aa r X i v : . [ m a t h . A T ] J u l ON CW-COMPLEXES OVER GROUPS WITH PERIODICCOHOMOLOGY
JOHNNY NICHOLSON
Abstract. If G has 4-periodic cohomology, then D2 complexes over G aredetermined up to polarised homotopy by their Euler characteristic if and onlyif G has at most two one-dimensional quaternionic representations. We use thisto solve Wall’s D2 problem for several infinite families of non-abelian groupsand, in these cases, also show that any finite Poincar´e 3-complex X with π ( X ) = G admits a cell structure with a single 3-cell. The proof involvescancellation theorems for Z G modules where G has periodic cohomology. Introduction
In the 1960s, C. T. C. Wall [49] considered the question of whether a finitePoincar´e n -complex X could be given a cell structure with a single n -cell. Bysubtracting an n -cell e n representing the fundamental class, it can be verified that X = K ∪ e n where K is a finite n -complex which is cohomologically ( n − H n ( K ; M ) = H n ( K ; M ) = 0 for all finitely generated Z [ π ( K )]-modules M , knownas a D( n − K were homotopy equivalent to a finite ( n − X = K ∪ e n would give a cell structure with a single n -cell.Wall proved this for arbitrary D n complexes provided n > n = 1 [35], [36]. In particular, every finite Poincar´e n -complex admits a cell structure with a single n -cell except possibly if n = 3. Thequestion for D2 complexes remains open and is known as Wall’s D2 problem: D2 problem.
Is every D2 complex homotopy equivalent to a finite -complex? We can view this as being parametrised by groups G by saying that G hasthe D2 property if every D2 complex X with π ( X ) = G is homotopic to a finite2-complex. So, if the fundamental groups of Poincar´e 3-complexes had the D2property, then every finite Poincar´e 3-complex would have a cell structure witha single 3-cell. Since the finite fundamental groups of Poincar´e 3-complexes have4-periodic cohomology, the D2 property for such groups is of special interest.Our main result is the following partial classification of D2 complexes whosefundamental group has 4-periodic cohomology. Let m H ( G ) denote the number ofcopies of H in the Wedderburn decomposition of R G for a finite group G , i.e. thenumber of one-dimensional quaternionic representations. Theorem A. If G has 4-periodic cohomology, then D2 complexes over G are de-termined up to (polarised) homotopy by their Euler characteristic if and only if m H ( G ) ≤ . Mathematics Subject Classification.
Primary 57M20; Secondary 57P10, 57Q12, 20C05.
Recall that every finite 2-complex over G is homotopy equivalent to the Cayleycomplex X P of some presentation P = h s , · · · , s n | r , · · · , r m i of G which hasthat χ ( X P ) = 1 − def( P ) where def( P ) = n − m is the deficiency of P .If G is 4-periodic then, as will see in Theorem 3.1, the minimal Euler character-istic of a D2 complex over G is one. If G satisfies the D2 property, then TheoremA implies that there exists a finite 2-complex X P over G with χ ( X P ) = 1 and so G has a presentation P with def( P ) = 0, i.e. a balanced presentation. If alternatively m H ( G ) ≤
2, then both D2 complexes and finite 2-complexes over G are determinedby their Euler characteristic. Since χ ( X ) ≥ χ takes the samevalues if and only if G has a balanced presentation. Hence we conclude: Theorem B.
Suppose G has -periodic cohomology. Then: (i) If G has the D2 property, then G has a balanced presentation (ii) If G has a balanced presentation and m H ( G ) ≤ , then G has the D2 property. Now suppose X is a finite Poincar´e 3-complex with G = π ( X ) finite. By thediscussion above and Theorem B, we know that X has a cell structure with a single3-cell provided m H ( G ) ≤ G has a balanced presentation. Conversely, if sucha cell structure exists, then X = K ∪ e for K = X (2) a finite 2-complex. Since χ ( X ) = 0 from Poincar´e duality, we know that χ ( K ) = 1 and so G has a balancedpresentation by the previous paragraph. In particular, we have: Theorem C. If X is a finite Poincar´e 3-complex with G = π ( X ) finite. Then: (i) If X has a cell structure with a single 3-cell, then G has a balanced presentation (ii) If G has a balanced presentation and m H ( G ) ≤ , then X has a cell structurewith a single 3-cell. Note that not all 4-periodic groups G are the fundamental groups of finitePoincar´e 3-complexes [9], [27] and so Theorem C can be deduced from a slightlyweaker statement than Theorem A.We show in Section 5 that the 4-periodic groups G for which m H ( G ) ≤ G × C n forany G listed with ( n, | G | ) = 1. We use the notation of Milnor [29]:(I) C n , D n +2 for n ≥ Q , Q , Q , Q , e T , e O, e I (III) D (2 n , D (2 n ,
5) for n ≥ P ′ · n for n ≥ P ′′ n for n ≥ Q (16; m, n ) for m > n ≥ Eichler condition , i.e. m H ( G ) = 0. The work of W. J. Browning [3] can be appliedin these cases and this has led to proofs of the D2 property for finite abelian groups[3], [5], [11], dihedral groups [25], the polyhedral groups T , O , I [15] (exhaustingthe finite subgroups of SO (3)) and various metacyclic groups [21], [33], [46]. It hasalso been shown for various infinite abelian groups [12], [13] and free groups [20].However, the case of finite groups which do not satisfy the Eichler conditionhas proven much more elusive. This includes all 4-periodic groups apart from thegroups in (I) and so consequently little progress has been made on applying theD2 problem to determine which finite Poincar´e 3-complexes admit cell structureswith a single 3-cell. The only result to date comes from F. E. A. Johnson [18] who N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 3 made use of cancellation results for Z G modules already in the literature [44], [42]to prove the D2 property for many of the groups in (II).The main aim of this article is an expansion of Johnson’s work, including acomplete resolution of the module-theoretic aspects of the problem. By consideringwhich groups have balanced presentations in Section 7, we prove the D2 property forthe groups in (I)-(IV) and many of the groups in (VI). The possibility remains thatsome 4-periodic group does not have a balanced presentation and so, by TheoremA, would be a counterexample to the D2 problem.We now proceed to outline the series of results which will lead to the proof ofTheorem A, many of which may be of independent interest.Let D2 G denote the graph whose vertices are the polarised homotopy types ofD2 complexes over G and whose edges connect X to X ∨ S with the inducedpolarisation. Also let Alg G be the graph whose vertices are the chain homotopytypes of algebraic 2-complexes E = ( F ∗ , ∂ ∗ ) over Z G and whose edges connect E to the complex Σ( E ) formed by replacing F with F ⊕ Z G . In Section 1, we show: Theorem 1.1. If G is a finitely presented group, then there exists an isomorphismof graded trees e C ∗ : D2 G → Alg G which is the same as the cellular chain map X C ∗ ( e X ) when X is a -complex. The generalises the Realisation Theorem of F. E. A. Johnson [19] since it impliesthat every algebraic 2-complex is geometrically realisable, i.e. chain homotopic to C ∗ ( e X ) for X a finite 2-complex, if and only if every ( X, p X ) ∈ D2 G is polarisedhomotopy equivalent to a finite 2-complex, i.e. if G has the D2 property.Recall that, if e K ( R ) is the projective class group, a class [ P ] ∈ e K ( R ) has cancellation if P ⊕ R ∼ = P ⊕ R implies P ∼ = P for all P , P ∈ [ P ]. In Section 2, wedefine the Wall finiteness obstruction χ ( g ) ∈ e K ( Z G ) for a generator g ∈ H ( G ; Z ).By adapting an approach of Johnson [18, Theorem 62.1], we will show: Theorem 3.1. If G has -periodic cohomology and g ∈ H ( G ; Z ) a generator, thenthere is an isomorphism of graded trees Φ :
Alg G → [ χ ( g )] . In order to distinguish it from the class χ ( g ) ∈ e K ( Z G ), we will write [ χ ( g )] torefer to the set of modules P for which P ⊕ Z G i ∼ = χ ( g ) ⊕ Z G j for some i, j ≥ χ ( g )] has cancellation. In Section 4, we prove a cancellation theoremfor projective modules over the integral group rings of finite groups, generalising aresult of the author [31, Theorem A]: Theorem 4.1.
Suppose G is a finite group with H = G/N , ¯ P ∈ LF ( Z G ) and P = ¯ P ⊗ ZN Z ∈ LF ( Z H ) . If m H ( G ) = m H ( H ) and the map Aut( P ) → K ( Z H ) is surjective, then [ ¯ P ] has cancellation if and only if [ P ] has cancellation. The forward direction will follow from [42, Theorem A10] but the converse ismuch more subtle and constitutes the main technical heart of the paper.
JOHNNY NICHOLSON
In Section 5, we show that 4-periodic groups G with m H ( G ) ≤ H for which m H ( G ) = m H ( H ) ≤
2. For such groups H , themap Z H × = Aut( Z H ) → K ( Z H ) is surjective [23, Theorems 7.15-7.18] and so wecan apply Theorem 4.1 in Section 6 to the case P = Z H .Our first application generalises the main result in [42]. Recall that a ring R has stably free cancellation (SFC) if it has cancellation in the class of R , i.e. everystably free module is free, and Q n denotes the quaternion group of order 4 n . Theorem 6.3. If G has periodic cohomology, then the following are equivalent: (i) Z G has SFC (ii) m H ( G ) ≤ G has no quotient of the form Q n for n ≥ . This completely determines the groups G with periodic cohomology for which Z G has SFC and also corrects a mistake in [18, p249] where it was suggested thatthe groups in (VI) did not have SFC. Our second application is the following: Theorem 6.9. If G has -periodic cohomology then, for any generator g ∈ H ( G ; Z ) , [ χ ( g )] has cancellation if and only if m H ( G ) ≤ . Since Theorems 1.1 and 3.1 imply that D2 G is isomorphic to [ χ ( g )] as a gradedtree, the theorem above shows that D2 G has cancellation if and only if m H ( G ) ≤ χ ( g )] gives us our D2 complex X over G with χ ( X ) = 1 which was needed for the proof of Theorem B.In light of this, examples of new presentations of 4-periodic groups is now a requirement for showing that any more of these groups have the D2 property. If m H ( G ) ≤ m H ( G ) ≥
3, we seekenough homotopically distinct balanced presentations of G to realise the minimalD2 complexes over G up to polarised homotopy. In Section 7, we show: Theorem 7.7. Q has the D2 property and m H ( Q ) = 3 . Our proof amounts to combining recent results of W. Mannan and T. Popiel [26]with Theorem 1.1. This group was proposed as a counterexample in [2].
Acknowledgment.
I would like to thank my supervisor F. E. A. Johnson for manyinteresting conversations on the D2 problem and stable modules, and W. H. Mannanfor drawing my attention to his recent work on complexes with fundamental group Q . I would also like to thank Jonathan Hillman for many helpful comments.1. Polarised homotopy types and algebraic 2-complexes
Recall that, if G is a finitely presented group, then a G -polarised space is apair ( X, p X ) where X is a topological space and p X : π ( X, ∗ ) → G is a givenisomorphism. We say that two G -polarised spaces ( X, p X ), ( Y, p Y ) are polarisedhomotopy equivalent if there exists a homotopy equivalence h : X → Y such that p X = p Y ◦ π ( h ).Let D2 G denote the polarised homotopy types of D2 complexes over G . This hasthe structure of a tree with vertices the polarised homotopy types of D2 complexesover G and an edge between each ( X, p X ) and ( X ∨ S , ( p X ) + ) where ( p X ) + isdefined via the collapse map X ∨ S → X which is an isomorphism on π . N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 5
Also define an algebraic -complex E = ( F ∗ , ∂ ∗ ) over Z G to be chain complexconsisting of an exact sequence E : F F F Z ∂ ∂ ∂ where the F i are stably free Z G -modules, i.e. F i ⊕ Z G i ∼ = Z G j for some i, j ≥ Z G also form a tree Alg G where the ver-tices are the chain homotopy classes of algebraic 2-complexes, with edges betweeneach E = ( F ∗ , ∂ ∗ ) and the corresponding stabilised complex Σ( E ) defined byΣ( E ) : F ⊕ Z G F F Z . ( ∂ , ∂ ∂ This tree inherits a grading by the Euler characteristic, i.e. the alternating sum ofthe ranks of the free modules χ ( E ) = rank( F ) − rank( F ) + rank( F ).Our aim is now to prove the following theorem from the introduction: Theorem 1.1. If G is a finitely presented group, then there exists an isomorphismof graded trees e C ∗ : D2 G → Alg G which is the same as the cellular chain map X C ∗ ( e X ) when X is a -complex. Our proof will amount to finding an explicit map e C ∗ : D2 G → Alg G , whichgeneralises the cellular chain map, and checking that it is an isomorphism of gradedgraphs. This would complete the proof since Alg G is a tree by [20, Corollary 8.10].First note that every D2 complex is a D3 complex and that, by Wall [47], everyD3 complex is homotopic to a finite 3-complex [47]. We therefore lose no generalityin assuming throughout that every D2 complex is a finite 3-complex.Let ( X, p X ) ∈ D2 G and recall that we can use p X to identify the augmentedcellular chain complex C ∗ ( X ) = ( C ( e X ) C ( e X ) C ( e X ) C ( e X ) Z ∂ ∂ ∂ ∂ as a chain complex of Z G modules. We now define e C ∗ ( X ) to be e C ∗ ( X ) = ( C ( e X ) / Im( ∂ ) C ( e X ) C ( e X ) Z e ∂ e ∂ e ∂ where e ∂ = ∂ and e ∂ is induced by ∂ since Im( ∂ ) ⊆ Ker( ∂ ). This is known asthe virtual -complex in [19] and is exact since Im( e ∂ ) = Im( ∂ ). Note that, if X isa finite 2-complex, Im( ∂ ) = 0 and so e C ∗ is just the cellular chain map X C ∗ ( X ).Recall that a Z G -module is a Z G -lattice if its underlying abelian group is torsion-free. We can now deduce the following, which implies that e C ∗ ( X ) ∈ Alg G : Lemma 1.2. If X is a D2 complex with C ∗ ( X ) = ( C ∗ ( e X ) , ∂ ∗ ) , then C ( e X ) / Im( ∂ ) is a stably free Z G -module.Proof. First note that Ker( e ∂ ) = Ker( ∂ ) / Im( ∂ ) = H ( e X ) and, by the universalcoefficients theorem, Tors( H ( e X )) ∼ = Tors( H ( e X )) = 0 since X is a D2 complex. If J = C ( e X ) / Im( ∂ ), then this implies thatTors( J ) ≤ Tors( J/ Ker( e ∂ )) = Tors(Im( e ∂ )) ≤ Tors( C ( e X )) = 0since C ( e X ) is free. In particular, J is a Z G lattice. JOHNNY NICHOLSON
Since X is a D2 complex, H ( e X ) = 0 and so Im( ∂ ) = C ( e X ) is free. The exactsequence 0 → C ( e X ) → C ( e X ) → J → Z G ( J, C ( e X )). This is known to vanish by [18, PropositionB.8]. This implies that the exact sequence splits and so J ⊕ C ( e X ) ∼ = C ( e X ). (cid:3) Note that, in the case where G is a finite group, we can get around relying onthe delicate argument given in [18, Proposition B.8] by using Lemma 3.10.For the rest of this section we will assume, where relevant, that e C ∗ ( X ) comesequipped with an additional map from 0 on the left, i.e. a co-augmentation. Lemma 1.3. If X is a D2 complex over G , there is a chain homotopy equivalence ϕ : C ∗ ( X ) → e C ∗ ( X ) .Proof. If ϕ : C ( e X ) → C ( e X ) / Im( ∂ ) is the quotient map, then we have thefollowing diagram C ∗ ( X ) e C ∗ ( X ) ϕ = C ( e X ) C ( e X ) C ( e X ) C ( e X ) Z C ( e X ) / Im( ∂ ) C ( e X ) C ( e X ) Z ∂ ∂ ϕ ∂ id ∂ id id e ∂ e ∂ e ∂ x Since H ( e X ) = Ker( e ∂ ) and H ( e X ) = 0, it is easy to see that ϕ is a homologyequivalence and hence a chain homotopy equivalence. (cid:3) By combining Lemmas 1.2 and 1.3, we get that e C ∗ ( X ) gives a well-defined ele-ment of Alg G . In particular: Proposition 1.4. If X is a D2 complex, then e C ∗ gives a well-defined map e C ∗ : D2 G → Alg G . To prove Theorem 1.1, i.e. that e C ∗ is an isomorphism of graded graphs, we willneed the following two lemmas. Lemma 1.5.
Let ( X, p X ) , ( Y, p Y ) ∈ D2 G be such that X (1) = Y (1) . If ν : C ∗ ( X ) → C ∗ ( Y ) is a chain map, then ν is chain homotopic to a chain map ϕ such that ϕ | C i ( e X ) = id for i ≤ . The case where X and Y are finite 2-complexes is proven in [19, Proposition2.2]. The proof in this case is similar and will be omitted for brevity. The followingis [19, Lemma 2.3]: Lemma 1.6.
Let ( X, p X ) , ( Y, p Y ) be polarised -complexes over G such that X (1) = Y (1) . If ϕ : C ∗ ( X ) → C ∗ ( Y ) is a chain map such that ϕ | C i ( e X ) = id for i ≤ , thenthere exists a map f : X → Y such that f ∗ = ϕ ∗ , f | X (1) = id and p X = p Y ◦ π ( f ) . We can now proceed to the proof of Theorem 1.1:
Proof of Theorem 1.1.
First note that χ ( X ) = χ ( C ∗ ( X )) and this is equal to χ ( e C ∗ ( X )) since χ is a chain homotopy invariant. This implies that e C ∗ respectsthe grading and so it suffices to show that it is a bijection.The fact that e C ∗ is surjective follows from the statement that every algebraic 2-complex is realisable by a D2 complex. For finitely presented groups, this is proven N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 7 in [24, Theorem 2.1] though an alternative proof can be found in [15, Corollary 2.4].It remains to show that e C ∗ is injective.Let ( X, p X ), ( Y, p Y ) ∈ D2 G and note that, by the argument of [19, Proposition2.1], we can assume that X (1) = Y (1) by replacing each space with a polarisedhomotopy equivalent space if necessary. Suppose there is a chain homotopy e ν : e C ∗ ( X ) → e C ∗ ( Y ) . By Lemma 1.3, this lifts to a chain homotopy ν : C ∗ ( X ) → C ∗ ( Y ) and, by Lemma1.5, this is chain homotopic to another chain homotopy ϕ : C ∗ ( X ) → C ∗ ( Y ) suchthat ϕ | C i ( e X ) = id for i ≤ i X : X (2) ֒ → X denote the inclusion and note that this induces a Z G chainmap ( i X ) ∗ : C ∗ ( X (2) ) → C ∗ ( X ) where the 2-skeleton X (2) comes equipped withthe polarisation p X (2) = p X ◦ π ( i X ), and similarly for Y (2) . Since ( ϕ ◦ i X ) = 0,the composition ϕ ∗ ◦ ( i X ) ∗ : C ∗ ( X (2) ) → C ∗ ( Y ) can be viewed as a chain map ϕ ∗ ◦ ( i X ) ∗ : C ∗ ( X (2) ) → C ∗≤ ( Y ) ∼ = C ∗ ( Y (2) ) . Since ( ϕ ◦ i X ) i = id for i ≤
1, Lemma 1.6 implies that there exists a map f : X (2) → Y (2) such that f ∗ = ϕ ∗ ◦ ( i X ) ∗ , f | X (1) = id and p X (2) = p Y (2) ◦ π ( f ). By composingwith i Y , we can assume f : X (2) → Y which instead has that p X (2) = p Y ◦ π ( f ).We now claim that f has an extension F : X → Y such that F ∗ = ϕ ∗ : H ( e X ) → H ( e Y ), which is an isomorphism since ϕ ∗ is a homology equivalence. Since X and Y are D2 complexes, we have that H i ( e X ) = H i ( e Y ) = 0 for i = 2. This impliesthat F is a homology equivalence and so is a homotopy equivalence by Whitehead’stheorem. Since F ◦ i X = f and p X (2) = p Y ◦ π ( f ), this implies that p X = p Y ◦ π ( F )and so F is the required polarised homotopy equivalence from ( X, p X ) to ( Y, p Y ).To find the extension F , first let X = X (2) ∪ α e ∪ α · · · ∪ α n e n for 3-cells e i ∼ = D and attaching maps α i ∈ π ( X (2) ), where such a decompositionexists since X is assumed to be a finite 3-complex.Using cellular chains, we have that ∂ ( e i ) = α i where we are using the identifi-cation Im( ∂ ) ⊆ Ker( ∂ ) ∼ = π ( X (2) ), and so α i ∈ Im( ∂ ) for all i = 1 , . . . , n . Notethat there is a commutative diagram π ( X (2) ) π ( X )Ker( ∂ ) Ker( ∂ ) / Im( ∂ ) ( i X ) ∗ ∼ = ∼ = q where q is the quotient map. This shows that Im( ∂ ) = Ker(( i X ) ∗ ). Consider thecomposition f ∗ = ϕ ∗ ◦ ( i X ) ∗ : π ( X (2) ) → π ( Y ). Since ϕ ∗ is a homology equiva-lence, this implies that Ker(( i X ) ∗ ) = Ker( f ∗ ). By combining with the above tworesults, we get that α i ∈ Ker( f ∗ ) and so the maps f ◦ α i ∈ π ( Y ) are nullhomotopicfor all i = 1 , . . . , n .By standard homotopy theory, this implies that there exists an extension F : X → Y . In particular, since f ◦ α i : S → Y is null-homotopic, there is a map f i : e i → Y for which f i ◦ i = f ◦ α i for i : S = ∂e i ֒ → e i and so we can get awell-defined map F : X → Y by defining F | e i = f i for each i = 1 , . . . , n . Finallynote that, by the above diagram, ( i X ) ∗ : π ( X (2) ) → π ( X ) is surjective. Since JOHNNY NICHOLSON F ∗ ◦ ( i X ) ∗ = ϕ ∗ ◦ ( i X ) ∗ for ∗ ≤
2, this implies that F ∗ = ϕ ∗ : π ( X ) → π ( Y ) or,equivalently, that F ∗ = ϕ ∗ : H ( e X ) → H ( e Y ). (cid:3) Note that, since we proved that D2 G and Alg G are isomorphic as graphs, theproof that Alg G is a tree implies that D2 G is a tree. Since D2 G contains theCayley complex of any presentation of G , this implies a stable solution to the D2problem in the following sense. This was first proven by J. M. Cohen [6]. Corollary 1.7. If X is a D2 complex, then there exists n ≥ for which X ∨ nS is homotopy equivalent to a finite -complex. Recall that an algebraic 2-complex of over Z G is geometrically realisable if it ischain homotopy equivalent to the cellular chain complex C ∗ ( X ) of a finite 2-complex X over G . This begs the following question: Realisation problem.
Let G be a finitely presented group. Then is every algebraic -complex over Z G geometrically realisable? Let CW G ⊆ D2 G be the subgraph corresponding to the polarised homotopytypes of finite 2-complexes over G . Then every algebraic 2-complex over Z G isgeometrically realisable if and only if the map C ∗ : CW G → Alg G is surjective.However, by Theorem 1.1, this is the case if and only if CW G = D2 G , i.e. if G has the D2 property. In particular, this shows the following which was proven byJohnson in the case of finite groups [19] and was later extended by Mannan [24]. Corollary 1.8.
A finitely presented group G has the D2 property if and only ifevery algebraic -complex over Z G is geometrically realisable. In particular, the D2problem and realisation problem are equivalent. We conclude this section by noting that one can get an algebraic classification ofthe homotopy types of D2 complexes over G by quotienting Alg G by the strongerequivalence relation. In particular, determining the number of polarised homo-topy types that correspond to a given homotopy type X is equivalent to findingwhich group automorphisms Aut( G ) are induced by self-homotopy equivalences E ( X ). This problem is discussed in [32] and [34]. The corresponding self-chainhomotopy equivalences in Alg G are described in [17, Section 3], and are dealt withalgebraically in terms of k -invariants in [10].2. The Swan-Wall finiteness obstruction
In this section, we give a brief summary of the Swan and Wall finiteness obstruc-tions for use in the rest of the article, much of which can be found in [8]. From thispoint onwards, all modules will be finitely-generated and all groups G will be finiteunless otherwise specified.Let R be a ring and define a projective extension to be an exact sequence of R -modules of the form E : 0 → B → P n − → P n − → · · · → P → P → A → , with the P i projective. This defines an extension class g ( E ) ∈ Ext nR ( A, B ) and itwas determined by Wall which classes correspond to the projective extensions [48]:
Theorem 2.1.
A class g ∈ Ext nR ( A, B ) is represented by a projective extension E if and only if, for any R -module C , the map − ∪ g : Ext iR ( B, C ) → Ext n + iR ( A, C ) N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 9 is an isomorphism for i > and is surjective for i = 0 . Recall that a projective extension E as above has an associated Euler class χ ( E ) = P n − i =0 ( − i [ P i ] ∈ e K ( R ). The following was also shown by Wall in [48]: Theorem 2.2.
The Euler class χ ( E ) depends only on g ∈ Ext nR ( A, B ) and not onthe choice of projective extension E . This is known as the
Wall finiteness obstruction and we denote it by χ ( g ) for g ∈ Ext nR ( A, B ). Recall also that a group G has n -periodic cohomology if there is agenerator g ∈ H n ( G ; Z ), for some n ≥
1, such that − ∪ g : H i ( G ; Z ) → H i + n ( G ; Z )is an isomorphism for all i > H i ( G ; Z ) = Ext i Z G ( Z , Z ),Theorem 2.1 can be used to show that G having n -periodic cohomology is equivalentto the existence of a projective extension of the form E : 0 → Z → P n − → P n − → · · · → P → P → Z → . If such a resolution exists with the P i free, then we say G has free period n .Recall that the Swan map S : ( Z / | G | ) × → e K ( Z G ) sends r [( r, Σ)], whereΣ = P g ∈ G g is the group norm and ( r, Σ) ⊆ Z G has finite index coprime to | G | and so is projective by [40]. We refer to the image T ( Z G ) = Im( S ) as the Swansubgroup . The following is a restatement of [41, Lemmas 7.3 and 7.4]:
Theorem 2.3.
Suppose G has n -periodic cohomology. Then: (i) If g , g ∈ H n ( G ; Z ) = Z / | G | are generators, then χ ( g ) − χ ( g ) ∈ T ( Z G )(ii) For any generator g ∈ H n ( G ; Z ) and any r ∈ ( Z / | G | ) × , there exists a gener-ator g ∈ H n ( G ; Z ) such χ ( g ) − χ ( g ) = [( r, Σ)] . This implies that the set of possible obstructions χ ( g ) ∈ e K ( Z G ) for generators g ∈ H n ( G ; Z ) is equal to the full coset χ ( g ) + T ( Z G ) for any generator g . Henceany generator g gives a well defined class σ n ( G ) = [ χ ( g )] ∈ e K ( Z G ) /T ( Z G )known as the Swan finiteness obstruction . The main result is as follows [41]:
Theorem 2.4. If G has n -periodic cohomology, then the following are equivalent: (i) G has free period n , i.e. there is an n -periodic free resolution of Z G modules (ii) σ n ( G ) = 0 ∈ e K ( Z G ) /T ( Z G )(iii) There is a generator g ∈ H n ( G ; Z ) for which χ ( g ) = 0 ∈ e K ( Z G )(iv) There is a finite complex X such that X ≃ S n − and G acts freely on X . The formulation (iv) has the following consequence for finite Poincar´e 3-complexeswhich is relevant to our discussion in the introduction:
Proposition 2.5.
A finite group G is the fundamental group of a finite Poincar´e -complex if and only if G has free period . The first example of a group with non-zero finiteness obstruction, i.e. withdiffering free period and cohomological period, was found by R. J. Milgram [27]around 20 years after Swan’s original paper [41]. It was later shown by J. F. Davis[9] that the 4-periodic group Q (16; 3 ,
1) of order 48 has free period 8, which is theexample of minimal order. For a definition, see Section 5. Classification of algebraic 2-complexes
This section will largely be dedicated to the proof of the following theorem fromthe introduction:
Theorem 3.1. If G has -periodic cohomology and g ∈ H ( G ; Z ) a generator, thenthere is an isomorphism of graded trees Φ :
Alg G → [ χ ( g )] . Recall that, if R is a ring, a class [ P ] ∈ e K ( R ) can be represented as a graph withvertices the isomorphism classes of non-zero modules P ′ ∈ [ P ] and edges betweeneach P ′ ∈ [ P ] and P ′ ⊕ R ∈ [ P ]. This graph inherits a grading from the rank ofeach projective module.Now let LF n ( R ) denote the set of isomorphism classes of (finitely-generated)locally-free modules of rank n . We will assume R = Z G/I for G a finite group and I an ideal, in which case this coincides with the rank n projective modules by [38,Theorems 2.21 and 4.2]. The map P P ⊕ R induces a sequence LF ( R ) LF ( R ) LF ( R ) · · · e K ( R ) ∼ = ∼ = ∼ = where all the maps are surjections by Serre’s Theorem and all but the first map areisomorphisms by Bass’ Cancellation Theorem [44, Section 2].It follows from [41] that LF ( Z G ) and e K ( Z G ) are finite. Hence, if P is aprojective Z G modules, then [ P ] has the structure of a fork: it has a single vertexat each non-minimal height (i.e. grade) and a finite set of vertices at the minimalheight corresponding to the fibre LF P ( Z G ) = { P ′ ∈ LF ( Z G ) : P ′ ∈ [ P ] } since wehave chosen to omit 0 even if 0 ∈ [ P ]. ... Figure 1.
Tree structure for [ P ] ∈ e K ( Z G )The proof of Theorem 3.1 will be broken into two distinct parts. Firstly, we willshow the following. Note that this is not quite the same as Theorem 1.1 since thepossibility remains that Φ is not surjective. Theorem 3.2. If G has -periodic cohomology and g ∈ H ( G ; Z ) = Z / | G | is agenerator, then there is a map of graded trees Φ :
Alg G → [ χ ( g )] which is a bijection when restricted to each grade in Alg G . N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 11
This was proven by Johnson in [18, Theorem 57.4]. We will give an overview ofthe proof below, noting that we can avoid reference to the derived module category.For a class c ∈ e K ( Z G ) and Z G -modules A and B , define Proj n,c Z G ( A, B ) to bethe set of chain homotopy types of exact sequences of the form E : 0 → B → P n − → P n − → · · · → P → P → A → , where the P i are projective and such that χ ( E ) = c . Also let Free n Z G ( A, B ) denotethe set of chain homotopy types of exact sequences E with the P i free.By addition of elementary complexes, it can be shown that every projectiveextension E with χ ( E ) = 0 is chain homotopy equivalent to an extension with the P i free, i.e. Proj n, Z G ( A, B ) =
Free n Z G ( A, B ).Let Ω cn ( A ) denote the set of Z G -modules B for which Proj n,c Z G ( A, B ) is non-empty. By Schanuel’s lemma [41], B , B ∈ Ω cn ( A ) implies that B ⊕ Z G i ∼ = B ⊕ Z G j for some i, j ≥ B ∼ B . It can similarly be shown that,if B ∼ B and B ∈ Ω cn ( A ), then B ∈ Ω cn ( A ), i.e. Ω cn ( A ) is a stable module .Note that Proj n,c Z G ( A, Ω cn ( A )) = F B ∈ Ω cn ( A ) Proj n,c Z G ( A, B ) can be given the struc-ture of a graded graph with edges from E to the complex Σ( E ) defined by:Σ( E ) : 0 → B → P n − ⊕ Z G → P n − → · · · → P → P → A → . Similarly to the remark made at the start of Section 1, the fact that
Proj n,c Z G ( A, Ω cn ( A ))is a tree follows from [20, Corollary 8.10]. The following is immediate by noting thatthe isomorphism class of Ker( ∂ ) is a chain homotopy invariant for E = ( F ∗ , ∂ ∗ ): Lemma 3.3.
There is an isomorphism of graded trees
Alg G ≃ Free Z G ( Z , Ω ( Z )) given by extending each algebraic n -complex E = ( F ∗ , ∂ ∗ ) by Ker( ∂ ) . We will now find a similar description for trees of projective modules. Thefollowing is a generalisation of [18, Corollary 56.5]:
Lemma 3.4.
Let E i = (0 → J → P i → Z → be exact sequences of Z G -modulesfor i = 1 , . Then there is a chain homotopy equivalence E ≃ E if and only if P ∼ = P . We can use this as follows:
Lemma 3.5.
There is an isomorphism of trees [ P ] ≃ Proj ,c Z G ( Z , Ω c ( Z )) . Proof. If P ∈ c then, by [43, Theorem 3], P ⊗ Z Q ∼ = Q G as a Q G -module. One canthen obtain a map ϕ : P → Q as the composition P → P ⊗ Z Q ∼ = Q G → Q where Q G → Q is the augmentation map. Since Im ϕ is a non-trivial finitely-generated subgroup of Q , we must have Im ϕ ∼ = Z and so we get a surjection ϕ : P → Z . By sending P to Ker( ϕ ), this defines a mapΨ : [ P ] → Ω c ( Z )by writing c = [ P ] for some P ∈ c . Since Ψ is surjective by definition, we have anisomorphism of trees [ P ] ≃ G J ∈ Ω c ( Z ) Ψ − ( J ) . By Lemma 3.4, there is a bijection Ψ − ( J ) ≃ Proj ,c Z G ( Z , J ). (cid:3) These decompositions can be compared due to the following duality result:
Lemma 3.6. If G has -periodic cohomology and c = χ ( g ) ∈ e K ( Z G ) for somegenerator g ∈ H ( G ; Z ) , then Ω ( Z ) ≃ Ω c ( Z ) ∗ .Proof. Since G has 4-periodic cohomology, the discussion in Section 2 implies thatthere exists a projective extension of the form E : 0 → Z → P → P → P → P → Z → , where χ ( E ) = χ ( g ) for a fixed generator g ∈ H ( G ; Z ). By addition of elementarycomplexes, this can be turned into an extension of the form E : 0 → Z → P ∂ −→ F ∂ −→ F ∂ −→ F → Z → , where the F i are free, so that P = χ ( P ) = χ ( g ).Let J = Ker( ∂ ) = Im( ∂ ). It is then clear that J ∈ Ω ( Z ) and also J ∈ Ω c ( Z ) ∗ . This implies that Ω ( Z ) = Ω c ( Z ) ∗ since two stable modules are equal ifthey intersect non-trivially. (cid:3) Recall that, if J is a Z G module, then an automorphism ϕ : J → J inducesa map ϕ ∗ : H ( G ; J ) → H ( G ; J ) induced by the coefficients [10, Section 2]. Byfixing an identification H ( G ; J ) ∼ = Z / | G | , this induces a map ν J : Aut Z G ( J ) → ( Z / | G | ) × . Also let S : ( Z / | G | ) × → e K ( Z G ) be the Swan map as defined in Section 2. Thefollowing is a generalisation of [18, Theorem 54.6] and [18, Theorem 56.10]: Lemma 3.7. If c ∈ e K ( Z G ) and J ∈ Ω c ( Z ) , then Im( ν J ) ⊆ Ker( S ) and there is abijection Proj n,c Z G ( Z , J ) ≃ Ker( S ) / Im( ν J ) . In particular, this shows that
Proj n,c Z G ( Z , J ) only depends on the isomorphismclass of J and not on n or c . We are now ready to prove Theorem 3.2: Proof of Theorem 3.2. If G has 4-periodic cohomology and g ∈ H ( G ; Z ) a gener-ator, then we can combine Lemmas 3.5 and 3.6 to get that[ χ ( g )] ≃ Proj ,c Z G ( Z , Ω ( Z ) ∗ ) . By Lemma 3.3, it therefore suffices to prove that there is a bijection
Free Z G ( Z , J ) ≃ Proj ,c Z G ( Z , J ∗ )for all J ∈ Ω ( Z ). To see this, note that, since there is a bijection Aut Z G ( J ) ≃ Aut Z G ( J ∗ ), there is also a bijection Im( ν J ) ≃ Im( ν J ∗ ). In particular, Free Z G ( Z , J ) ≃ Ker( S ) / Im( ν J ) ≃ Ker( S ) / Im( ν J ∗ ) ≃ Proj ,c Z G ( Z , J ∗ )when c = [ χ ( g )]. In particular, we have the chain of isomorphisms: Alg G ≃ Free Z G ( Z , Ω ( Z )) ≃ Proj ,c Z G ( Z , Ω ( Z ) ∗ ) ≃ [ χ ( g )] . (cid:3) To finish the proof of Theorem 3.1, it is necessary to show that Φ is a bijection.Let µ ( G ) be the minimum value of χ ( E ) over all E ∈ Alg G . Since the modules J ∈ [ χ ( g )] have minimal rank one, Φ is a bijection if and only if µ ( G ) = 1. It iseasy to see that µ ( G ) ≥
1. The second task is therefore to show:
N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 13
Theorem 3.8. If G has -periodic cohomology, then there exists E ∈ Alg G suchthat χ ( E ) = 1 . To show this, we will need the following two facts. Let I ∗ = Hom Z G ( I, Z G )denote the dual of the augmentation ideal I = Ker( ε : Z G → Z ) where ε : Z G → Z is the augmentation map which sends g g ∈ G . Lemma 3.9.
Let Z be the trivial Z G module and suppose that f : Z → Z G n isinjective and such that coker( f ) is a Z G -lattice. Then coker( f ) = I ∗ ⊕ Z G n − . One can show this easily by noting that coker( f ) is in the syzygy Ω − ( Z G ) [18].The following is standard [48, p514]: Lemma 3.10. If G is a finite group and J is a Z G -lattice, then Ext Z G ( J, Z G ) = 0 . Note that we could have used this to give a simpler proof of Lemma 1.2 in thecase where G is finite. We will use it now to prove the following: Lemma 3.11. If P is a projective Z G -module of rank r ≥ , then there exists a Z G -lattice J for which I ⊕ P = J ⊕ Z G r and so rk Z ( J ) = rk Z ( I ) = | G | − .Proof. First note that, by [40, Theorem A], P is of the form P = P ′ ⊕ Z G r − forsome rank one projective P ′ and so it suffices to prove the case r = 1.Now note that, by [43, Theorem 3], P ⊗ Z Q ∼ = Q G as a Q G -module. One canthen obtain a map ϕ : P → Q as the composition P → P ⊗ Z Q ∼ = Q G → Q where Q G → Q is the augmentation map. Since Im ϕ is a non-trivial finitely-generated subgroup of Q , we must have Im ϕ ∼ = Z and so we get a surjection ϕ : P → Z . Let J = Ker( ϕ ). By applying Schanuel’s lemma to the exact sequences0 → I → Z G → Z → , → J → P → Z → , we then get that I ⊕ P ∼ = J ⊕ Z G . (cid:3) Proof of 3.8.
First note that µ ( G ) ≥ G finite [15], [39], so it suffices to showthat there exists an algebraic 2-complex E with µ ( E ) = 1.Since G has 4-periodic cohomology, the discussion in Section 2 implies that thereexists an exact sequence of Z G -modules0 → Z f −→ F → P → F → F → Z → P projective and we can assume the F i are free by, where necessary,forming the direct sum with pairs of projective modules. By Lemma 3.9, coker( f ) = I ∗ ⊕ F ′ for some F ′ free. This gives an exact sequence:0 → I ∗ ⊕ F ′ → P → F → F → Z → . Now let ¯ P be a projective for which F = P ⊕ ¯ P is free. By forming the directsum with length two exact sequence, we get0 → I ∗ ⊕ P → F → F → Λ → Z → P = F ′ ⊕ ¯ P projective. By dualising the result in Lemma 3.11, we can write I ∗ ⊕ P = J ⊕ F for some F free and some J with rk Z ( J ) = rk Z ( I ∗ ) = | G | − If i denotes the injection i : J ⊕ F ∼ = I ∗ ⊕ P → F , we can form exact sequences:0 → J → F /i ( F ) → F → F → Z → , → F → F → F /i ( F ) → . The first exact sequence shows that F /i ( F ) is a Z G -lattice and, by Lemma 3.10,this implies that Ext Z G ( F /i ( F ) , F ) = 0 where r is the rank of F , and so F = F /i ( F ) ⊕ F . Hence we get an exact sequence0 → J → F → F ⊕ F → Λ → Z → E formed by removing J has µ ( E ) equal to | G | · (rk Z ( J ) + rk Z ( Z )) = 1. (cid:3) This completes the proof of Theorem 3.1.4.
Cancellation for projective modules over integral group rings
Recall that K ( R ) = GL( R ) ab where GL( R ) = S n GL n ( R ) with respect to thenatural inclusions GL n ( R ) ֒ → GL n +1 ( R ). The aim of this section will be to provethe following theorem from the introduction. Theorem 4.1.
Let H = G/N , ¯ P ∈ LF ( Z G ) and P = ¯ P ⊗ ZN Z ∈ LF ( Z H ) . If m H ( G ) = m H ( H ) and the map Aut( P ) → K ( Z H ) is surjective, then [ ¯ P ] has cancellation if and only if [ P ] has cancellation. Here the map Aut( P ) → K ( Z H ) is induced by picking a projective Q such that P ⊕ Q is free of rank r and then lettingAut( P ) ⊆ Aut( P ⊕ Q ) ∼ = GL r ( Z H ) ⊆ GL( Z H ) → K ( Z H ) , which is well-defined by [28, Lemma 3.2]. This is the analogue of [31, Theorem A]for projective modules, and the proof will follow a similar outline.Recall that, by the discussion in the previous section, LF ( R ) ։ e K ( R ), andthis is bijective precisely when R has projective cancellation. Furthermore, R hascancellation in the class of [ P ] precisely when the fibre over [ P ] ∈ e K ( R ) is trivial.One direction of Theorem 4.1 follows from the following refinement of Fr¨ohlich’sresult [14] that Z G has projective cancellation implies Z H has projective cancella-tion if H = G/N . The proof follows from [42, Theorem A10] in exactly the sameway as was shown in [31, Theorem 1.1] in the stably free case:
Theorem 4.2.
Let H = G/N , let ¯ P ∈ LF ( Z G ) and P = ¯ P ⊗ Z N Z ∈ LF ( Z H ) .If [ ¯ P ] has cancellation, then [ P ] has cancellation. We now state a general version of the Jacobinski cancellation theorem which wewill need for the rest of the proof of Theorem 4.1. Let A be a semisimple separable Q -algebra which is finite-dimensional over Q and let Λ be a Z -order in A , i.e. afinitely-generated subring of A such that Q · Λ = A . For example, if Λ = Z G and A = Q G for G a finite group. Since Λ R = Λ ⊗ R has a real Wedderburndecomposition, the Eichler condition generalises to Z -orders Λ in the natural way.The following can be found in [44, Theorem 9.3]: Theorem 4.3 (Jacobinski) . If Λ satisfies the Eichler condition, then Λ has pro-jective cancellation. N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 15
Let H = G/N and suppose m H ( G ) = m H ( H ) and that Z H has cancellation inthe class of P = ¯ P ⊗ Z N Z ∈ LF ( Z H ). Since the other direction was proven inTheorem 4.2, it will suffice to prove that Z G has cancellation in the class of P subject to the conditions of the theorem. Consider the following pullback diagramfor Z G induced by the normal subgroup N : Z G Λ + Z H ( Z /n Z )[ H ]where Λ + = Z G/ b N , b N = P g ∈ N g and n = | N | . This is the standard pullbackconstruction for the ring Z G and trivially intersecting ideals I = ker( Z G → Z H ) = I ( N ) · G and J = b N · Z G, where I ( N ) = ker( Z N → Z ) is the augmentation ideal [7, Example 42.3]. Applying R ⊗ − to the diagram shows that R G ∼ = R H × Λ + R , from which we can deduce: Proposition 4.4. Λ + satisfies the Eichler condition. Now note that Λ + is a Z -order in Λ + Q which is a semisimple separable Q -algebra offinite dimension over Q since Q G ∼ = Q H × Λ + Q . Hence we can apply Theorem 4.3 toget that Λ + has cancellation. In particular, LF (Λ + ) = e K (Λ + ) and LF ( Z H ) → e K ( Z H ) has trivial fibre over [ P ].Consider the following diagram induced by the maps on projective modules. LF ( Z G ) LF ( Z H ) × LF (Λ + ) e K ( Z G ) e K ( Z H ) × e K (Λ + ) ϕ ϕ If Q is the image of ¯ P in LF (Λ + ), then proving cancellation in the class of [ ¯ P ]amounts to proving that the fibres of ϕ , ϕ over ( P, Q ) are in bijection.Now observe that the pullback diagram above is a Milnor square [28] and soinduces an exact sequence K ( Z H ) × K (Λ + ) K (( Z /n Z )[ H ]) e K ( Z G ) e K ( Z H ) × e K (Λ + ) ϕ which is part of the Mayer-Vietoris sequence for the diagram, where K ( R ) =GL( R ) ab for GL( R ) = S n GL n ( R ) with respect to the obvious inclusion GL n ( R ) ֒ → GL n +1 ( R ). This sequence extends ϕ and so the fibres of ϕ are in correspondencewith Im( K (( Z /n Z )[ H ]) → e K ( Z G )) ∼ = K (( Z /n Z )[ H ]) K ( Z H ) × K (Λ + ) . Furthermore, by Theorem 8.1 of [41], we know that P ⊗ ( Z /n Z )[ H ] ∼ = ( Z /n Z )[ H ] ∼ = Q ⊗ ( Z /n Z )[ H ] . Hence, by the more general construction of projectives modules over Milnor squares[44, Proposition 4.1] the fibre ϕ − ( P, Q ) is in correspondence with the double cosetAut( P ) \ ( Z /n Z )[ H ] × / Aut( Q )where we have used that Aut ( Z /n Z )[ H ] (( Z /n Z )[ H ]) = ( Z /n Z )[ H ] × . One result which helps towards comparing these fibres is due to Swan [44]. Sup-pose Λ is a Z -order in A as defined previously and let I be a two-sided ideal offinite index in Λ. Theorem 4.5. If Λ satisfies the Eichler condition then, with respect to the quotientmap Λ → Λ /I , we have Λ × E (Λ /I ) × and the map (Λ /I ) × → K (Λ /I ) induces (Λ /I ) × Λ × ∼ = K (Λ /I ) K (Λ) . We apply this to Λ = End( Q ) which satisfies the Eichler condition sinceEnd( Q ) ⊗ R ∼ = End( Q ⊗ R ) ∼ = End(Λ + R ) ∼ = Λ + R . Suppose J = Ker(Λ + ։ ( Z /n Z )[ H ]) and note that there is a map End( Q ) ։ Λ + /J induced by localisation [42, p146]. Since Λ + /J is finite, I = Ker(End( Q ) ։ Λ + /J )has finite index in End( Q ) and is naturally a two-sided ideal such that Λ /I ∼ =Λ + /J ∼ = ( Z /n Z )[ H ] . Combining this with the above Theorem gives that( Z /n Z )[ H ] × Aut( Q ) ∼ = K (( Z /n Z )[ H ]) K (End( Q )) . by using that Aut( Q ) = End( Q ) × . Since there is a commutative diagram K (Λ + ) K (Λ + /J ) K (End( Q )) ∼ = by [42, Corollary A17], we get that A = ( Z /n Z )[ H ] × Aut( Q ) ∼ = K (( Z /n Z )[ H ]) K (Λ + ) . and so Aut( P ) \ ( Z /n Z )[ H ] × / Aut( Q ) and K (( Z /n Z )[ H ]) K ( Z H ) × K (Λ + ) are in correspondence ifand only if the maps Aut( P ) → A and K ( Z H ) → A have the same images.Assume now that the map ϕ : Aut( P ) → K ( Z H ) is surjective, i.e. we now takethe full hypothesis of Theorem 4.1. Since Aut( P ) → Aut ( Z /n Z )[ H ] ( P ⊗ ( Z /n Z )[ H ]) ∼ =( Z /n Z )[ H ] × , we can place this into the following commutative diagram.Aut( P ) ( Z /n Z )[ H ] × ( Z /n Z )[ H ] × Aut( Q ) ∼ = AK ( Z H ) K (( Z /n Z )[ H ]) K (( Z /n Z )[ H ]) K (Λ + ) ∼ = A ϕ To see that the left hand square commutes, suppose Aut( P ) → K ( Z H ) isdefined via P ⊕ P ′ ∼ = f Z G r for some r ≥
1. Since ( P ⊕ P ′ ) ⊗ ( Z /n Z )[ H ] ∼ =( Z /n Z )[ H ] r and Z G r ⊗ ( Z /n Z )[ H ] ∼ = ( Z /n Z )[ H ] r , this induces an automorphism F = f ⊗ ( Z /n Z )[ H ] : ( Z /n Z )[ H ] r → ( Z /n Z )[ H ] r from which we can define a map( Z /n Z )[ H ] ֒ → ( Z /n Z )[ H ] ⊕ ( P ′ ⊗ ( Z /n Z )[ H ]) ∼ = ( Z /n Z )[ H ] r F −→ ( Z /n Z )[ H ] r . N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 17
Since this induces a map of units ( Z /n Z )[ H ] × → GL r (( Z /n Z )[ H ]), we can use thisto define a map ( Z /n Z )[ H ] × → K (( Z /n Z )[ H ]). It follows from [28, Lemma 3.2]that this is the same as the map defined using the inclusion( Z /n Z )[ H ] × ֒ → GL r (( Z /n Z )[ H ])and so is the same as the middle vertical map in the diagram above. The left handsquare then commutes by construction of the map F .If ψ : Aut( P ) → A denotes the map along the top row and ψ : K ( Z H ) → A denotes the map along the bottom row, then commutativity shows that ψ = ψ ◦ ϕ .Since ϕ is surjective, Im ψ = Im ψ and so [ P ] has cancellation. This completesthe proof of Theorem 4.1.5. Groups with periodic cohomology
The aim of this section will be to find restrictions on the quotients of groups withperiodic cohomology which will allow us to apply Theorem 4.1 in the Section 6. Wewill also classify the groups G with 4-periodic cohomology for which m H ( G ) ≤ Theorem 5.1. If G is a finite group, then the following are equivalent: (i) G has periodic cohomology (ii) G has no subgroup of the form C p for p prime (iii) The Sylow subgroups of G are cyclic or (generalised) quaternionic Q n . Recall also that a binary polyhedral group is a finite non-cyclic subgroup of S and consists of the quaternion groups Q n for n ≥ e T , e O , e I . We note that [31, Proposition 1.3]: Proposition 5.2.
A finite group G satisfies the Eichler condition if and only if G has no quotient which is a binary polyhedral group. We begin by establishing the following series of lemmas. Let Syl p ( G ) denote theisomorphism class of the Sylow p -subgroup of G for p prime. Lemma 5.3.
The proper quotients of D n and Q n are either C or of the form D m for ≤ m ≤ n − . Lemma 5.4. If → N → G → H → is an extension, then there is an extensionof abstract groups → Syl p ( N ) → Syl p ( G ) → Syl p ( H ) → for every prime p . Note that, by combining these two lemmas with Theorem 5.1 (iii), we see thatany quotient H of a group G with periodic cohomology has Syl p ( H ) cyclic for p odd and Syl ( H ) cyclic, dihedral or quaternionic. Lemma 5.5.
Suppose G has periodic cohomology and N , N ′ ≤ G are non-trivialdisjoint normal subgroups such that H = G/N and H ′ = G/N ′ are binary polyhedralgroups. Then G has a quotient of the form Q n for some n ≥ . Furthermore H and H ′ are not of the form Q n for any n ≥ , e T , e O or e I .Proof. Suppose for contradiction that H and H ′ are not of the form Q n for n ≥ H and H ′ are each of the form Q , Q , Q , Q , e T , e O or e I . Since N and N ′ are disjoint, N × N ′ ≤ G is a normal subgroup and so H and H ′ have a common quotient G/ ( N × N ′ ) ∼ = H/N ′ ∼ = H ′ /N . Furthermore, since G has periodic cohomology, Theorem 5.1 (ii) implies that | N | and | N ′ | are coprime.The remainder of the proof will be split into two cases.First suppose that Syl ( G ) is cyclic. By Lemma 5.4, Syl ( H ) and Syl ( H ′ ) mustbe quotients of Syl ( G ) and so are also cyclic. This implies H and H ′ are each of theform Q or Q . Since | N | and | N ′ | are non-trivial and coprime, | H | 6 = | H ′ | and sowe can assume that H = Q and H ′ = Q . These groups have common quotients1, C and C and the restriction that | N | and | N ′ | be coprime implies that N = C and N ′ = C . This implies, for example, that G/C ∼ = Q . Since (3 ,
20) = 1, thisextension must split and so G ∼ = C ⋊ ϕ Q for some map ϕ : Q → Aut( C ) ∼ = C .If ϕ = 1, then G ∼ = Q × C which does not have Q as a quotient. The onlyother option is that ϕ is the quotient by C which implies that G ∼ = Q .Now suppose that Syl ( G ) = Q n for some n ≥
3. Similarly Lemma 5.4 impliesthat Syl ( H ) and Syl ( H ′ ) are quotients of Q n . Since 4 | | H | , | H ′ | we can deduce,by Lemma 5.3, that Syl ( H ), Syl ( H ′ ) = Q n and so | N | , | N ′ | are odd. Now H and H ′ are each of the form Q , Q , e T , e O or e I and it is easy to verify that thesegroups have no non-trivial normal subgroups of odd order, which is a contradiction.For the last part note that, if H or H ′ were of the form Q n , e T , e O or e I , then wecan get a contradiction using the same argument in the previous paragraph. (cid:3) Recall the following, which is proven in [31, Proposition 3.3]:
Proposition 5.6.
Let N be a normal subgroup of G and let H = G/N . Then N is contained in all normal subgroups N ′ for which G/N ′ is binary polyhedral if andonly if m H ( G ) = m H ( H ) . We are now ready to prove the main results of this section.
Theorem 5.7. If G has periodic cohomology and fails the Eichler condition, then G either has a quotient of the form Q n for n ≥ or a binary polyhedral quotient H for which m H ( G ) = m H ( H ) .Proof. Suppose G fails the Eichler condition and has no binary polyhedral quotient H for which m H ( G ) = m H ( H ). Since G fails the Eichler condition, there exists abinary polyhedral quotient H = G/N which we can pick to have maximal order.Since m H ( G ) = m H ( H ), Proposition 5.6 implies that there exists a binary polyhe-dral quotient H ′ = G/N ′ for which N N ′ , and N ′ N also by maximality of | H | . Now G has quotient b G = G/ ( N ∩ N ′ ) which has H = G/K and H ′ = G/K ′ for K = N/ ( N ∩ N ′ ) and K ′ = N ′ / ( N ∩ N ′ ) disjoint normal subgroups, by the thirdisomorphism theorem. In addition, K and K ′ are non-trivial since N ∩ N ′ = N, N ′ .If b G has periodic cohomology, then Lemma 5.5 implies that G has a quotientof the form Q n for n ≥
6. Suppose that b G does not have periodic cohomology.Since Syl p ( b G ) is a quotient of Syl p ( G ) for all p prime by Lemma 5.4, combiningLemma 5.3 and Theorem 5.1 (iii) shows that we must have Syl ( b G ) = D n for some n ≥
2. Now Lemma 5.4 also implies that Syl ( H ) and Syl ( H ′ ) are quotients of D n . However, by Lemma 5.3, the only cyclic or generalised quaternionic quotientsof D n are 1 or C which contradicts the fact that 4 | | H | , | H ′ | . (cid:3) Recall that m H ( Q n ) = ⌊ n/ ⌋ by [18, Section 12] and, by considering theirquotients, it can be shown that m H ( e T ) = 1, m H ( e O ) = 2 and m H ( e I ) = 2. N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 19
This has the following corollary, which is also part of Theorem 6.3.
Corollary 5.8. If G has periodic cohomology, then the following are equivalent: (i) G has no quotient of the form Q n for n ≥ m H ( G ) ≤ G has a binary polyhedral quotient H for which m H ( G ) = m H ( H ) ≤ .Proof. If G has quotient Q n for n ≥
6, then m H ( G ) ≥ m H ( Q n ) ≥ G has no quotient ofthe form Q n for n ≥
6, then Theorem 5.7 implies that either m H ( G ) = 0 or G hasbinary polyhedral quotient H for which m H ( G ) = m H ( H ). Since H is not of theform Q n for n ≥
6, the results stated above imply that m H ( H ) ≤ (cid:3) Theorem 5.9. If G has periodic cohomology and has quotient H = e T , e O or e I ,then m H ( G ) = m H ( H ) ≤ . In particular, G has no quotient of the form Q n for n ≥ .Proof. If m H ( G ) = m H ( H ), then Proposition 5.2 implies that there exists a binarypolyhedral quotient H ′ = G/N ′ for which N N ′ . Since e T , e O and e I have noproper quotients which are binary polyhedral groups, we must also have N ′ N and so the group b G = G/ ( N ∩ N ′ ) satisfies the conditions of Lemma 5.5 provided b G has periodic cohomology. Note that Syl ( b G ) has quotient Syl ( H ) = Q or Q by Lemma 5.4 and so Syl ( b G ) is not dihedral by Lemma 5.3. This implies that b G has periodic cohomology and Lemma 5.5 then contradicts the fact that H is of theform e T , e O or e I . The second part now follows by Corollary 5.8. (cid:3) We will determine the groups with 4-periodic cohomology for which m H ( G ) ≤ C n ⋊ ( r ) C m to denote the semi-direct product wherethe generator x ∈ C m acts on the generator y ∈ C n by xyx − = y r for some r ∈ Z .We also assume each family contains G × C n for any G listed with ( n, | G | ) = 1.(I) ′ C n , D n +2 for n ≥
1, the cyclic group and the dihedral group of order 4 n + 2(II) ′ Q n = h x, y | x n = y , yxy − = x − i for n ≥ e T , e O, e I (III) ′ D (2 n , m ) = C m ⋊ ( − C n for n ≥ m ≥ ′ P ′ · n = Q ⋊ ϕ C n for n ≥
2, where ϕ : C n → Aut Q sends the generator z ∈ C n to ϕ ( z ) : x y, y xy (V) ′ P ′′ n = C n · e O for n ≥ e O ։ e O/ e T = C ≤ Aut C n ,(VI) ′ Q (2 n a ; b, c ) = ( C a × C b × C c ) ⋊ ϕ Q n for n ≥ a, b, c ≥ b > c . If C a = h p i , C b = h q i and C c = h r i , then the action is given by ϕ ( x ) : p p − , q q − , r r ϕ ( y ) : p p − , q q, r r − Theorem 5.10.
The groups G with -periodic cohomology for which m H ( G ) ≤ are as follows where each family contains G × C n for any G listed with ( n, | G | ) = 1 . (I) C n , D n +2 for n ≥ Q , Q , Q , Q , e T , e O, e I (III) D (2 n , , D (2 n , for n ≥ P ′ · n for n ≥ P ′′ n for n ≥ odd (VI) Q (16; m, n ) for m > n ≥ odd coprimeProof. First note that we can ignore the groups of the form G × C n for G listedand ( n, | G | ) = 1 since m H ( G × C n ) = m H ( G ) in these cases.It can be shown that the groups in (I) ′ satisfy the Eichler condition [18, Section12]. For the groups G in (II) ′ , we use that m H ( Q n ) = ⌊ n/ ⌋ , m H ( e T ) = 1, m H ( e O ) =2 and m H ( e I ) = 2 as mentioned previously.In case (III) ′ , suppose G has a binary polyhedral quotient H . Explicit compu-tation shows that Z ( H ) = C and so the quotient map f : G ։ H must have f ( Z ( G )) ⊆ Z ( H ) = C . If x ∈ C m and y ∈ C n are generators, it is easy to seethat Z ( D (2 n , m )) = h y i = C n − which has index two subgroup N = h y i . Hence f factors through G/ h y i = C m ⋊ ( − C = Q m . By Proposition 5.6, we have that m H ( G ) = m H ( Q m ) = ( m − / m is odd and so m H ( G ) ≤ m = 3 or 5 and any n ≥ ′ all have quotient e T and so m H ( P ′ · n ) = 1 by Theorem 5.9.Similarly the groups in (V) ′ have quotient e O and so m H ( P ′′ n ) = 2.For the groups in (VI) ′ , suppose G = Q (2 k a ; b, c ) has m H ( G ) ≤ a, b, c ≥ b > c . Recall that, by Corollary 5.8, m H ( G ) ≤ G has no quotient of the form Q n for n ≥
6. Since G has quotient Q k a for k ≥
3, this implies that a = 1 and k = 3 or 4. If k = 3, then it is easy to see that G ∼ = Q (8 c ; 1 , b ) ∼ = Q (8 b ; c,
1) and so has quotients Q b and Q c . Hence b = c = 1which contradicts the fact that b > c .Now suppose k = 4, which we write as G = Q (16; m, n ) = ( C n × C m ) ⋊ Q for m > n ≥ N ′ = C n × C m , then Q = G/N ′ . If m H ( G ) = 2, thenProposition 5.6 implies that G has another binary polyhedral quotient H = G/N such that N ′ N . If b G = G/ ( N ∩ N ′ ), then Syl ( b G ) has quotient Q by 5.4 whichimplies that Syl ( b G ) is not dihedral and so b G has periodic cohomology. If N N ′ ,we could then apply Lemma 5.5 to get a contradiction since Q = G/N ′ .Hence we can assume that N ⊆ N ′ = C n × C m = C nm and so N is of the form C n ′ × C m ′ for n ′ ≤ n and m ′ ≤ m . It is easy to see that H = G/N = ( C a × C b ) ⋊ ϕ ′ Q = Q (16; a, b )where a = n/n ′ and b = n/n ′ . It is then a straightforward exercise to check thatthis is not a binary polyhedral group unless a = b = 1. This implies that N = N ′ which contradicts the fact that N ′ N . (cid:3) Note also that the case m H ( G ) = 0 corresponds to (I) and the case m H ( G ) = 1corresponds to Q × C n , Q × C n and e T × C n from (II) as well as (IV). All othergroups have m H ( G ) = 2.We conclude this section by noting the following which we will use in Section 6. Lemma 5.11. If G has -periodic cohomology and m H ( G ) ≥ , then either G has aquotient Q n for some n ≥ or G = Q × C n where n ≥ is such that ( n,
24) = 1 .Proof. If m H ( G ) ≥ G does not have a quotient Q n for n ≥
7, then Theorem5.7 implies that G has a quotient Q . This rules out the groups in (I) ′ , (III) ′ , (IV) ′ and (V) ′ by the proof of Theorem 5.10. If G is in (VI) ′ , then G = Q (2 k a ; b, c ) × C n for k ≥ a, b, c, n ≥ b > c . Since G has no quotient Q n for n ≥
7, we must have that k = 3. Since Q (8 a ; b, c ) ∼ = Q (8 b ; c, a ) ∼ = Q (8 c ; a, b ), G has quotients Q a , Q b and Q c and so a, b, c ≤ G is in (II) ′ which implies that G = Q × C n for some n ≥ n,
24) = 1. (cid:3)
N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 21 Cancellation over groups with periodic cohomology
Using the results in [31] and the previous section, we will now prove Theorems6.3 and 6.9.First note that Theorem 4.3 implies that, if G satisfies the Eichler condition,then Z G has SFC. In the case where G does not satisfy the Eichler condition, wehave the following theorem of the author [31, Theorem B]: Theorem 6.1.
Suppose G has a binary polyhedral quotient H such that m H ( G ) = m H ( H ) . Then Z G has SFC if and only if Z H has SFC. This can be proven as an application of [31, Theorem A], which is the weakerversion of Theorem 4.1 in the case where P is free. The case of binary polyhedralgroups was dealt with by Swan [42, Theorem I]: Theorem 6.2. If G is a binary polyhedral group, then Z G has SFC if and only if G is of the form Q , Q , Q , Q , e T , e O, e I . In particular, the case where Z H does not have SFC follows from Theorem 4.2.The case where Z H has SFC is equivalent to H = Q , Q , Q , Q , e T , e O, e I byTheorem 6.2 and so can be dealt with using results of [23, Theorems 7.15-7.18]which show that the map of units Z H × → K ( Z H ) is surjective in all these cases. Theorem 6.3. If G has periodic cohomology, then the following are equivalent: (i) Z G has SFC (ii) m H ( G ) ≤ G has no quotient of the form Q n for n ≥ .Proof. The equivalence of (ii) and (iii) is proven in Corollary 5.8 and the fact that(i) implies (iii) follows from Theorem 4.2 since Z Q n does not have SFC for all n ≥ m H ( G ) =0 or there exists a binary polyhedral quotient H for which m H ( G ) = m H ( H ). Inthe first case, we are done by Theorem 4.3 and, in the second case, (ii) implies that m H ( H ) ≤ Z H has SFC by Swan’s determination of the binary polyhedralgroups H with SFC [42], [31, Remark 3.2]. Hence we are done by Theorem 6.1. (cid:3) We now turn to the proof of Theorem 6.9. First recall the following result ofSwan on projective cancellation [42, p66], building on work of M. F. Vigneras [45]:
Lemma 6.4. If G = Q n for n ≥ , then [ P ] does not have cancellation for everyprojective Z G -module P . To deal with the finiteness obstructions it will be useful to note that, by theclassification of J. A. Wolf [50], the groups in (I) ′ -(IV) ′ are the fixed-point freefinite subgroups of SO (4) and so are 3-manifold groups, i.e. fundamental groups ofclosed 3-manifolds. Therefore: Lemma 6.5. If G is in (I) ′ - (IV) ′ , then σ ( G ) = 0 ∈ e K ( Z G ) /T ( Z G ) . In particular Theorem 2.3 implies that, for every P ∈ T ( Z G ), there exists agenerator g ∈ H ( G ; Z ) for which χ ( g ) = [ P ] ∈ e K ( Z G ).The following two lemmas will be useful in applying Theorem 4.1 to the groupsin (V) and (VI): Lemma 6.6. If N E G and N ≤ H ≤ G , then there is a commutative diagram: e K ( Z G ) e K ( Z [ G/N ]) e K ( Z H ) e K ( Z [ H/N ]) Res GH Res
G/NH/N where the horizontal maps are induced by − ⊗ Z N Z . Lemma 6.7.
The map T ( Z G ) → T ( Z [ G/N ]) induced by − ⊗ Z N Z is surjective. Our main tool will be Theorem 4.1 which we will use via the following lemma:
Lemma 6.8.
Suppose G has -periodic cohomology and H = G/N is a binarypolyhedral group such that m H ( G ) = m H ( H ) ≤ . If g ∈ H ( G ; Z ) is a generatorsuch that χ ( g ) ⊗ Z N Z ∈ T ( Z H ) , then [ χ ( g )] has cancellation.Proof. First note that, as is implicit in Theorem 3.1 or Theorem 3.2, the propertythat [ χ ( g )] has cancellation is independent of the choice of generator g ∈ H ( G ; Z )and so it suffices to determine cancellation for any choice of generator.By Lemma 6.7, there exists P ∈ T ( Z G ) for which P ⊗ Z N Z = − χ ( g ) ⊗ Z N Z . Since χ ( g ) ⊕ P ∈ χ ( g ) + T ( Z G ), Theorem 2.3 implies that we can find another generator g ′ ∈ H ( G ; Z ) for which χ ( g ′ ) = χ ( g ) ⊕ [ P ]. Hence χ ( g ′ ) ⊗ Z N Z = 0 ∈ e K ( Z H ) andso χ ( g ′ ) ⊗ Z N Z = Z H since Z H has SFC by Theorem 6.3 and we can assume χ ( g ′ )has rank one. As discussed at the start of this section, [23, Theorems 7.15 - 7.18]implies that Aut( Z H ) = Z H × ։ K ( Z H ). The conditions of Theorem 4.1 are metand so [ χ ( g ′ )], and hence also [ χ ( g )], must have cancellation. (cid:3) We are now ready to prove the following theorem from the introduction:
Theorem 6.9. If G has -periodic cohomology then, for any generator g ∈ H ( G ; Z ) , [ χ ( g )] has cancellation if and only if m H ( G ) ≤ .Proof. To prove the forward direction, suppose m H ( G ) ≥
3. By Lemma 5.11,there are two cases to consider. Firstly, if G has a quotient Q n = G/N for n ≥
7, and P = χ ( g ) ⊗ Z N Z for some generator g ∈ H ( G ; Z ), then [ P ] does nothave cancellation by Theorem 6.4 and so neither does [ χ ( g )] by Theorem 4.2. If G = Q × C n , then G has free period 4 by Lemma 6.5 and so Theorem 2.3 implieswe can find a generator g such that χ ( g ) = [ Z G ] which does not have cancellationby Theorem 6.2.For the converse, suppose m H ( G ) ≤
2. If G has free period 4, then similarlyTheorem 2.3 implies there exists a generator g ∈ H ( G ; Z ) for which χ ( g ) = [ Z G ]and so has cancellation by Theorem 6.3. If G does not have free period 4, i.e. σ ( G ) = 0, then Lemma 6.5 implies that G is one of the groups in (V) or (VI).If G is one of the groups in (VI), then G = Q (16; m, n ) × C r for m, n odd coprimeand ( r, mn ) = 1. By Theorem 5.10, the quotient Q = G/N has m H ( G ) = m H ( Q ) = 2. It is also proven in [42, Theorems III, VI] that e K ( Z Q ) = T ( Z Q )and so χ ( g ) ⊗ Z N Z ∈ T ( Z Q ) for any generator g ∈ H ( G ; Z ). Hence [ χ ( g )] hascancellation by Lemma 6.8.Finally if G is one of the groups in (V), then G = P ′′ n × C m for n ≥ m ≥ n, m ) = 1. Theorem 5.10 implies that the quotient e O = G/N has m H ( G ) = m H ( e O ) = 2 where N = C n × C m . It is proven in [42, Theorem N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 23 e K ( Z e O ) → e K ( Z Q ) ⊕ e K ( Z Q ) is bijective.This restricts to an injection Res | T ( Z e O ) : T ( Z e O ) → T ( Z Q ) ⊕ T ( Z Q ) which isnecessarily bijective since, for example, [42, Theorem IV] implies that T ( Z e O ) ∼ = Z / T ( Z Q ) ∼ = Z / T ( Z Q ) = 0. Since e K ( Z Q ) = T ( Z Q ), any P ∈ e K ( Z e O )in contained in T ( Z e O ) if and only if Res e OQ ( P ) = 0 ∈ e K ( Z Q ).Note that G has a (hyperelementary) subgroup H = Q n × C m which contains N and has Q = H/N . By Lemma 6.6, we now have a commutative diagram: e K ( Z G ) e K ( Z e O ) e K ( Z H ) e K ( Z Q ) Res GH Res e OQ Let g ∈ H ( G ; Z ) be a generator and note that Res GH ( χ ( g )) = χ (¯ g ), where ¯ g is the image of g under the induced map H ( G ; Z ) → H ( H ; Z ). Since H is in(II), σ ( H ) = 0 and so χ (¯ g ) ∈ T ( Z H ). This implies that χ (¯ g ) ⊗ Z N Z = 0 since itis contained in T ( Z Q ). Hence Res GH ( χ ( g )) ⊗ Z N Z = 0 ∈ e K ( Z Q ) and so, bythe commutativity of the diagram above, we get that Res e OQ ( χ ( g ) ⊗ Z N Z ) = 0 ∈ e K ( Z Q ). By the discussion above, this implies that χ ( g ) ⊗ Z N Z ∈ T ( Z e O ) whichimplies that [ χ ( g )] has cancellation by Lemma 6.8. (cid:3) Group presentations and the D2 problem
We conclude this article with a discussion of the work that remains to be doneon D2 complexes over groups with 4-periodic cohomology. In fact, as we shallsee, all further work will require the uncovering of new balanced presentations forthese groups. Since H ( G ) = 0 for groups with periodic cohomology [1], [37],these correspond to efficient presentations and so the groups G with 4-periodiccohomology are either counterexamples to the D2 problem by Theorem B (i), orgive a new supply of groups with efficient presentations.This gives some response to comments made by L. G. Kov´acs [22, p212] andJ. Harlender [16, p167] on the scarcity of efficient finite groups. In contrast, onemight be tempted to conjecture that every group with periodic cohomology has abalanced, and hence efficient, presentation.As discussed in Section 6, the groups in (I) ′ -(IV) ′ are all 3-manifold groups. Infact, it follows from Perelman’s solution to the Geometrisation Conjecture thatthese are the only finite 3-manifold groups. By Morse theory, 3-manifolds M havecell structures with single 3-cells. This implies that χ ( X (2) ) = 1 which induces abalanced presentation of π ( M ). This shows the following: Proposition 7.1. If G is a finite -manifold group, then G has a balanced presen-tation. By Theorem B (ii) this implies that, if G is a finite 3-manifold group with m H ( G ) ≤
2, then G has the D2 property. In particular, this shows the following: Corollary 7.2. If G is in (I) - (IV) , then G has the D2 property.Remark . Whilst the cases (I) and (II) were covered in [18], the use of Proposition7.1 avoids the need to find explicit balanced presentations.
We now turn our attention to the groups in (V) and (VI). We have not been ableto find balanced presentations for any of the groups in (V), but have succeeded fora few of the groups in (VI). These groups overlap in the case where k = n = 1 withthe groups in [30, Theorem 3.1]. Proposition 7.4. If n ≥ and a, b, k ≥ odd coprime, then: Q (2 n a ; b, × C k ∼ = h x, y | y k x b y k = x b , xyx = y n − a − i . In particular, Q (16; n, × C k has the D2 property. The simplest groups that we have not been able to find balanced presentations forare P ′′ · and Q (16; 3 , Conjecture 7.5. If G has -periodic cohomology, then G has a balanced presenta-tion. We now consider the case m H ( G ) ≥
3. By Theorem A (i), there always exists atleast two D2 complexes over G with Euler characteristic one up to polarised homo-topy. In particular, a necessary condition for such a group to have the D2 propertyis the existence of at least two homotopically distinct balanced presentations.We will now demonstrate how one might go about proving the D2 property fora group with m H ( G ) ≥ G = Q which has m H ( Q ) = 3.Recall the following recent result of Mannan and Popiel [26]. This is, to date, theonly known example of an exotic presentation for a finite non-abelian group. Theorem 7.6.
The quaternion group Q has presentations P = h x, y | x = y , xyx = y i , P = h x, | x = y , y − xyx = x y − x y i such that π ( X P ) = π ( X P ) as Z Q -modules. In particular, X P X P . Now note that, by combining Theorem 3.1 with the fact that Q has free period4, there is a one-to-one correspondence between minimal D2 complexes over Q and modules J ∈ [ Z Q ] with rank Z Q ( J ) = 1, i.e. the stably free modules over Z Q of rank one.By [42, Theorem III], Z Q has exactly two rank one stably free modules andso there are two minimal D2-complexes. Since non-minimal D2-complexes are de-termined up to homotopy by their Euler characteristic, this shows that the tree ofhomotopy types of D2 complexes over Q is a fork with two prongs.By Theorem 7.6, there are two non-homotopic finite 2-complexes over Q withminimal Euler characteristic which implies that both minimal D2-complexes over Q are geometrically realisable. In particular, this proves the following: Theorem 7.7. Q has the D2 property and m H ( Q ) = 3 . In [2], this is proposed as a counterexample by F. R. Beyl and N. Waller and sothis answers their question in the negative.It should not be too difficult to replicate this proof for more examples with 4-periodic cohomology and m H ( G ) ≥
3, though difficulties arise in the general case.For example, to replicate this proof for all quaternion groups G = Q n would requirea more general method of distinguishing presentations for Q n than the method in[26], and also require an explicit computation of the number of rank one stably free Z Q n -modules, extending Swan’s calculations in the case n ≤
10 [42, Theorem III].
N CW-COMPLEXES OVER GROUPS WITH PERIODIC COHOMOLOGY 25
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Department of Mathematics, UCL, Gower Street, London, WC1E 6BT, U.K.
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