have a reduction S → W (which may not be injective2 ) that is useful in describing certain elements in Mp,q by constructing their counterparts in S. For an α ∈ πp−1 (SO(q)) (resp. β ∈πq−1 (SO(p))), let ξ be the Euclidean q-bundle over S p (resp. p-bundle over S q ) whose characteristic map is α (resp. β), and let Mα (Mβ )∈ S be the total space of the sphere bundle of ξ ⊕ , where  is the trivial line bundle over S p ( S q ). If 2 ≤ p < q, Mα (Mβ ) is 1-connected. From the Gysin sequence one finds that Mα (Mβ )∈ Mp,q . This yields a map s : πp−1 (SO(q)) → Mp,q , α → Mα (resp.s  : πq−1 (SO(p)) → Mp,q , β → Mβ ). Clearly one has Lemma 3. Let g : S p → Mα (resp. g : S q → Mβ ) be the cross section given by the unit section of  and the zero section of ξ . Then 1) g is an embedding that represents a generator of Hp (Mα ) =Z (resp. Hq (Mβ ) = Z); 2) the characteristic map of the normal bundle of g(S p ) (resp. g(S q )) is α (resp. β). We use the correspondence s  : πq−1 (SO(p)) → Mp,q to show Lemma 4. If #Mp,q = 1, then i∗ : πq−1 (SO(p)) → πq−1 (SO(p + 1)) is trivial. Proof. Since #Mp,q = 1, one has a W -homeomorphism F : (Mβ , j (D)) → (S p × S q , j  (D)), dim D = p + q, for every β ∈ πq−1 (SO(p)), where j and j  are smooth embeddings which can be taken arbitrarily by the remark at the beginning of this section. In particular, we can assume that the embedding g : S q → Mβ specified in Lemma 3 has no intersection with j (D). Since F : Mβ → S p × S q is a homotopy equivalence, we can assume [F ◦ g] = θ ⊕ (±lq ) ∈ πq (S p × S q ), θ ∈ πq (S p ). By composing F with a self diffeomorphism of S p × S q that fixes the first factor space and alters the orientation of the second if necessary, we can assume further [F ◦ g] = θ ⊕ lq . According to 1) of Lemma 2, ψ(F ◦ g) ∈ Ker{i∗ : πq−1 (SO(p)) → πq−1 (SO(p + 1)}. 2

The operation of connected sum defines an action of the group of homotopy spheres on S. The reduction S → W clearly factors through the set of orbits.

Poincar´e manifolds

717

On the other hand, since F restricts to a diffeomorphism from a tubular neighborhood of g in Mβ to a tubular neighborhood of F ◦ g in S p × S q , β = ψ(F ◦ g). This completes the proof. 

5. Constructions in Mp,q : Attaching handles Suppose that 3 ≤ p < q. Consider a pair σ = (S p × D q , f ) in which D q is the q-dimensional disc, and f : D p × S q−1 → S p × S q−1 = ∂(S p × D q ) is a smooth embedding. From σ one forms a smooth manifold N (σ ) by attaching D p × D q to S p × D q using f , and smoothing round the corners. We recall a set of invariants that can be employed to enumerate all diffeomorphism types of N(σ ). By the diffeotopy extension theorem of Palais [P], isotopic f ’s give rise to diffeomorphic N(σ )’s. Clearly, the isotopy classes of f = f | 0 × S q−1 are parameterized by the subset {S q−1 , S p ×S q−1 }0 = {h ∈ {S q−1 , S p ×S q−1 } | the normal bundle of h is trivial}. It remains to find isotopy invariant of f corresponding to a fixed f ∈ {S q−1 , S p × S q−1 }0 . For this purpose we fix, for each f ∈ {S q−1 , S p × S q−1 }0 , a tubular neighborhood f0 : D p × S q−1 → S p × S q−1 . By the tubular neighborhood theorem of Milnor, after the diffeotopy, each embedding f : D p × S q−1 → S p × S q−1 corresponding to f is given by f (u, v) = f0 (β(v)u, v) for some β ∈ πq−1 (SO(p)). So we may write fβ instead of f . Summarizing we get (in analogue with Lemma 1 in [W1 ]): Lemma 5. All possible diffeomorphism types of N (σ ) (i.e. isotopy classes of embeddings D p × S q−1 → S p × S q−1 ) can be enumerated by 1) a choice of f ∈ {S q−1 , S p × S q−1 }0 with a fixed tubular neighborhood f0 ; 2) a choice of β ∈ πq−1 (SO(p)). Suppose that σ = (S p × D q , f ), and assume, with respect to the splitting πq−1 (S p × S q−1 ) = πq−1 (S p ) ⊕ Z, that [f | 0 × S q−1 ] = θ ⊕ klq−1 , θ ∈ πq−1 (S p ), k ∈ Z We look for the conditions under which ∂N (σ )is a homotopy sphere; and  Z for r = 0, p, q Hr (N (σ )) = 0 otherwise.

(5.1) (5.2)

718

H. Duan, B. Li

Lemma 6. If 3 ≤ p < q, then 1) ∂N (σ ) is a homotopy sphere if and only if k = ±1; 2) (5.2) holds if and only if either p + 1 < q or, θ = 0 when q = p + 1. Proof. ∂N (σ ) is constructed from S p ×S q−1 and f by first removing the interior of Im f then pasting in S p−1 ×D q using f | ∂(D p ×S q−1 ). Since 3 ≤ p < q, ∂N (σ ) is 1-connected by the Van Kampen Theorem. We get 1) from the Mayer-Vietoris exact sequence. 2) follows from the fact that N (σ ) is homotopy equivalent to the two cells complex S p ∪ eq with attaching map θ . 

6. Proof of Theorem 1 In view of the exactness of the homotopy sequence of the fibration SO(p + 1)) → Sp ∂p,q−1

i∗

· · · πq−1 (SO(p)) → πq−1 (SO(p + 1)) → πq−1 (S p ) → πq−2 (SO(p)) · · · , Theorem 1 is clearly equivalent to Lemma 7. Assume that 3 ≤ p < q < 2p −3. Then S p ×S q is a Poincar´e manifold with respect to W if and only if 1) p ≡ 3, 5, 6, 7 mod 8; 2) Ker∂p,q−1 = 0 if p + 1 < q; 3) the induced map i∗ : πq−1 (SO(p)) → πq−1 (SO(p + 1)) is trivial. Proof of Lemma 7. Consider first the necessities: under the assumptions 3 ≤ p < q < 2p − 3 and #Mp,q = 1, we show that conditions 1), 2) and 3) must be met by the pair (p, q) of integers. Fix, for each (M, DM ) ∈ Mp,q , a generator x ∈ Hp (M\ intDM ) = Z. Since M\ intDM is p − 1 connected and since 2 ≤ p < q, x is representable by a differential embedding g : S p → M\ intDM whose diffeotopy class is unique (Lemma 1). Thus, we can define a map t : Mp,q → πp−1 (SO(q)) by setting t (M, DM ) = the characteristic map for the normal bundle of g(S p ) in M\ intDM . Since the composition t ◦s : πp−1 (SO(q)) → πp−1 (SO(q)) is the identity by Lemma 3, t is surjective. Now #Mp,q = 1 implies that πp−1 (SO(q)) = 0. We get 1) from the Bott periodicity theorem. For 2) suppose that p + 1 < q < 2p − 3. Given an θ ∈Ker∂p,q−1 , we have an h ∈ Kp,q−1 with [h] = θ ⊕ lq−1 ∈ πq−1 (S p × S q−1 ) by Lemma 1. Since the normal bundle of h is trivial by 2) of Lemma 2, we have a pair σ = (S p × D q , f ) with f | 0 × S q−1 = h. Further, ∂N (σ ) is a homotopy sphere and  Z for r = 0, p, q Hr (N (σ )) = 0 otherwise

Poincar´e manifolds

719

by Lemma 6. Since p + q ≥ 7 (recall that 3 ≤ p < q < 2p − 3), a (p + q)dimensional disc D can be added to N (σ ) to yield a class (M(σ ), D) ∈ Mp,q , M(σ ) = N(σ ) ∪ D. By the construction above, M(σ ) admits a cell decomposition S p ∪ eq ∪ ep+q and the attaching map for the q-cell is θ . Therefore πq−1 (M(σ )) = πq−1 (S p )/ < θ >, where < θ > is the subgroup generated by θ . On the other hand, since #Mp,q = 1, M(σ ) must be homeomorphic to S p × S q . In particular πq−1 (M(σ )) ∼ = πq−1 (S p × S q ) = πq−1 (S p ). This verifies θ = 0, i.e. Ker∂p,q−1 = 0. The necessity of condition 3) has been shown by Lemma 4. Turn now to the sufficiency of the conditions 1), 2) and 3). Given an (M, DM ) ∈ Mp,q , let g : S p → M\ int(DM ) be a smooth embedding representing a generator of πp (M) = Hp (M) = Z (cf. Lemma 1). Since p ≡ 3, 5, 6, 7 mod 8, the tubular neighborhood N(S p ) of g must be trivial N (S p ) ∼ = S p × D q . We may assume p further that N(S ) ∩ DM = ∅. Let K = M\ intDM . Since 3 ≤ p < q and since  Z, r = q p Hr (K, N (S )) = 0, otherwise, the cancellation theorems in Morse theory can be applied to show that K\ intN (S p ) is an elementary cobordism (cf. the proof of Theorem 7.8 in [Mil, p. 97]). Therefore K = N(σ ) for some σ = (S p × D q , f ) (cf. Theorem 3.13 in [Mil, p. 31]). Moreover we must have, in πq−1 (S p × S q−1 ), that  ±lq−1 if q = p + 1 q−1 ]= [f | 0 × S θ ⊕ (±lq−1 ) if p + 1 < q by 1) of Lemma 6. We may also assume that the sign of lq−1 is + (by changing the orientation of S q−1 if necessary). Since the tubular neighborhood of f | 0 × S q−1 is trivial, θ ∈Ker∂p,q−1 when p + 1 < q < 2p − 3 by 2) of Lemma 2. It follows now from condition 2) that [f | 0 × S q−1 ] = lq−1 ∈ πq−1 (S p × S q−1 ), p < q < 2p − 3. So we may assume below that f | 0 × S q−1 = lq−1 by Lemma 1. Let R p+1 be the Euclidean space spanned by the orthonormal vectors e1 , . . . , ep+1 , and regard S p as the set of unit vectors in R p+1 . Take ∗ = ep+1 ∈ S p as the p base point, and let D+ ⊂ S p be the hemisphere centered at ∗, i.e. p

D+ = {u ∈ S p | u = (a1 , . . . , ap+1 ), ap+1 ≥ 0}. p

The natural inclusion f0 : D+ × S q−1 → S p × S q−1 is taken to be a fixed tubular neighborhood of f | 0 × S q−1 = lq−1 : S q−1 → S p × S q−1 , v → (∗, v).

720

H. Duan, B. Li

By Lemma 5, σ = (S p × D q , fβ ) for some β ∈ πq−1 (SO(p)). Regard SO(p) as the subgroup of SO(p + 1) that fixes ep+1 . Since i∗ (β) = 0  | ∂D q = β. Consider  : D q → SO(p + 1) with β by condition 3), we find an β the composed map p

Gj

p

L : S p × D q  D+ × D q → S p × D q  S p × D q → S p × S q , where −1 (v)u, v), (u, v) ∈ S p × D q ; i) G(u, v) = (β p p ii) j : D+ × D q → S p × D q the inclusion induced by D+ ⊂ S p ; iii) p the quotient map that identifies the boundaries of the two copies of S p × D q by the identity, and where  means disjoint union. L factors through K = N (σ ) to yield a smooth embedding F  : N(σ ) → S p × S q which can, clearly, be extended to a W -homeomorphism F : (M, DM ) → (S p × S q , D), D = S p × S q \I nt (F  (K)). This completes the proof of Lemma 7, hence of Theorem 1.



7. Preliminaries in the PL setting We develop the preliminary results (from Lemma 1’ to Lemma 4’) required by the proof of Theorem 2, in such a way in analogue to Lemma 1-Lemma 4. Given two closed PL-manifolds N and M of dimensions m and m + n respectively, let (N, M) denote the set of isotopy classes of locally flat piecewise linear embeddings N → M. Reduction from the isotopy classes to the homotopy classes yields the correspondence ϕ  : (N, M) → [N, M]. In analogue with Lemma 1 (cf. [BS2 , p. 96] and [Mi]) we have Lemma 1’. If N and M are k -connected and n ≥ 3, then ϕ  is a bijection whenever n > m − k and 2n − 3 > m. In particular, if 3 ≤ n ≤ m < 2n − 3, the correspondence

ϕ  : (S m , S n × S m ) → πm (S n × S m ) = πm (S n ) ⊕ πm (S m ) is a bijection.

Let lm ∈ πm (S m ) = Z be the class of identity, and set Jn,m = {h ∈ (S m , S n × S m ) | ϕ  (h) = θ ⊕ lm , θ ∈ πm (S n )}.
Consider the representation ψ  : Jn,m → πm−1 (P Ln ) (in analogue to ψ : Kn,m → πm−1 (SO(n)) in the W -category) defined by ψ  (h) = the characteristic map of the normal block bundle of h[BS1 ]

Poincar´e manifolds

721

We note that, when n ≥ 3, the ψ  (h) is independent of the choice of an embedding S m → S n × S m representing h since by Hudson [Hu], in codimension ≥ 3, homotopic embeddings are also ambient isotopic under the condition of Lemma 1’. In term of ψ  we introduce the correspondence 
ψn,m : πm (S n ) → πm−1 (P Ln ), 3 ≤ n ≤ m < 2n − 3,  (θ ) = ψ  ϕ −1 (θ ⊕ l ), θ ∈ π (S n ). by ψn,m m m

Ln ) one can replace the In the homotopy exact sequence of the pair (P Ln+1 , P n

groups πr (P Ln+1 , P Ln ) by πr (S ) whenever r ≤ 2n−3, via the canonical isomor

phisms w : πr (P Ln+1 , P Ln ) → πr (S n ), r ≤ 2n − 3, due to Rourke-Sanderson [BS1 ] and Kato [K2 ]. That is, one has the sequence p∗

∂n,r




· · · → πr (P Ln ) → πr (P Ln+1 ) → πr (S n ) → πr−1 (P Ln ) → · · · i∗

which is exact up to r ≤ 2n − 3. In terms of this we have the following result instead of Lemma 2. Lemma 2’. Assume as the above.

1) If n ≥ 3, Im ψ  ⊆Ker{i∗ : πm−1 (P Ln ) → πm−1 (P Ln+1 )}.  2) If 3 ≤ n ≤ m < 2n − 3, Kerψn,m =Ker∂n,m .
Proof. For an α ∈ Im ψ  ⊆ πm−1 (P Ln ) we have an h ∈ Jn,m so that α : S m−1 →
P Ln is the characteristic map for the normal block bundle ξ of the embedding h : S m → S n × S m . Consider the composed embedding h

j

d : S m → S n × S m → S n+m+1 = D n+1 × S m ∪ S n × D m+1 , where j is the isomorphism onto the subspace S n ×S m = D n+1 ×S m ∩S n ×D m+1 . A normal block bundle of d is seen to be ξ ⊕ε, where ε is the trivial 1-bundle over S m . Since n + 1 ≥ 4, d is extendible to a locally flat embedding d : D m+1 → n+m+1 S by the unknotting theorem in the PL category (cf. [BS2 , Theorem 7.1 and Corollary 7.2]) whose normal block bundle must be trivial. Further, can be so chosen such that there is a bundle isometry ⊕ ε | S m = ξ ⊕ ε by [BS1 , Theorem 4.3]. This verifies that ξ ⊕ ε = εn+1 , hence proves 1). For 2) assume that 3 ≤ n ≤ m < 2n − 3. If θ ∈ Ker∂n,m , one has an β ∈ πm (P Ln+1 ) such that p∗ (β) = θ by the exactness of the sequence. As in [K2 ] we regard β as an automorphism Fβ of D n+1 × S m . Let fβ be the automorphism of S n × S m obtained by restricting Fβ to ∂(D n+1 × S m ). Take base points x0 ∈ S n , y0 ∈ S m . Then i) [fβ | x0 × S m ] = p∗ (β) ⊕ lm = θ ⊕ lm (cf. [K2 , proof of Lemma 3.4]). Since fβ restricts to an isomorphism from the trivial normal block bundle of  . x0 ×S m ⊂ S n ×S m to the normal block bundle of fβ | x0 ×S m we find θ ∈Kerψn,m  . This verifies that Ker∂n,m ⊆Kerψn,m

722

H. Duan, B. Li  . We have then an embedding Conversely, assume that θ ∈Kerψn,m

f : D n × S m → S n × S m with [f | x0 × S m ] = θ ⊕ lm , where D n is regarded as an embedded PL disc centered at x0 ∈ S n . Since the homotopy class of [f | x0 × S m ] factors through the subspace S n ∨ S m (one point union over the point (x0 , y0 ) ∈ S n × S m ), we can assume, after isotopy, that ii) f is an embedding into the subspace D n × S m ∪ S n × D m ⊂ S n × S m , here m D ⊂ S m is a PL disc centered at y0 (by Lemma 1’); Moreover since the class [f | x0 × S m ] = θ ⊕ lm falls on the fixed generator lm by the projection onto the second factor sphere we can assume further iii) f maps D n ×(S m \IntD m ) identically onto the subspace D n ×(S m \IntD m ). Let C ⊂ S m \IntD m be an embedded PL disc and let i : D n+1 ×C → D n+1 ×S m be the natural inclusion. The map f ∪ i : D n × S m ∪ D n+1 × C → D n+1 × S m is easily seen to be an embedding which can be extended to an automorphism F of D n+1 × S m (since D n+1 × S m can be obtained from D n × S m ∪ D n+1 × C by attaching an n + m + 1 ball B along a half hemisphere of the boundary sphere ∂B). According to [K2 , Lemma 3.1], module automorphisms concordant to the identity,
F is congruent to Fβ for some β ∈ πm (P Ln+1 ). Put fβ = Fβ | ∂(D n+1 × S m ). Since F extends f , we have [fβ | x0 × S m ] = [f | x0 × S m ]. That is p∗ (β) ⊕ lm = θ ⊕ lm (cf.i)). The shows that θ ∈Ker∂n,m by the exactness of the sequence. Summarizing we get  Kerψn,m =Ker∂n,m . This completes the proof. 

By discussion in [BS1 , I. section 5] we may assign to each block bundle with an

associated sphere bundle (by fixing a deformation retraction P Ln → P Ln (
) for every n in the notion of [BS1 , Theorem 5.1]). Thus, assuming 2 ≤ p < q, similar constructions as that in Section 4 yields the correspondences
s : πp−1 (P Lq ) → Ep,q , α → Mα
(resp. s  : πq−1 (P Lp ) → Ep,q , β → Mβ ). In place of Lemma 3 we have Lemma 3’. Let g : S p → Mα (resp. g : S q → Mβ ) be the cross section given by the unit section of  and the zero section of ξ . Then 1) g is an embedding that represents a generator of Hp (Mα ) = Z (resp. Hq (Mβ ) = Z); 2) the characteristic map of the normal block bundle of g(S p ) (resp. g(S q )) is α (resp. β).

Poincar´e manifolds

723

Using the correspondence s  and 1) of Lemma 2’, one can show, by the same argument as that in the proof of Lemma 4, that
Lp ) → Lemma 4’. Suppose that 3 ≤ p < q, and that #Ep,q = 1. Then i∗ : πq−1 (P
πq−1 (P Lp+1 ) is trivial.

8. Proofs of Theorem 2 and 3 We can now prove Theorem 2 by showing Lemma 7’. Assume that 3 ≤ p < q < 2p − 3. Then S p × S q is a Poincar´e manifold with respect to P if and only if
1) πp−1 (P Lq ) = 0; 2) Ker∂p,q−1 = 0 when p + 1 < q;

Lp ) → πq−1 (P Lp+1 ) is trivial. 3) the induced map i∗ : πq−1 (P
Proof. Consider first the necessities. We have a map t  : Ep,q → πp−1 (P Lq ) that sends each M ∈ Ep,q to the characteristic map of the normal block bundle of the embedding g : S p → M that represents a fixed generator of Hp (M) = Z. From

t  ◦ s = id : πp−1 (P Lq ) → πp−1 (P Lq ) (by 1) of Lemma 3’) one finds that t  is
Lq ) = 0. surjective. #Ep,q = 1 yields πp−1 (P Suppose that p + 1 < q < 2p − 3. From an θ ∈Ker∂p,q−1 , one finds an embedding h ∈ Jp,q−1 with [h] = θ ⊕ lq−1 ∈ πq−1 (S p × S q−1 ) by Lemma 1’. Further, since a normal block bundle of h is trivial by 2) of Lemma 2’, one may construct an M ∈ Ep,q with πq−1 (M) = πq−1 (S p )/ < θ > in the same way as that in the weakly smooth case. Now #Ep,q = 1 restricts to Ker∂p,q−1 = 0. The necessity of 3) is contained in Lemma 4’. The sufficient part of Lemma 7’ follows from the same arguments as that in the proof of sufficient part of Lemma 7. To avoid repetition the details are omitted. We remark that Chapter 6 in [BS2 ] contains preliminaries concerning handle decomposition and simplification in the PL category, required by the proof of sufficiency of the condition 2), beside 1) and 3). The proof of Theorem 3 is already contained in the proofs of the necessary parts of Theorem 1 and 2. 

Acknowledgements. We are very grateful to M. Kreck who brought our attention to this interesting question in a lecture given at the Academy of Mathematics and Systems Sciences, Beijing, in April, 2001. Thanks are also due to our referee for many helpful improvements and valuable criticisms on an earlier version of this paper. In particular, we benefit a lot from discussion with him on the normal block bundle to a PL embedding.

References [H1 ]

Haefliger, A.: Plongements differentiable de varietes dan varietes. Comment. Math. Helv. 36, 47–82 (1961)

724

H. Duan, B. Li

[H2 ] Haefliger, A.: Differential imbeddings. Bull. Am. Math. Soc. 67, 109–112 (1961) [HW] Haefliger, A., Wall, C.T.C.: Piecewise linear bundles in the stable range. Topology 4, 209–214 (1965) [HM] Hoo, C.S., Mahowald, M.: Some homotopy groups of Stiefel manifolds. Bull. Am. Math. Soc. 71, 661–667 (1965) [Hu] Hudson, J.: Concordance, isotopy, and diffeotopy. Ann. of Math. (2) 91, 425–448 (1970) [K1 ] Kato, M.: Mitsuyoshi Combinatorial prebundles. I, II. Osaka J. Math. 4, 289–303 (1967); ibid. 4, 305–311 (1967) [K2 ] Kato, M.: A concordance classification of P L homeomorphisms of S p ×S q . Topology 8, 371–383 (1969) [Ke] Kervaire, M.A.: Some nonstable homotopy groups of Lie groups. Illinois J. Math. 4, 161–169 (1960) [Kr] Kreck, M.: An inverse to the Poincar´e conjecture, Festschrift: Erich Lamprecht. Arch. Math. (Basel) 77, 98–106 (2001) [M] Mahowald, M.: The metastable homotopy of S n . Memoirs of the American Mathematical Society, No. 72 American Mathematical Society, Providence, R.I. 1967 [Mi] Miller, R.T.: Closed isotopy on picewise linear manifolds. Trans. Am. Math. Soc. 151, 597–682 (1970) [Mil] Milnor, J.: Lectures on the h-corbordism Theorem. Princeton University Press, 1965 [P] Palais, R.: Local triviality of the restriction map for embeddings. Comment. Math. Helv. 34, 305–312 (1960) [BS1 ] Rourke, C.P., Sanderson, B.J.: Block bundles. I; II; III. Ann. Math. (2) 87, 1–28; 256–278; 431–483 (1968) [BS2 ] Rourke, C.P., Sanderson, B.J.: Introduction to piecewise-linear topology. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69. Springer-Verlag, New York-Heidelberg, 1972 [S] Smale, S.: On the structure of manifolds. Am. J. Math. 84, 387–399 (1962) [W1 ] Wall, C.T.C.: Classification of (n-1) connected 2n-manifolds. Ann. Math 75, 163– 198 (1962) [W2 ] Wall, C.T.C.: Classification problem in differential topology-part I. Topology 2, 253–261 (1963)