FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS

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1 Communications in Contemporary Mathematics, Vol., No. 1 (000) c World Scientific Publishing Company FIXED POINT FREE CIRCLE ACTIONS AND FINITENESS THEOREMS FUQUAN FANG Nankai Institute of Mathematics, Nankai University, Tianjin , People s Republic of China ffang@sun.nankai.edu.cn XIAOCHUN RONG Mathematics Department, Rutgers University, New Brunswick, NJ 08903, USA rong@math.rutgers.edu Received 7 September 1999 We prove a vanishing theorem of certain cohomology classes for an n-manifold of finite fundamental group which admits a fixed point free circle action. In particular, it implies that any T k -action on a compact symplectic manifold of finite fundamental group has a non-empty fixed point set. The vanishing theorem is used to prove two finiteness results in which no lower bound on volume is assumed. (i) The set of symplectic n-manifolds of finite fundamental groups with curvature, λ sec Λ, and diameter, diam d, contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. (ii) The set of simply connected n-manifolds (n 6) with λ sec Λ and diam d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. 1. Introduction The classical Cheeger s finiteness theorem asserts that the collection, M Λ,d, λ,,v (n), of compact n-manifolds whose sectional curvature, diameters and volume satisfy λ sec Λ, diam d, vol v>0 contains only finitely many diffeomorphism types depending only on n, λ, Λ, d and v [3]. Note that counterexamples will occur if any of the bounds but Λ is removed without imposing further restrictions (cf. [11, 1, 18]). A partial motivation of this paper is to prove some finiteness theorems which do not require a uniform lower bound on volume and which require, instead, certain topological restrictions (compare [8, 19]). In view of the Cheeger s finiteness theorem, it suffices to consider the collapsed manifolds i.e whose volumes are sufficiently small depending only on n, Λ,λ and d. According to the fibration theorem of Cheeger Fukaya Gromov, the above collapsed manifold M admits a so-called pure nilpotent Killing structure of some nearby 75

2 76 F. Fang & X. Rong metric [4]. If π 1 (M) is also finite, then the pure nilpotent Killing structure is given by an almost isometric π 1 -invariant T k -action without fixed point on M i.e. a T k -action on the universal covering space without fixed point which extends to a π 1 (M) ρ Aut(T k )-action (see Sec. 1, cf. [4, 1]). The existence of a T k -action without fixed points puts constraints on the underlying topology; such as the vanishing of characteristic numbers and the Â-genus (by the Chern Gauss Bonnet type theorems and Atiyah Singer G-index theorem). By making use of the Atiyah Singer index theorem for the twisted signature operator (see Sec. 1), we find new constraints from a fixed point free S 1 -action (see Theorem 1.1). In the comparison with the previous vanishing theorems, the difference is ours provides certain restrictions on the cohomology ring. As applications, we use this to prove some finiteness theorems without assuming a lower bound on volume (see Theorems 1.4 and 1.5). Let L = L 0 + L 1 + L + +L [ n ] denote the Hirzebruch L-polynomial of M which is the universal polynomial on the Pontryagin classes of M associated to x the power series tanh x,wherel 0 =1,L 1 = 1 3 p 1,L = 1 45 (7p p 1 )andl 3 = (6p 3 13p 1 p +p 3 1 ),... (cf. [16]). Theorem 1.1. Let M be a compact n-manifold of finite fundamental group. If M admits a fixed point free S 1 -action, then for all c H (M) and 0 m [ n ], In particular, (c n )[M] =0. (c n m L m )[M] =0. The above ([ n ] + 1) vanishing equations constitute certain constraints on the cohomology ring structure. In this paper, we will restrict ourself only to applications related to the first two equations i.e. m =0,1 (see Theorems 1.4 and 1.5). Since a T k -action without fixed point implies a fixed point free circle subgroup action, one immediate consequence of Theorem 1.1 is Theorem 1.. Let M be a compact symplectic manifold of finite fundamental group. If M admits a (π 1 -invariant) T k -action, then the fixed point set is not empty. A special case of Theorem 1. (i.e. if the S 1 -action also preserves the symplectic structure) is a well known result (cf. p. 150 [17], Theorem 5.5). Together with the fibration theorem mentioned above, we conclude Theorem 1.3. Let M be a compact symplectic n-manifold of finite fundamental group. If M admits a metric with sectional curvature, λ sec M Λ, and diameter, diam (M) d, then vol (M) v(n, λ, Λ,d)>0. By the Cheeger s finiteness theorem, we conclude

3 Fixed Point Free Circle Actions and Finiteness Theorems 77 Theorem 1.4. The collection of compact symplectic n-manifolds of finite fundamental groups such that λ sec Λ and diam d contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. Theorem 1.1 also plays a crucial role in the proof of the following finiteness theorem (see Remark 1.3). Let S Λ,d λ, (n) be the subset of MΛ,d λ, (n) consisting of simply connected manifolds, where M Λ,d λ, (n) denote the set of n-manifolds with λ sec Λ and diam d. Note that S Λ,d λ, (n) may contain infinitely many homeomorphic types for n 7(cf.[1,7]). Theorem 1.5. For n 6, S Λ,d λ, (n) contains only finitely many diffeomorphism types depending only on n, λ, Λ and d. Except Theorem 1.1, the other crucial ingredient of Theorem 1.5 is that H (M) has only finitely many possible isomorphism classes depending only on n, λ, Λ, d (Theorem.4, cf. [9, 0]). Note that for the collapsed manifolds in Theorem 1.5, this implies a bound on the total Betti number (compare [10]). We point it out that a bound on the total Betti number is inadequate for Theorem 1.5 (see Remark 1.3). Theorem 1.5 for n = 3,4,5 are indeed easy consequence from the above discussion and the classification for simply connected 5-manifolds []. First, a compact simply connected 3-manifold is a sphere if it admits a free point free circle action. Secondly, M S Λ,d λ, (4) has volume bounded below by a positive constant and thus the Cheeger s finiteness theorem applies. This is because otherwise M admits a T k -action without fixed point and thus zero Euler number, not possible. By the classification for compact simply connected (smooth) 5-manifolds [], the second homology group and the Stiefel Whitney class w completely determines the diffeomorphism type. Clearly, the finiteness isomorphism classes for the second homotopy groups [0] implies the finiteness for w (Hurewicz theorem). Compact simply connected (smooth) 6-manifolds has been classified by [7] and a complete list of invariants, I(M), is given. By employing Theorem 1.1, we show that if a collection, C, of compact simply connected 6-manifolds which admit circle actions without fixed point has a uniform bound on total Betti numbers and only finitely many (isomorphic) second homology groups, then {I(M),M C}is a finite set and therefore C contains finitely many diffeomorphism types. We conclude the introduction with the following remarks. Remark 1.1. Theorem 1.1 is false if one removes the condition of finite fundamental group without imposing further restriction (e.g. S T will be a counterexample to the assertion that c =0forallc H (S T )). Remark 1.. From the above discussion, it is clear that Theorem 1.5 for n =4 is valid if the fundamental groups are finite. If one removes the upper bound on curvature, then S,d λ, (n) (n 4) contains finitely many homeomorphism types (see

4 78 F. Fang & X. Rong Lemma.1, [9]). Recently, [13] show that S,d 0, (6) contains infinitely many diffeomorphic types ([13], Corollary 3.9). Remark 1.3. There are compact simply connected 5-manifolds whose second homology group is isomorphic to Z p Z p for all p 1 [3]. This implies that to prove Theorem 1.5 for n = 5, a bound on the total Betti number is inadequate information. Moreover, from Remark 1. [13] one sees that information in Theorem 1.1 is required to prove Theorem 1.5 for n =6. Finally, we mention that after this work is complete, we were informed that the case of Theorem 1.1 for n = 3 (for simply connected) was also proved in [14] by a different method; using which W. Tuschmann recently gave a proof of Theorem 1.5 [4]. The rest of the paper is organized as follows. In Sec. 1, we prove Theorem 1.1. In Sec., we will prove Theorem 1.5 (for n =6).. Proof of Theorem The twisted signature operator and the G-index The general reference for this subsection is [16] (Chapter 3). Let Cl(M) denote the Clifford algebra of a Riemannian n-manifold M. Wefirst assume M is orientable and n is even. Then the volume element, w, introduces a Z -graded algebra on Cl(M), Cl(M) =Cl + (M) Cl (M), with Cl ± (M) = (1 ± w) Cl(M). Let E be a vector bundle over M, andletddenote the Dirac operator on Γ(Cl(M) E)=Γ(Cl + (M) E) Γ(Cl (M) E). The twisted signature operator, D + E, is the restriction of D, D + E :Γ(Cl+ (M) E) Γ(Cl (M) E) and whose index, ind (D + E )=dim(ker(d+ E )) dim(coker(d+ E )). By the Atiyah Singer index theory, ind(d + E )=sig(m:e)={ch(e) L(M)}[M], where ch(e) is the Chern character of E and L(M) isthel-genus of M which is the universal polynomial on the Pontryagin classes of M associated to the power x tanh x. series We now assume that M admits a G-action (G is a compact Lie group) and E is G-equivariant, i.e., G acts on E such that for each x M, the fiber, E x,is mapped linearly by g G into the fiber E g x. Then, the Dirac operator satisfies D(gs) =gd(s) fors Γ(Cl(M) E) and thus G acts on ker(d + E )andcoker(d+ E ). The G-index of D + E is (formally) defined by (as a representation of G) ind G (D + E )=ker(d+ E ) coker(d+ E )

5 Fixed Point Free Circle Actions and Finiteness Theorems 79 and ind G (D + E ) = 0 implies ind(d+ E )=dim(ker(d+ E )) dim(coker(d+ E )) = 0. For all g G,theg-index of D + E is ind g(d + E )=tr g(ind G (D + E )) = tr g(ker(d + E )) tr g (coker(d + E )). By the Atiyah Singer G-index theorem, ind g(d + E ) = 0 if the fixed point set by g is empty (cf. [16], p. 61). Note that ind G (D + E )=0ifind g(d + E )=0 for all g G (cf. [], p. 16, p. 3). Lemma.1. Let M be a compact orientable even-dimensional manifold. Suppose M admits a fixed point free S 1 -action. If E is a S 1 -invariant complex vector bundle over M, then ind (D + E )=0. Proof. From the above discussion, it suffices to show that for all t S 1,ind t (D + E ) = 0. Since M is compact, there are only finitely many elements, t 1,...,t k in S 1 whose fixed point sets are not empty. By the Atiyah Singer G-index theorem, for all t t i,ind t (D + E ) = 0. By the continuity, ind t i (D + E ) = 0 for all 1 i k... Proof of Theorem 1.1 Lemma.. Let M be a compact simply connected manifold. Assume that M admits a S 1 -action. Then for all c H (M), there exists a S 1 -equivariant complex line bundle, E c,over M, whose Chern class is c 1 (E c )=c. Note that Lemma. is false if one removes the condition of simply connect without imposing further restriction. Proof of Lemma.. Let Vect (S 1,U(1))(M) denote the set of equivalence classes of S 1 -equivariant complex line bundles over M. ForaspaceY, let Vect U(1) (Y ) denote the equivalence classes of complex line bundles over Y.Asusual,letM S 1 = M S 1 ES 1,whereES 1 BS 1 is the universal S 1 principal bundle. There is a natural transformation φ : Vect (S 1,U(1))(M) Vect U(1) (M S 1) by sending a S 1 -equivariant complex line bundle p : E M to the complex line bundle p S 1 : E S 1 M S 1. Theorem 1.1 of [15] implies that φ is an isomorphism. By choose a base point of ES 1, we get a natural injection i : M M S 1. This induces a natural transformation i : Vect U(1) (M S 1) Vect U(1) (M). Note that the composition i φ : Vect (S1,U(1))(M) Vect U(1) (M) coincides with the forgetful transformation from S 1 -equivariant complex line bundles to complex line bundles. There is a commutative diagram: Vect (S 1,U(1))(M) c 1 φ H (M S 1) i φ Vect U(1) (M) c1 i H (M )

6 80 F. Fang & X. Rong where c 1 is the first Chern class. It is well-known that Vect U(1) (M) =H (M)and Vect U(1) (M S 1)=H (M S 1). The isomorphisms are exactly given by assigning a complex line bundle to its first Chern class. Therefore, the desired result follows if we can verify that the bottom homomorphism i in the above diagram is surjective. The rest of the proof will use the condition of M (and M S 1) simply connected. From the homotopy exact sequence of the fibration, M ES 1 M S 1 BS 1, we get that π (M S 1) = π (M) Z. By the universal coefficients theorem and the Hurewicz theorem, H (M S 1) = Hom(H (M S 1),Z) = Hom(π (M S 1),Z) = Hom(π (M) Z,Z) = Hom(H (M) Z,Z) = H (M) Z. Observe that i can be identified with the projection to the first factor by the above isomorphism. Now the desired conclusion follows. Proof of Theorem 1.1. Case 1. Assume M is simply connected. For all c H (M), by Lemma. we get a S 1 -invariant complex line bundle, E c,overmwith c 1 (E c )=c. By applying Lemma.1 to E c,weget ind(d + E c )={ch(e c ) L(M)}[M]={e c L(M)}[M]=0 (note that for any complex line bundle L, ch(e)=e c1(e) ). To see the desired vanishing equations, substitute c, in the above equation, by kc and obtain {e kc L(M)}[M] = (1+kc + (kc)! Consider the sum of the top terms: + (kc)n n! ) + (L 0 +L 1 + +L [ n ] )[M] =0. (b 0 k n c n + b 1 k n c n L 1 + b k n 4 c n 4 L + +b [ n ] k n [ n ] c n [ n ] L [ n ] )[M] =0, (.1) 1 where b i = (n i)!. Evaluating (.1) for k =1,,...,[n+ ], we get a system of linear equations in (c n )[M], (c n L 1 )[M],...,(c n [ n ] L [ n ] )[M]. It is easy to see the desired vanishing result follows if the coefficient matrix, A, is invertible. Indeed, the determinant of A is proportional to that of the so-called Vandermonde matrix and thus is non-zero. The following is a computation in the case of n odd (the case

7 Fixed Point Free Circle Actions and Finiteness Theorems 81 of n even is similar): det(a) = [ n+ b 0 b 0 b 1 b b [ n ] b 0 n b 1 n b n 4 b [ n ] b 0 3 n b 1 3 n b 3 n 4 b [ n ] 3 ] n b 1 [ n+ ] n [ ] n 4 [ n+ n+ b b [ n ] n 1 4 n 3 4 n n 1 9 n 3 9 n 5 1 = ( [n+ )n 1 ] ( [n+ )n 3 ] ( [n+ )n 5 ] 1 ([ ]) n +!b 0 b 1 b b [ n ] ([ ]) n + =±!b 0 b 1 b b [ n ] Π (j i ) 0. 1 i<j [ n+ ] Case. In general, given a metric g on M with curvature tensor Ω, and let π : M M be the Riemannian universal covering map of order r. Clearly, the curvature tensor on M is Ω =π (Ω). By the Chern Weil isomorphism, we can write the ith Pontryagin class as p i = P i (Ω), where P i is the corresponding invariant polynomial. Then the relation between the Pontryagin classes of M and M are: p i = P i ( Ω) = P i (π (Ω)) = π (P i (Ω)) = π (p i ) and thus their L-polynomials satisfy L m = π (L m ). For all c H (M), c = π (c) H ( M). Since the lifted S 1 -action on M is almost free, by Case 1 we get, (c n m L m )[M] = c n m L m = 1 c L m = 1 r r ( cn m L m )[ M] =0. M 3. Proof of Theorem The case n =4 We first recall the notion of π 1 -invariant T k -action [1]. Let π : M M be a universal covering map. A π 1 -invariant T k -action on M is a T k -action on M M ]

8 8 F. Fang & X. Rong which extends to a π 1 (M) ρ T k -action on M, whereρ:π 1 (M) Aut(T k )isa homomorphism and π 1 (M) ρ T k is the skew-product. Note that ρ = id if and only if the T k -action on M is the lift of a T k -action on M. Ifπ 1 (M) is finite, then the notion of π 1 -invariant T k -action is equivalent to that of pure F -structure defined by [4, 5]. Theorem 3.1 [4]. Let F Λ,d λ, (n) MΛ,d λ, (n) consist of manifolds with finite fun- (n) has damental groups. There exists ɛ = ɛ(n, Λ,λ,d) > 0 such that if M F Λ,d λ, volume less than ɛ, then M admits an almost isometric π 1 -invariant T k -action without fixed point. Note that in general, if M M Λ,d λ, (n) has volume less than ɛ, thenmadmits a so-called pure nilpotent Killing structure (cf. [4]). Corollary 3.. Any M F Λ,d λ, (4) has volume ɛ(4, Λ,λ,d). Proof. If false, then by Theorem 3.1 the universal covering, M, ofm F Λ,d λ, (4) admits a T k -action without fixed point. This implies the Euler characteristic of M vanishes; a contradiction. 3.. The case n =5 By [0], the set of the qth rational homotopy groups, {π q (M) Q, M M Λ,d λ, (n)}, contains only finitely many isomorphism classes depending only on n, Λ, λ, d and q. Indeed, by a trivial observation one sees the exactly same proof in [0] shows the isomorphism finiteness for {π q (M), M M Λ,d λ, (n)} with a few exceptional value of q (precisely when π q (O(n)) is not finite), provided π q (M) is finitely generated. In [9], we obtain the following: Theorem 3.3 ([9, 0]). {π q (M), M M Λ,d λ, (n)}, contains only finitely many isomorphism classes depending on n, λ, Λ,d and q, provided π q (M) is finitely generated (e.g. λ 0 or π 1 (M) is finite). By the classification theorem of simply connected 5-manifold [], the complete invariants determining diffeomorphism type are H and the Stiefel Whitney class w. On the other hand, for a simply connected manifold M, π (M) = H (M) (Hurewicz theorem). Hence, by Theorem.4 {H (M), M S Λ,d λ,. (5)} contains only finitely many isomorphism classes depending only on λ, λ, and d. Notethat, w (M) H (M,Z ) has at most r -different values, r = rank H (M,Z ). Therefore, S Λ,d λ,. (5) contains only finitely many diffeomorphic types The case n =6 First, the homology groups alone cannot determine the diffeomorphism type up to finite ambiguity (see Remark 1.3). A complete classification of simply connected

9 Fixed Point Free Circle Actions and Finiteness Theorems 83 6-manifolds was given by Zǔbr [7]. For convenience we briefly describe the main results below. Notations. The set of natural integers is denoted by N, the same set with added element, byˆn. For any integer m N, letρ m (resp. i m ) denote the mod m (resp. multiplying m) homomorphism. Suppose G is a finitely generated abelian group and w : G Z a homomorphism. By h(w) we denote the largest m ˆN for which there exists ω Hom (G, Z m) with ρ (ω) =w. Clearly, if h(w) =,thenw tor (G) =0and otherwise h(w) tor (G). Wedenoteby(w) the set of all such ω s with m = h(w). For any simply connected closed smooth 6-manifold M, we have the following set of invariants: (3.4.1) a group G = H (M); (3.4.) a number b = rank H 3 (M); (3.4.3) an abstract orientation class µ H 6 (K(G, )), the image of the orientation class [M] by the canonical homomorphism H 6 (M) H 6 (K(G, )), where K(G, ) is a Eilenberg Maclane space. (3.4.4) a cohomology class(the second Stiefel Whitney class) w = w (M) H (M,Z )=Hom(G, Z ) (3.4.5) a cohomology class (the first Pontryagin class) p = p 1 (M) H 4 (M) in view of the Poincaré duality, p will be regarded as a homology class p G. (3.4.6) two exotic invariants a number and a cohomology class E = E ω (M) Z h(w) 1 e = e ω (M) H 4 (M,Z h(w) 1), depending on the choice of the class ω (w). The class e can be also regarded as an element of G/ h(w) 1, in view of Poincaré duality. Suppose that (G, b, µ, w, p, e, E) and(g,b,µ,w,p,e,e ) are two sets of invariants as above so that b = b,andthatgis isomorphic to G. Following [7], we say that an isomorphism φ : G G induces an isomorphism of invariants if, identifying G and G by φ, µ = µ, w = w, p = p and also e = e + µ (ωx + x ) and E = E + e, x µ, ω x +3i 1 (ωp(x)) + x3

10 84 F. Fang & X. Rong where e, E(resp. e,e ) correspond to some ω(resp. ω ), x is defined by ω ω = i x,andp :H (M,Z h(w) 1) H (M,Z h(w)) the Pontryagin square. In particular, if h(w) =,P(x)=x. This defines an equivalence relations about the above list of invariants. Let I(M) = (H (M),b 3 (M),µ M,w (M),p 1 (M),E ω (M),e ω (M)) denote the equivalence classes of the invariants. It turns out that I(M) is the invariant for simply connected closed smooth 6-manifolds, up to oriented diffeomorphism. Theorem 3.5 ([7]). If the sets of invariants, corresponding to simply connected closed oriented smooth 6-manifolds M and M, are isomorphic, then these manifolds are oriented-diffeomorphic. Now we will apply the above theorem to prove Proposition 3.6. Let S k,s denote the collection of simply connected 6-manifolds, M, satisfying (3.6.1) M admits a circle action without fixed point. (3.6.) b (M)+b 3 (M) k and tor H (M) s. Then S k,s is a finite set whose order is upper bounded by c(s, k). Lemma 3.7. Let M be a compact simply connected 6-manifold. If M admits a circle action without fixed point, then (3.7.1) For all α, β H (M),αβ =0. (3.7.) the first Pontryagin class p 1 is a torsion element. (3.7.3) µ is a torsion element in H 6 (K(G, )), where G = H (M). Proof. For all α H (M ), by Theorem 1.1 we get α 3 = 0. Hence, for α and β H (M), (α + β) 3 (α 3 + β 3 )=3αβ(α + β) = 0. Similarly, 3αβ(α β) =0. Sum these we get α β = αβ =0foranyα, β. By duality this implies that α =0 for all α. Therefore, αβ = 1 {(α + β) (α + β )} =0forallα, β. Also by Theorem 1.1, p 1 (M)α =0foranyα H (M). Clearly, from Poincare duality this shows that the image of p 1 (M) inh 4 (M,Q) vanishes and thus (3.7.). If µ =, we will show that there are x, y, z H (K(G, )) such that xyz, µ 0, a contradiction to (3.7.1). Put G = Z m T,whereT is the torsion subgroup. Note that K(G, ) = K(Z m, ) K(T,) and H (K(T,)) = 0. Let e 1,e,...,e m be a basis for H (K(Z m, )). Clearly, {e i e j e k, 1 i, j, k m} forms a basis for H 6 (K(Z m, )). Moreover, an algebraic dual basis, {e ijk, 1 i, j, k m}, forh 6 (K(Z m,)) may be uniquely defined by e a e b e c,e ijk =1ifand only if (a, b, c) =(i, j, k) up to a possible permutation. By the Kunneth formula, {e ijk, 1 i, j, k m} generates the torsion free part of H 6 (K(G, )). Therefore, µ = a ijk e ijk for some not all zero integers a ijk modulo torsion. Let x = e i,y = e j and z = e k if a ijk 0. By the definition, xyz, µ = a ijk 0. A contradiction.

11 Fixed Point Free Circle Actions and Finiteness Theorems 85 Proof of Proposition 3.6. By Theorem 3.5, it suffices to prove that the set of invariants, {I(M), M S k,s }, is a finite set of order less than c(k, s) (hence the number of the equivalent classes in S k,s is bounded by c(k, s)). Case 1. h(w) <. Then h(w) tor(g). By Lemma 3.7, it is easy to see that all invariants defined in (3.4.1) (3.4.6) have finitely many possibilities of number depending only on k and s except for µ. By the Kunneth formula, the order of tor(h 6 (K(G, ))) is bounded above by a constant depending only on k and s. Case. h(w) =. Clearly, we only need to consider invariants in (3.4.6). From (7) and (1) in [7], one sees, for ω H (M), that the invariant E ω (M) =(p 1 (X)ω ω 3 )[M] and i 4 e ω (M) =p 1 (M)+ω =p 1 (M). By Theorem 1.1 and (3.6.1), from the above we get E ω (M)=0ande ω (M) is a torsion element in H 4 (M) = H (M). Now we are ready to complete the proof of Theorem 1.5. Proof of Theorem 1.5. By (3.1) and (3.), it remains to check Theorem 1.5 for n = 6. We divide S Λ,d λ, (6) into two parts: SΛ,d λ, (6) = S ɛ S f,wherem S ɛ has volume less than ɛ = ɛ(6, Λ,λ,d) as in Theorem 3.1 and S f is the complement of S ɛ. First, by the Cheeger s finiteness theorem, S f contains only finitely many diffeomorphism types depending on Λ,λ and d. Since M S ɛ admits a T k -action without fixed points (Theorem 3.1), M admits an almost free circle action. Since the Euler characteristic of M vanishes, b 3 = +b. Hence, by Theorem 3.3 we can assume S ɛ S k,s for some k and s depending on Λ,λ,d. By now, the proof is complete by Proposition 3.6. Acknowledgments The first author is supported by NSFC Grant , RFDP, CNPq and the Qiu-Shi Foundation and the second author is supported partially by NSF Grant DMS and Alfred P. Sloan Research Fellowship. References [1] S. Aloff and N. R. Wallach, An infinite family of 7-manifolds admitting positive curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975) [] D. Barden, Simply connected five manifolds curvature, Ann. of Math 8 (1965) [3] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 9 (1970) [4] J. Cheeger, K. Fukaya and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J.A.M.S. 5 (199) [5] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bound I, J. Diff. Geom. 3 (1986) [6] J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded II, J. Differential Geom. 3 (1990)

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