aa r X i v : . [ m a t h . G T ] S e p AN ASPHERICAL 5-MANIFOLD WITH PERFECTFUNDAMENTAL GROUP
J.A. HILLMAN
Abstract.
We construct aspherical closed orientable 5-manifoldswith perfect fundamental group. This completes earlier work on
P D n -groups with pro- p completion a pro- p Poincar´e duality groupof dimension ≤ n −
2. We also consider the question of whetherthere are any examples with “dimension drop” 1.
The paper [6] considers the phenomenon of dimension drop on pro- p completion for orientable Poincar´e duality groups. Products of aspher-ical homology spheres, copies of S and copies of the 3-manifold M ( K )obtained by 0-framed surgery on a nontrivial prime knot give exam-ples of aspherical closed orientable n -manifolds N such that the pro- p completion of π ( N ) is a pro- p Poincar´e duality group of dimension r ,for all n ≥ r + 2 and r ≥
0, except when n = 5 and r = 0. (This gapreflects the fact that 5 is not in the additive semigroup generated by 3and 4, dimensions in which aspherical homology spheres are known.)We fill this gap in Theorem 2 below. Modifying the constructionof [9] gives an aspherical closed 4-manifold with perfect fundamentalgroup and non-trivial second cohomology. The total space of a suitable S -bundle over this 4-manifold has the required properties. We thenapply Theorem 2 to refine the final result of [6]. No such examples withdimension drop n − r = 1 are known as yet. The lowest dimension inwhich there might be such examples is n = 4, and we consider this casein the final section. If they exist, products with copies of S would giveexamples in all higher dimensions.In this paper all manifolds and P D n -groups shall be orientable. If G is a group then G ′ and G [ n ] shall denote the commutator subgroupand the n th term of the lower central series, respectively. Let G ( Z ) = ∩ λ ∈ Hom ( G, Z ) Ker( λ ). Then G/G ( Z ) is the maximal torsion-free abelianquotient of G . If p ≥ X p ( G ) be the verbal subgroup generated byall p th powers in G . Mathematics Subject Classification.
Primary 57M05, Secondary 20F99,20J99.
Key words and phrases. aspherical, perfect group, Poincar´e duality group, pro- p completion. an aspherical 5-manifold with π perfect Let X be a compact 4-manifold whose boundary components arediffeomorphic to the 3-torus T . A Dehn filling of a component Y of ∂X is the adjunction of T × D to X via a diffeomorphism ∂ ( T × D ) ∼ = Y .If the interior of X has a complete hyperbolic metric then “most”systems of Dehn fillings on some or all of the boundary componentsgive manifolds which admit metrics of non-positive curvature, and thefundamental groups of the cores of the solid tori T × D map injectivelyto the fundamental group of the filling of X , by the Gromov-Thurston2 π -Theorem. (Here “most” means “excluding finitely many fillings ofeach boundary component”. See [1].) Theorem 1.
There are aspherical closed -manifolds M with perfectfundamental group and H ( M ; Z ) = 0 .Proof. Let M = S \ T be the complete hyperbolic 4-manifold withfinite volume and five cusps considered in [7] and [9], and let M be acompact core, with interior diffeomorphic to M . Then H ( M ; Z ) ∼ = Z , χ ( M ) = 2 and the boundary components of M are all diffeomorphic tothe 3-torus T . There are infinitely many quintuples of Dehn fillings ofthe components of ∂M such that the resulting closed 4-manifold is anaspherical homology 4-sphere [9]. Let c M be one such closed 4-manifold,and let N ⊂ c M be the compact 4-manifold obtained by leaving oneboundary component of X unfilled. We may assume that the interiorof N has a non-positively curved metric, and so N is aspherical. TheMayer-Vietoris sequence for M = N ∪ T × D gives an isomorphism H ( T ; Z ) ∼ = H ( N ; Z ) ⊕ H ( T ; Z ) . Let { x, y, z } be a basis for H ( T ; Z ) compatible with this splitting.Thus x represents a generator of H ( N ; Z ) and maps to 0 in the sec-ond summand, while { y, z } has image 0 in H ( N ; Z ) but generates thesecond summand. Since the subgroup generated by { y, z } maps injec-tively to π ( c M ) [1], the inclusion of ∂N into N is π -injective. Let φ be the automorphism of ∂N = T which swaps the generators x and y , and let P = N ∪ φ N . Then P is aspherical and χ ( P ) = 2 χ ( N ) = 4.A Mayer-Vietoris calculation gives H ( P ; Z ) = 0, and so π = π ( P ) isperfect and H ( P ; Z ) ∼ = Z . (cid:3) Are there compact complex surfaces with perfect fundamental groupand which are uniformized by the unit ball or the bidisc? (Such man-ifolds are aspherical, and have middle cohomology of rank >
0, sincethey are K¨ahler.)
N ASPHERICAL 5-MANIFOLD WITH PERFECT FUNDAMENTAL GROUP 3
Theorem 2.
There are aspherical closed -manifolds with perfect fun-damental group.Proof. Let P be an aspherical closed 4-manifold with π ( P ) perfect and H ( P ; Z ) = 0, as in Theorem 1. Let e generate a direct summand of H ( π ; Z ) = H ( P ; Z ), and let E be the total space of the S -bundleover P with Euler class e . Then E is an aspherical 5-manifold, and G = π ( E ) is the central extension of π ( P ) by Z corresponding to e ∈ H ( π ( P ); Z ). The Gysin sequence for the bundle (with coefficientsin F p ) has a subsequence0 → H ( E ; F p ) → H ( P ; F p ) → H ( P ; F p ) → H ( E ; F p ) → . . . in which the mod - p reduction of e generates the image of H ( P ; F p ).Since e is indivisible this image is nonzero, for all primes p . Therefore H ( G ; F p ) = H ( E ; F p ) = 0, for all p , and so G is perfect. (cid:3) (From the algebraic point of view, G is a quotient of the universalcentral extension of π ( P ), which is perfect.)Is there an aspherical 5-dimensional homology sphere? If there is anaspherical 4-manifold X with the integral homology of CP then thetotal space of the S -bundle with Euler class a generator of H ( X ; Z )would be such an example. The fake projective planes of [8] are asperi-cal and have the rational homology of CP , but they have nonzero firsthomology. The total spaces of the S -bundles over such fake projec-tive planes and with Euler class of infinite order are aspherical rationalhomology 5-spheres.2. pro- p completions and dimension drop ≥ P D n groupswith pro- p completion a pro- p Poincar´e duality group of lower dimen-sion uses the fact that finite p -groups are nilpotent. Thus if G is agroup with G ′ = [ G, G ′ ] the abelianization homomorphism induces iso-morphisms on pro- p completions, for all primes p . Hence products ofperfect groups with free abelian groups have pro- p completion a freeabelian pro- p group. In this section we shall use Theorem 2 to removea minor constraint on the final result of [6], which excluded a family ofsuch examples. Theorem 3.
For each r ≥ and n ≥ max { r + 2 , } there is an aspher-ical closed n -manifold with fundamental group π such that π/π ′ ∼ = Z r and π ′ = π ′′ .Proof. Let Σ be an aspherical homology 3-sphere (such as the Brieskorn3-manifold Σ(2 , , P and E be as in Theorems 1 and 2. J.A. HILLMAN
Taking suitable products of copies of Σ, P , E and S with each otherrealizes all the possibilities with n ≥ r + 3, for all r ≥ M = M ( K ) be the 3-manifold obtained by 0-framed surgeryon a nontrivial prime knot K with Alexander polynomial ∆( K ) = 1(such as the Kinoshita-Terasaka knot 11 n ). Then M is aspherical,since K is nontrivial [2], and if µ = π ( M ) then µ/µ ′ ∼ = Z and µ ′ isperfect, since ∆( K ) = 1. Hence products M × ( S ) r − give exampleswith n = r + 2, for all r ≥ (cid:3) In particular, the dimension hypotheses in Theorem 6.3 of [6] maybe simplified, so that it now asserts:
Let m ≥ and r ≥ . Then there is an aspherical closed ( m + r ) -manifold M with fundamental group G = K × Z r , where K = K ′ . If m = 4 we may assume that χ ( M ) = 0 , and if r > this must be so. This is best possible, as no
P D - or P D -group is perfect, and noperfect P D -group H has χ ( H ) = 0.3. dimension drop ≤ ? Can we extend Theorem 3 to give examples of
P D n -groups π with π/π ′ ∼ = Z n − and π [2] = π [3] , realizing dimension drop 1 on all pro- p completions? In this section we shall weaken some of these conditions,by considering maps to P D n − groups other than Z n − and requiringonly that pro- p completion be well-behaved for some primes p .There are clearly no such examples with n = 2, since the only pro- p Poincar´e duality group of dimension 1 is b Z p . The next lemma rules outa direct analogue of Theorem 3 with n = 3 and r = n − Lemma 4.
Let π be a P D -group such that π/π ′ ∼ = Z r and π ′ = π ′′ .Then r = 0 , or .Proof. We may assume that r >
1. The augmentation π -module Z has a finitely generated projective resolution C ∗ of length 3. Let Λ = Z [ π/π ′ ], and let D ∗ = Λ ⊗ π C ∗ and D ∗ = Hom Λ ( D −∗ , Λ). Then H p ( D ∗ ) ∼ = H − p ( D ∗ ), by Poincar´e duality for π . We have H ( D ∗ ) = 0,since π ′ is perfect, H ( D ∗ ) ∼ = H ( D ∗ ) ∼ = Ext ( Z , Λ) = 0, since r > H ( D ∗ ) ∼ = H ( D ∗ ) ∼ = Hom Λ ( Z , Λ) = 0, since r >
0. Therefore D ∗ is a finitely generated free resolution of the augmentation π/π ′ -module. Since D ∗ has length 3 and H ( D ∗ ) ∼ = H ( D ∗ ) = Z , we musthave r = 3. (cid:3) The values r = 0 , , , M (11 n ) and the 3-torus ( S ) , respectively. N ASPHERICAL 5-MANIFOLD WITH PERFECT FUNDAMENTAL GROUP 5
Pro- p Poincar´e duality groups of dimension 2 are also well under-stood. This class (of so-called
Demuˇskin groups ) includes all pro- p completions of P D -groups, but is somewhat larger. Theorem 5.
The pro- p completion of a P D -group G is not a Demuˇskingroup, for any odd prime p .Proof. If G is a discrete group and p is an odd prime then the ker-nel of cup product from ∧ H ( G ; F p ) to H ( G ; F p ) is isomorphic to G [2] X p ( G ) /G [3] X p ( G ) [4], and a similar result holds for pro- p groups[11]. Hence the dimension of the image of this cup product is deter-mined by the p -lower central series. In particular, if the pro- p comple-tion of G is a Demuˇskin group the image of ∧ H ( G ; F p ) in H ( G ; F p )is 1-dimensional. If G is a P D -group this is impossible, by the non-singularity of Poincar´e duality. (cid:3) This argument can be modified to apply for p = 2 also.In higher dimensions the most convenient candidates for quotientsare torsion-free nilpotent groups. A finitely generated nilpotent group ν of Hirsch length h has a maximal finite normal subgroup T ( ν ), withquotient a P D h -group. Moreover, ν/T ( ν ) has nilpotency class < h ,and is residually a finite p -group for all p , by Theorem 4 of Chapter1 of [10]. Thus the pro- p completion of ν is a pro- p Poincar´e dualitygroup for all p prime to the order of T ( ν ).We shall focus on the first undecided case, n = 4. If G [ k ] /G [ k +1] isfinite, of exponent e , say, then so are all subsequent subquotients ofthe lower central series, by Proposition 11 of Chapter 1 of [10]. Thus if G is a P D -group such that G/G [3] has Hirsch length 3 and G [3] /G [4] is finite then, setting ν = G/G [3] , the canonical projection to ν/T ( ν )induces isomorphisms on pro- p completions, for almost all primes p .Taking products of one such group with copies of Z would give similarexamples with dimension drop 1 in all higher dimensions.We consider first the case when the quotient P D -group is abelian. Theorem 6.
Let G be a P D -group. Then there is an epimorphismfrom G to Z which induces isomorphisms on pro- p completions, foralmost all p , if and only if β ( G ) = 3 and the homomorphism from ∧ H ( G ; Z ) to H ( G ; Z ) induced by cup product is injective.If these conditions hold then χ ( G ) ≥ and G ( Z ) is not F P .Proof. The homomorphism from ∧ H ( G ; Z ) to H ( G ; Z ) induced bycup product is a monomorphism if and only if G [2] /G [3] is finite [4].Suppose that there is such an epimorphism. Since G/G ′ and G [2] /G [3] are finitely generated, it follows easily that β ( G ) = 3 and G [2] /G [3] isfinite. Thus the conditions in the first assertion are necessary. J.A. HILLMAN
If they hold then G [2] /G [3] is finite, and the rational lower centralseries for G terminates at G ( Z ) [4]. Therefore G ( Z ) /G [3] is the tor-sion subgroup of G/G [3] . Thus if p is prime to the order of G ( Z ) /G [3] then the canonical epimorphism from G to G/G ( Z ) ∼ = Z induces anisomorphism of pro- p completions.The image of cup product from ∧ H ( G ; Z ) to H ( G ; Z ) has rank 3,and must be self-annihilating, since ∧ ( Z ) = 0. Hence β ( G ) ≥
6, bythe non-singularity of Poincar´e duality, and so χ ( G ) ≥ G ( Z ) were F P then it would be a P D -group, and so F P , byTheorem 1.19 of [3]. But then χ ( G ) = 0, since χ is multiplicative inexact sequences of groups of type F P . (cid:3) The conditions in this theorem are detected by de Rham cohomol-ogy, which suggests that we should perhaps seek examples among thefundamental groups of smooth manifolds with metrics of negative cur-vature. Are there any such groups? Since χ ( G ) = 0, no such group issolvable or a semidirect product H ⋊ Z with H of type F P .There are parallel criteria in the nilpotent case.
Theorem 7.
Let G be a P D -group. Then there is an epimorphismfrom G to a nonabelian nilpotent P D -group which induces isomor-phisms on pro- p completions, for almost all p , if and only if β ( G ) = 2 ,cup product from ∧ H ( G ; Z ) to H ( G ; Z ) is , and G [3] /G [4] is finite.If these conditions hold then χ ( G ) ≥ .Proof. The conditions are clearly necessary. Suppose that they hold.Then
G/G ( Z ) ∼ = Z . The homomorphism from the free group F (2) to G determined by elements of G representing a basis for this quotient in-duces a monomorphism from F (2) /F (2) [3] to G/G [3] with image of finiteindex, by the cup-product condition [4]. Thus G ( Z ) /G [3] is nilpotentand virtually Z . Let T be the preimage in G of the torsion subgroupof G ( Z ) /G [3] . This is characteristic in G , and G/T is a non-abelianextension of Z by Z . Hence it is a nilpotent P D -group. If G [3] /G [4] is finite, then the quotient epimorphism to G/T induces isomorphismson pro- p completions, for almost all p .If these conditions hold then the natural map from H ( G ; Q ) to H ( G/G [3] ; Q ) ∼ = Q is an epimorphism, by the 5-term exact sequenceof low degree for the homology of G as an extension of G/G [3] by G [3] .Hence β ( G ) ≥
2, and so χ ( G ) ≥ (cid:3) Are there any such groups? Theorem 1.19 of [3] again implies that T cannot be F P . If χ ( G ) = 0 then T cannot even be finitely generated,by Corollary 6.1 of [5] (used twice). For otherwise T would be Z , so G would be nilpotent, and G [3] /G [4] would be infinite. (If Γ is a lattice in N ASPHERICAL 5-MANIFOLD WITH PERFECT FUNDAMENTAL GROUP 7 the nilpotent Lie group
N il then the first two conditions of Theorem6 hold, and χ (Γ) = 0, but Γ [3] ∼ = Z and Γ [4] = 1.) Is there such a groupwith T free of infinite rank?The possibility of no dimension drop ( n = r ) is realized by the n -torus ( S ) n , for any n . Are there any examples in which the dimensionincreases on pro- p completion, i.e., with n < r ? It again follows fromLemma 4 that if π/π ′ ∼ = Z r and π ′ is perfect then n ≥
4. Moreover, ifthere is such a
P D -group G then χ ( G ) ≥
2, by the non-singularity ofPoincar´e duality.
Acknowledgment.
We would like to thank Bruno Martelli for sug-gesting the use of manifolds such as the manifold N of Theorem 1. J.A. HILLMAN
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