Cancellation for ( G , n ) -complexes and the Swan Finiteness Obstruction

In previous work, we related homotopy types of finite ( G , n ) -complexes when G has periodic cohomology to projective Z G -modules representing the Swan finiteness obstruction. We use this to determine when X ∨ S n (cid:3) Y ∨ S n implies X (cid:3) Y for finite ( G , n ) -complexes X and Y , and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective Z G - modules as lifts of locally free modules over orders in products of quaternion algebras,whose existence follows from the Eichler mass formula. In the case n = 2 , difficulties arise that lead to a new approach to finding a counterexample to Wall’s D2 problem.


Introduction
For an integer n 2, a (G, n)-complex is a connected n-dimensional CW-complex X with fundamental group G and whose universal cover X is (n − 1)-connected.Equivalently, it is the n-skeleton of a K(G, 1).Given a group G and n 2, a finite (G, n)-complex exists if and only if G has type F n in the sense of Wall [44].For a group G and n 2, let HT(G, n) be the set of homotopy types of finite (G, n)-complexes that is a graded tree with edges between each X and X ∨ S n and with grading coming from the directed Euler characteristic that is defined to be − → χ (X) = (−1) n χ(X).The first basic question is whether or not HT(G, n) has cancellation, that is, has the property that X ∨ S n Y ∨ S n implies that X Y for all X, Y ∈ HT(G, n).This is equivalent to the existence of a finite (G, n)-complex X 0 such that every finite (G, n)-complex is of the form X 0 ∨ rS n for some r 0.
This question is completely inaccessible in general.However, a solution in the case where G is finite abelian, which includes non-cancellation examples, follows from work of Browning [4], Dyer-Sieradski [13], and Metzler [27].Examples of non-cancellation are also known over certain infinite groups, with the first coming from work of Dunwoody [12].These examples are of special interest due to their applications to smooth 4-manifolds [3,19,23], Wall's D2 problem [22,33], and combinatorial group theory [28].
The aim of this article is to study the cancellation problem over groups with periodic cohomology.This is the second part of a two-part series in which part one [31] reduces the classification of HT(G, n) to problems about projective ZG-modules that we resolve in the present article.

Main results
Let PHT(G, n) denote the set of polarised homotopy types of finite (G, n)-complexes, that is, the homotopy types of pairs (X, ρ) where X is a finite (G, n)-complex and ρ : π 1 (X) → G is an isomorphism.Recall that a D2 complex is a connected CW-complex that is cohomologically 2-dimensional and a group G has the D2 property if every finite D2 complex X with π 1 (X) ∼ = G is homotopy equivalent to a 2-complex [44].
We say that a non-trivial finite group G has k-periodic cohomology if its Tate cohomology groups satisfy H i (G; Z) ∼ = H i+k (G; Z) for all i ∈ Z.
Our main result is that cancellation for HT(G, n) is completely determined by m H (G).
but whose action is given by g • m = θ(g) • P m for g ∈ G and m ∈ P. Note that this does not necessarily give an action of Aut(G) on [P (G,n) ] since we may have that P ∈ [P (G,n) ] but P θ ∈ [P (G,n) ].Recall that, for G a finite group, every projective ZG-module is of the form P ⊕ ZG r where P has rank one and r 0 [38].If I ⊆ ZG is the augmentation ideal and r is an integer such that (r, |G|) = 1, then (I, r) is a projective ZG-module [39].If G has k-periodic cohomology, then the action of θ ∈ Aut(G) on [P (G,n) ] is then given by θ : P ⊕ ZG r → ((I, ψ k (θ ) i ) ⊗ P θ ) ⊕ ZG r , where P has rank one and ψ k : Aut(G) → (Z/|G|) × is a map that depends only on G [31, Section 7].
In particular, in order to prove Theorem A, it suffices to determine when [P (G,n) ] and [P (G,n) ]/ Aut(G) have cancellation for G a group with k-periodic cohomology and an appropriate integer k.In order to prove Theorem B, it suffices to compute lower bounds on the number of equivalence classes of rank one projective modules in [P (G,n) ]/ Aut(G).These tasks will be achieved by combining detailed information on the modules P (G,n) with the cancellation theorems for projective ZG-modules recently established by the author in [32] and [33].

Applications to Wall's D2 problem
We now consider the case n = 2 in more detail.Recall that every finite presentation P for a group G has an associated presentation complex X P which is a finite 2-complex with π 1 (X P ) ∼ = G.Conversely, every (connected) finite 2-complex X with π 1 (X) ∼ = G is homotopy equivalent to X P for some finite presentation P for G.We say that two finite presentations P, Q are homotopy equivalent if X P , X Q are homotopy equivalent (see Section 9 for more details).
For a finite presentation P define its deficiency def(P) to be the number of generators minus the number of relators.For a finitely presented group G, define the deficiency def(G) to be the maximum deficiency across all finite presentations for G.Note that χ(X P ) = 1 − def(P) and so N(G, 2) is the number of homotopy classes of presentations of G with maximal deficiency.
Let G be a group with 4-periodic cohomology.It is a consequence Theorems A and B that, if G has the D2 property, then we would expect non-cancellation examples for finite 2-complexes over G provided m H (G) is sufficiently large.For example, this applies when G = Q 28 is the quaternion group of order 28 since G has the D2 property and m H (G) = 3 [33,Theorem 8.11].
However, if we are interested in the D2 property itself, we could instead view Theorems A and B as a constraint that needs to be satisfied in order for the D2 property to hold.For example, let Q 4n denote the quaternion group of order 4n which has standard presentation P std = x, y | x n = y 2 , y −1 xy = x −1 .Since def(Q 4n ) = 0, the presentations P with maximal deficiency are balanced in that def(P) = 0.If Q 4n has the D2 property for all n 2, then m H (Q 4n ) = n/2 implies that N(Q 4n , 2) e λn for all λ > 0 and n sufficiently large.In [25, Section 3], Mannan-Popiel introduced a family of presentations P r MP = x, y | x n = y 2 , y −1 xyx r−1 = x r y −1 x 2 y , which for certain r ∈ Z present Q 4n .Note that, if r = 0 or 1, then P r MP is homotopy equivalent to P std .It was proposed in [25] that the P r MP contain all presentations for Q 4n up to homotopy.In Proposition 9.2, we will show that r ≡ s mod n implies P r MP P s MP and so there are at most n presentations of the form P r MP up to homotopy.Hence, for n sufficiently large, either Q 4n does not have the D2 property or Q 4n has a presentation that is not of the form P r MP up to homotopy.This has the following broad generalisation.For a finite presentation For the presentations for Q 4n above, we have (P std ) = n + 6 and (P r MP ) = n + 2r + 8 3n + 6 for 2 r < n.The following is a consequence of the fact that, if Q 4n has the D2 property, then N(Q 4n , 2) grows super-exponentially in n.
Corollary C. Let λ > 0.Then, for all n sufficiently large, at least one of the following holds: (i) Q 4n does not have the D2 property.(ii) Q 4n has a balanced presentation which is not homotopy equivalent to a two-generator two-relator presentation P with (P) λn.This suggests that, if Q 4n has the D2 property for all n, then there would need to exist a family of presentations for Q 4n for which the maximal relation length growths super-linearly in n.This leads to a new approach to finding a counterexample to the D2 problem; namely, by obstructing the existence of such a large number of presentations for Q 4n that are distinct up to homotopy.
This would be in contrast to the situation for finite abelian groups G where, from results of Metzler [27] and Browning [4], we know that every presentation for G with maximal deficiency is homotopy equivalent to a presentation P with (P) < (P std ) + t where t |G| is the greatest common divisor of the invariant factors of G and P std denotes the standard presentation for G.

Applications to the classification of 4-manifolds
In [31, Section 1.3], we discussed how a key motivation behind the homotopy classification of finite (G, n)-complexes is the homotopy classification of 2n-dimensional manifolds.Each finite (G, n)-complex X has an associated closed smooth 2n-manifold M(X), which is well defined up to homotopy equivalence [23].For fixed G and n, the manifolds M(X) are stably diffeomorphic in the sense that, for finite (G, n)complexes X, Y, there exists r 0 such that M(X)#r(S n × S n ) ∼ = M(Y)#r(S n × S n ) are diffeomorphic.The case n = 2 is of particular interest since the only known examples of closed smooth 4-manifolds which are stably diffeomorphic but not homotopy equivalent are of the form M(X), M(Y) where X, Y are homotopically distinct finite 2-complexes with certain finite abelian fundamental groups [23].
Let G be a finite group with 4-periodic cohomology and m H (G) 3.Then, conditional on G having the D2 property, Theorem A implies that there exists homotopically distinct finite 2-complexes One difficulty with tackling this problem is that, if G has 4-periodic cohomology and m H (G) 3 then we do not know whether homotopically distinct finite 2-complexes X, Y with fundamental group G and χ(X) = χ(Y) actually exist unless we (somehow) know that G has the D2 property.
In general, we can still apply Theorem 7.5 (see also Remark 1.1 (b)), which is the same statement as Theorem A but with HT(G, 2) replaced by the set of homotopy types of finite D2-complexes X with π 1 (X) ∼ = G.Recently, Adem-Hambleton showed that every finite D2-complex X with π 1 (X) ∼ = G finite has an associated closed topological 4-manifold [1,Theorem 3.8].It may therefore be possible to use Theorem 7.5 to obtain new examples of closed topological 4-manifolds that are stably homeomorphic but not homotopy equivalent.Such examples are known to exist by results of Freedman [15] though by a different construction.We will not explore Problem 1.4 further in this article.

The action of Aut(G) on general projective modules
We will conclude by considering two special properties that the projective ZG-modules P (G,n) have, and ask whether these properties are shared by arbitrary projective ZG-modules.This is of purely algebraic interest.
In order to compute we needed to quotient by a certain action of Aut(G) on [P (G,n) ] that was defined using ψ : Aut(G) → (Z/|G|) × .Let C(ZG) denote the projective class group and T G the Swan subgroup which is generated by the (I, r) where (r, |G|) = 1.The existence of the action was dependent on the following property of P (G,n) : Given such an action, the second property that the P (G,n) have is a consequence of Theorem A: In contrast to this, we show that properties (P1) and (P2) fail for arbitrary projective ZG-modules, even in the case where G has periodic cohomology.For (P1), we show that, if G = C p for p 23 prime, then there exists [P] ∈ C(ZG) and θ ∈ Aut(G) such that [P θ ] = [P] ∈ C(ZG)/T G (see Theorem 10.3).For (P2) note that, if G = Q 28 is the quaternion group of order 28, then T G = 0 and Aut(G) acts trivially on C(ZG).We will prove the following, which is stated later as Theorem 10.4.Theorem 1.5.There exists a projective ZQ 28 -module P such that [P] has non-cancellation and [P]/ Aut(Q 28 ) has cancellation.Here θ ∈ Aut(Q 28 ) acts on [P] by sending P 0 → (P 0 ) θ where P 0 ∈ [P].

Organisation
The paper will be structured as follows.In Section 2, we begin by establishing the necessary grouptheoretic facts on groups with periodic cohomology.This includes calculating m H (G) for each group and relating its value to the vanishing of σ k (G).In Section 3, we establish preliminaries on induced representations and show that, if f : G Q 4n then there is an induced map f * : Aut(G) → Aut(Q 4n ) and a surjection [P (G,n) ]/ Aut(G) [ P (G,n) ]/ Im(f * ).This allows us to show non-cancellation occurs for G by considering the case Q 4n .
In Section 4, we will combine the results in Section 2 with [33,Theorem 5.1] to show that, if m H (G) 2, then [P (G,n) ] has cancellation.Since the converse also holds, this leads to a complete determination of when cancellation occurs for a representative of σ k (G) and implies (ii) ⇔ (iii) in Theorem A.
In Section 5, we discuss locally free modules over orders in quaternion algebras and the Eichler mass formula.In Section 6 we study the orders which arise as quotients of ZQ 4n .In Theorem 6.13, we classify the for which every stably free -module is free, completing the classification started by Swan in [42, Section 8].In Sections 7 and 8, we then apply these results to prove Theorems A and B.
In Section 9, we discuss applications to the classification of group presentations, culminating in a proof of Corollary C. In Section 10, we explore the special properties of the modules P (G,n) , as described in Section 1.4.
This work can be viewed as an attempt to properly amalgamate the techniques and results obtained by Swan in [42] with the wider literature on applications of the Swan finiteness obstruction [22,29,33].As such, we will rely heavily on calculations from [42] but will give alternate proofs where possible.

Groups with Periodic Cohomology
Recall that a binary polyhedral group is a non-cyclic finite subgroup of H × where H is the real quaternions.They are the generalised quaternion groups for n 2 and the binary tetrahedral, octahedral and icosahedral groups T, O, I, which are the preimages of the dihedral groups D 2n and the symmetry groups T, O, I under the double cover of Lie groups f : We say that a group G has k-periodic cohomology for some k 1 if its Tate cohomology groups satisfy Ĥi (G; Z) ∼ = Ĥi+k (G; Z) for all i ∈ Z.Note that, if this is the case, then the periodicity must be induced by cupping with a class g ∈ H k (G; Z) (this follows from [5,XII.11.1]).It is easy to show that the binary polyhedral groups have 4-periodic cohomology.The following can be found in [5,XII.11 For G a finite group, let O(G) be the unique maximal normal subgroup of odd order.If G has periodic cohomology, then the type of G is determined by G/O(G) as follows [46,Corollary 3.6].For reasons that will become apparent later, we will split II and V into two classes.
For the rest of this section, we will assume all groups are finite and will write f : G H to denote a surjective group homomorphism.We will also assume basic facts about quaternion groups; for example, Q 2 n has proper quotients C 2 and the dihedral groups D 2 m for 1 < m < n.We begin with the following observation.We will split the remainder of this section into three parts.Firstly, we will determine the binary polyhedral quotients of groups G with periodic cohomology.We will then use this to determine m H (G) and finally we compare this with the Swan finiteness obstruction σ k (G).

Binary polyhedral quotients
If G is a finite group, we say that two quotients For a prime p, let G p be the isomorphism class of the Sylow p-subgroup of G.
H 2 factors through Ḡ2 , and so Ḡ has periodic cohomology.
In particular, this shows that B(G) is in one-to-one correspondence with the isomorphism classes of binary polyhedral groups H that are quotients of G.For brevity, we will often write H ∈ B(G) when there exists f : G H with f ∈ B(G).In order to determine B(G), it suffices to determine the set of maximal binary polyhedral quotients B max (G), that is, the subset containing those f ∈ B(G) such that f does not factor through any other g ∈ B(G).The rest of this section will be devoted to proving the following: Theorem 2.5.If G has periodic cohomology, then the type and the number of maximal binary polyhedral quotients # B max (G) are related as follows.
The entries in the table above denote the possible values that # B max (G) can attain for a group G of each type.When multiple values are listed, all values listed are attained by some group of the respective type.

Type I
Recall that G has type I if and only if its Sylow subgroups are cyclic, and G has a presentation for some r ∈ Z/m where r n ≡ 1 mod m [46, Lemma 4.1] and (n, m) = 1.We will write C m (r) C n to denote this presentation, where C n = u and C m = v .By [22, p. 165], we can assume that m is odd.
If G has a binary polyhedral quotient H, then Proposition 2.2 implies that H = Q 4a for a > 1 odd and 4 | n since m is odd.

Type II
Recall that, if G has type II, then O(G) G has cyclic Sylow subgroups and so there exists n 3 and t, s odd coprime such that Theorem 4.6].In what follows we will write If G has a binary polyhedral quotient H, then the proof of Proposition 2.
Proof.It follows easily from the standard presentation that A similar argument works in the case where G has type IIa, that is, , then Lemmas 2.7 and 2.8 imply there exists i, j for which (a, b) ≡ (1, −1), (−1, 1) and (−1, −1) mod m i , m j .By a similar argument to the above, this implies that f i , f j factors through f : G Q 8m where m = lcm(m i , m j ), which is a contradiction since m i = m j and f i , f j are maximal.Hence, 1 # B max (G) 3.
Furthermore, if G has quotients Q 8m i and Q 8m j , then this implies that (a, b) mod m i and (a, b) mod m j are distinct, which is a contradiction unless (m i , m j ) = 1.

Types Vb, VI
Suppose G has type Vb or VI.Since no binary polyhedral groups have type Vb or VI, Proposition 2.2 implies that G has no binary polyhedral quotients.Hence, # B max (G), #B(G) = 0.This completes the proof of Theorem 2.5.

Quaternionic representations
Recall that m H (G) denotes the number of copies of H in the Wedderburn decomposition of RG for a finite group G.This coincides with the number of irreducible quaternionic representations of G of (quaternionic) dimension one.We say that a finite group G is said to satisfy the For example, this shows that G satisfies the Eichler condition if and only if G has no quotient which is a binary polyhedral group.It also follows that, if G has a unique maximal binary polyhedral quotient We now show how to use this to deduce the following from Theorem 2.5.
Theorem 2.10.If G has periodic cohomology, then type and m H (G) are related as in Table 4 below.

Type IIa
If G has type IIa, then Theorem 2.5 implies that # B max (G) = 1, 2 or 3.If b = # B max (G), let f i : G Q 8m i denote the maximal binary polyhedral quotients for 1 i b.It follows from the proof of Theorem 2.5 that the m i are coprime and so the maximal quotient factoring through any two of the f i is the unique quotient which is odd since the m i are odd.This completes the proof of Theorem 2.10.

Vanishing of the Swan finiteness obstruction
Recall that a group G has k-periodic cohomology if and only if there exists a k-periodic projective resolution over ZG (see, e.g., [31,Proposition 3.9]).If G has k-periodic cohomology, then Swan [39] defined an obstruction σ k (G) ∈ C(ZG)/T G , where T G is generated by (I, r) for r ∈ (Z/|G|) × , which vanishes if and only if there exists a k-periodic resolution of free ZG-modules.Determining which groups have σ k (G) = 0 remains a difficult open problem, and has applications to the classification of spherical space forms [10].
The main result of this section will be the following extension of Theorem 2.10, which shows how m H (G) and the vanishing of σ k (G) are related to the type of G.
Theorem 2.11.The columns of the following table list the triples of type, m H (G) and σ k (G), which occur for groups G with periodic cohomology.
In order to prove Theorem 2.11, we will begin by noting the following, which is proven in [ Theorem 2.13.Let p, q be distinct odd primes.

Type IIb
Let n 1.By Dirichlet's theorem on primes in arithmetic progression, there exists a prime p such that p n, p ≡ 1 mod 8 and p ≡ ±1 mod 2 k+3 where k = ν 2 (n) is the highest power of 2 dividing n.If G = Q(16n; p, 1), then Theorem 2.13 (ii) implies that σ 4 (G) = 0.It is easy to see that B max (G) = {Q 16n } and so m H (G) = m H (Q 8n ) = n by Proposition 2.9.

Induced Representations and the Action of Aut(G)
If G is a finite group and P ∈ P(ZG), we say that Aut(G) acts on [P] if there exists a group homomorphism ψ : Aut(G) → (Z/|G|) × for which [P] − [P θ ] = [(I, ψ(θ))] ∈ C(ZG) where I is the augmentation ideal.The action is then given by sending P 0 → (P 0 ) θ ⊗ (I, ψ(θ)) for any P 0 ∈ [P] of rank one.For example, Aut(G) acts on the stably free class [ZG] for all finite groups G via the trivial map ψ(θ) = 1 for all θ ∈ Aut(G).More generally, if G has periodic cohomology, then Aut(G) acts on [P (G,n) ] by the action defined in the introduction (see also [31,Section 7]).
Recall that, if R and S are rings and f : R → S is a ring homomorphism, then S is an (S, R)-bimodule, with right-multiplication by r ∈ R given by x • r = xf (r) for any x ∈ S. If M is an R-module, we can define the extension of scalars of M by f as the tensor product which is defined since S as a right R-module and M as a left R-module.We will view this as a left Smodule where left-multiplication by s ∈ S is given by s • (x ⊗ m) = (sx) ⊗ m for any x ∈ S and m ∈ M.
For example, a group homomorphism f : G → H can be viewed as a ring homomorphism f : ZG → ZH by sending g∈G x g g → g∈G x g f (g) where Proof.Since f is characteristic, there is a map    By combining this with Theorem 3.3, we have: 3. Then there is a surjection of graded trees where f : G Q 4n for some n max{ 2 3 m H (G), 6} and where the action of Note that the two bounds for n have distinct uses.If m H (G) is small, as is the case when dealing with cancellation in Theorem A, then the bound n 6 will be most useful.If m H (G) is large, as in the asymptotic estimates in Theorem B, then we will use the bound n

Cancellation for the Swan Finiteness Obstruction
The aim of this section will be to use the results in Section 2 to prove the following cancellation result.

Type IV
O and that this is unique up to conjugacy [42,Lemma 14.3].We will need the following lemma, the proof of which is contained in the proof of [33,

The Eichler Mass Formula
Let K be a number field with ring of integers O K and let be an O K -order in a finite-dimensional semisimple K-algebra A. It is a standard fact (see, e.g., [41,Lemma 2.1]) that, if M is a finitely generated -module, then M is projective if and only if M is locally projective, that is, for all p prime, M p = M ⊗ Z p is projective over p = ⊗ Z p where Z p is the p-adic integers.
In the case where K = Q, = ZG and A = QG for G a finite group, then M projective implies that M p is a free Z p G-module for all p prime [40,Theorem 2.21,4.2].In particular, in this case, M is projective if and only if M is locally free.
Define the locally free class group C( ) to be the equivalence classes of locally free modules up to the relation P Q if P ⊕ i ∼ = Q ⊕ j for some i, j 0. By abuse of notation, we write [P] to denote both the class [P] ∈ C( ) and, where convenient, the set of isomorphism classes of projective modules P 0 where Define the class set Cls as the set of isomorphism classes of rank one locally free -modules, which is finite by the Jordan-Zassenhaus theorem [7, Section 24].Equivalently, this is the set of locally principal fractional -ideals, under the relation I ∼ J if there exists α ∈ A × such that I = αJ (see [37]).This comes with the stable class map Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnae141/7704607 by guest on 04 July 2024 which sends P → [P] and is surjective since every locally free -module P is of the form P 0 ⊕ i where P 0 ∈ Cls and i 0 [16].Define Cls [P] ( ) to be [ • ] −1 ([P]), that is, the rank one locally free modules in [P], and let SF( ) be Cls [ ] ( ), that is, the set of rank one stably free modules.
We say that has locally free cancellation if P ⊕ ∼ = Q ⊕ implies P ∼ = Q for all locally free -modules P and Q.It follows from the discussion above that has locally free cancellation if and only if Similarly we say that has stably free cancellation when P ⊕ i ∼ = j implies that P ∼ = j−i , or equivalently, if # SF( ) = 1.
If X ⊆ Cls , then we can define the mass of X to be .
Recall that a quaternion algebra A over K is totally definite if A is ramified over all archimedean places ν, that is, A ⊗ K ν is a division algebra over K ν .Note that every complex place ν splits since the only quaternion algebra over C is M 2 (C).In particular, if A is totally definite, then K must be a totally real field.
Let ζ K (s) be the Dedekind zeta function, let h K = |C(O K )| be the class number of K and let K be the discriminant of K.The following was proven in [14].
Theorem 5.1 (Eichler mass formula).Let A be a totally definite quaternion algebra over K and let be a maximal The following was first shown by Vignéras in [43], though a simplified proof can be found in [37,Theorem 5.11].
Theorem 5.2.Let A be a totally definite quaternion algebra over K and let be a maximal O Korder in A. If P, Q are locally free -modules, then In particular, this implies that mass(Cls , where C( ) denotes the class group of locally free -modules.
It was shown by Eichler that where (O × K ) + denotes the group of totally positive units, that is, those units u ∈ O × K for which σ (u) > 0 for all embeddings σ : K → R. The following can be shown using the results above as well as lower bounds on Theorem 5.3.Let A be a totally definite quaternion algebra over K and let be a maximal O Korder in A. If has stably free cancellation, then [K : Q] 6.
Remark 5.4.This was proven by Hallouin-Maire [18, Theorem 1], though it is worth noting that part of their result was incorrect as stated (see [36]).
In the notation of [42, Section 3], define the Eichler constant where d = [K : Q] and where the second equality comes from the functional equation for ζ K (s).This is rational since ζ K (−1) ∈ Q.Another constraint on the fields K over which stably free cancellation can occur is as follows.
Proposition 5.5.Let A be a totally definite quaternion algebra over K and let be a maximal O Korder in A. If has stably free cancellation, then the numerator of ζ K (−1) (or, equivalently, ei K ) is a power of 2.
Proof.If has stably free cancellation, then mass(SF( )) = [ × : O × K ] −1 since a maximal order implies O L ( ) = , that is, the numerator is 1.By Theorems 5.1 and 5.2, we also have that 6 Orders in Quaternionic Components of QG Recall that, for a finite group G, the rational group ring QG is semisimple and so admits a decomposition into simple Q-algebras.For the quaternion groups of order 4n 8, we have and where Q ab 4n denotes the abelianisation of Q 4n , where ζ d = e 2π i/d ∈ C is a primitive dth root of unity, and Q[ζ d , j] ⊆ H sits inside the real quaternions.This is stated on [42, p. 75] though a more detailed proof can be found in [22, p. 48-51].In order to apply the results of Section 5, it will be helpful to note that Further details, as well as a proof, can be found in [22, p. 51].It is straightforward to check that Q[ζ n , j] is totally definite for n 3 (see, e.g., [42,Lemma 4.3]).
For the rest of this section, we will consider the cancellation problem for orders of the form n1,••• ,nk .We begin by considering the case k = 1.It is possible to show that 2n has stably free cancellation in all other cases.It can be shown using [37,Table 2] that 2n has cancellation in all classes for all remaining cases other than 2n = 20, 24.The cases 2n = 20, 24 can then be dealt with either using a Magma program, or by explicitly computing mass(SF( 2n )) and showing it is equal to [ × : O × K ] −1 .We also note the following bounds that we will use in the proof of Theorem B. where

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Proof.To get the first inequality, note that # Cls [P] ( 2n ) mass(Cls [P] ( 2n )).We can then apply Theorems 5.1 and 5.2 and note that N K/Q (p) 2 for all p | disc(Q[ζ 2n , j]) to get that: where we note that ζ K (2) 1 by the Euler product formula.
Recall that In order to compute K = K/Q , we will use that , we get that where where The following is immediate from Theorem 6.13.Note that this shows that ZQ 4m does not have stably free cancellation for m 6 as was shown in [42,

Recall that, if
⊆ A is a Z-order in a finite-dimensional semisimple separable Q-algebra A and ⊆ ⊆ A is a maximal order, then the kernel group is defined as D( ) = Ker(i * : C( ) → C( )) where i : → , and note that i * is surjective by [42,Theorem A10].By [42, Theorem A24], this is independent of the choice of and, if f : 1 → 2 is a map of Z-orders, then f induces a map   This was necessary in order to define our action of Aut(G) on the class [P (G,n) ] since θ ∈ Aut(G) sends P → (I, ψ(θ)) ⊗ P θ for P of rank one and some ψ : The second property follows from Theorem A: The aim of this section will be to give examples to show that both properties are false for general projective modules over groups with periodic cohomology.

Cyclic group of order p
Here we will consider the case G = C p for a prime p. Recall that T Cp = 0 [39, Corollary 6.1] and so we need only find Proof.It follows from the Chevalley's ambiguous class number formula [6] (see also [17,Remark 6.2.3]) that , where  Our approach will be to use the action of Aut(G) on Milnor squares computed in [31,Lemma 8.6], and our computations will be similar to the case Q 24 that was worked out in [31,Section 9].

Problem 1 . 4 .
This raises the question of whether the M(X i ) contain further examples of stably diffeomorphic closed smooth 4-manifolds which are homotopically distinct.Let G have 4-periodic cohomology with m H (G) 3. Do there exist homotopically distinct finite 2-complexes X, Y with fundamental group G and χ(X) = χ(Y) such that M(X), M(Y) are homotopically distinct?
the recognition criteria for direct products.Hence, K × K Ḡ and, since Ḡ has periodic cohomology, Proposition 2.1 (ii) implies that |K| and |K | are coprime.Since |N| = |N |, this implies that |K| = |K | = 1 and so |N| = |N ∩ N | = |N | and N = N as required.Now let B(G) denote the set of equivalence classes of quotients f : G H where H is a binary polyhedral group.Since 4 | |H|, applying Lemma 2.3 again gives: Corollary 2.4.Let G have periodic cohomology and let f 1 , some a, b > 1 odd, and we can assume a is maximal.Then Lemma 2.6 implies that a, b | m and r ≡ −1 mod a and r ≡ −1 mod b.If d = lcm(a, b), then d | m and r ≡ −1 mod d and so there is a quotient f d : G Q 4d by Lemma 2.6.By Corollary 2.4 (or the proof of Lemma 2.6), f a and f b factor through f d , which implies that a = b = d as f a and f b are maximal.By Corollary 2.4 again, this implies that f a and f b are equivalent.In particular, this shows that # B max (G) 1.

Lemma 2 . 12 .
If G has k-periodic cohomology and type I, III or V, then σ k (G) = 0.This gives the restrictions on the vanishing of σ k (G) given in the above table.It now suffices to construct examples that realise the constraints in each column.We begin by constructing the examples with σ k (G) = 0. Firstly the groups C n , Q 4n and SL 2 (F p ), TL 2 (F p ) for p 3 can all be shown to have vanishing finiteness obstruction [10, Theorem 3.19 (c)].Now note that m H (Q 4n ) = n/2 [22, Section 12].In particular, m H (Q 8n+4 ) = m H (Q 8n ) = n for all n 1 and Q 8n+4 has type I and Q 8n has type II.Finally, the cyclic groups G = C n have type I and m H (C n ) = 0.If n 3 and a, b, c 1 are odd coprime, then define Q(2 n a; b, c) = C abc (r,s) Q 2 n , where (r, s) ≡ (1, −1) mod a, (r, s) ≡ (−1, −1) mod b and (r, s) ≡ (−1, 1) mod c.The following was shown by Milgram [29, Theorem D] and Davis [9, Corollary 6.2].

2 n 2
and so n 2m.If b = 2, 3, then G has type IIa.If Q 8m i are the maximal binary polyhedral quotients for 1 i b, then the proof of Theorem 2.5 implies that m

2 and m 2 m 3 + 3 2 n − 8 if b = 3 . Hence, n m + 3 and n 2 3Proposition 3 . 5 .
2 since the m i are odd coprime.Since n = 2m 1 , this implies that m n − 3 if b = 2 and m (m + 8) in the two cases respectively.The bound now follows since min{2m, m + 3, 2 3 (m + 8)} max{6, 2 3 m} for all m 3.In fact, the quotient f : G Q 4n is always characteristic due to the following.If f : G H where G has periodic cohomology and H is a binary polyhedral group, then f is characteristic.Proof.Let ϕ ∈ Aut(G) and consider N = ϕ(Ker(f )) G. Then N is a normal subgroup with |N| = | Ker(f )|.Since H is a binary polyhedral group, it has 4 | |H| and so Lemma 2.3 implies that N = Ker(f ).

Lemma 6 . 1 .
where θ a,b (x) = x a and θ a,b (y) = x b y.Let 4n 12.If n i are distinct positive integers such that n i n and n i | 2n for 1 i k, then the map f : ZQ 4n n1,••• ,nk induces a map
has cancellation if and only if [P (G,n) ]/ Aut(G) has cancellation.These properties each hold for free modules since the Aut(G) action on a free module is trivial.This implies that the class ZG ∈ [ZG] is a fixed point under the Aut(G)-action and so [ZG] has cancellation if and only if [ZG]/ Aut(G) has cancellation.

Table 1 .
[46,G has no subgroup of the form C p × C p for p prime; and (iii) The Sylow subgroups of G are cyclic or generalised quaternionic Q 2 n .The types of finite groups with periodic cohomology Let SL 2 (F p ) denote the special linear group of degree 2 over F p .Let TL 2 (F p ) denote the unique extension with kernel SL 2 (F p ) and quotient C 2 which has periodic cohomology.An explicit description of TL 2 (F p ) can be found in[46, p. 384] and the proof that it is characterised by the given properties can be found in[46, Proposition 2.2 (iii)].Recall that we have

Proposition 2.2. Let
f : G H where G and H have periodic cohomology.If |H| > 2, then G and H have the same type.
and C 2 , which are not in F .It is easy to verify that the quotients of SL 2 (F 3 ) are C 3 , A 4 and the quotients of TL 2 (F 3 ) are C 2 , S 3 , S 4 , none of which are in F . has normal subgroups C 2 , SL 2 (F p ) with quotients PGL 2 (F p ), C 2 .To see this, note that they each surject onto PSL 2 (F p ) and PGL 2 (F p ) respectively with kernel C 2 [46, p. 384].That PSL 2 (F p ) is a simple group then implies that any normal subgroups are of the required form.That these groups are not in F follows, for example, from [46, Proposition 2.3].
For p 5, SL 2 (F p ) has one (proper) normal subgroup C 2 with quotient PSL 2 (F p ) and similarly TL 2 (F p ) Since there are successive quotients G Ḡ H, we have G p Ḡp H p for all primes p.If G p is cyclic, then this implies Ḡp is cyclic.If not, then p = 2 and G 2 = Q 2 n , which has proper quotients D 2 m for 2 m n − 1 and C 2 .Since H has periodic cohomology, H 2 is cyclic or generalised quaternionic and so Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnae141/7704607 by guest on 04 July 2024

Table 3 .
Now suppose G has type IIb, that is, G/O(G) = Q 2 n for some n 4. By combining Lemmas 2.7 and 2.8, we get that G has a quotient Q 2 n m if and only if m | t, r ≡ 1 mod m and (a, b) = (1, −1) mod m.If G has two distinct maximal binary polyhedral quotients f Downloaded from https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnae141/7704607 by guest on 04 July 2024 Type IIb, III, IV, Va If G has type IIb, III, IV or Va, then Theorem 2.5 implies that # B max (G) = 1, that is, G has a unique maximal binary polyhedral quotient H.By Proposition 2.9, we must have that m H (G) = m H (H). By Proposition 2.2, H has the same type as G. Recall that m H (Q 4n ) = n/2 [22, Section 12].If G has type IIa, then H = Q 2 n m for n 4, m 1 odd and m H (Q 2 n m ) = 2 n−3 m 2 is even.If G has type III, IV or Va, then H = T, O or I respectively, which have m H If G has type Vb or VI, then Theorem 2.5 implies that G has no binary polyhedral quotients and so m H (G) = 0 by Proposition 2.9.
[46,a 4.3].Also define O v for v 1 to be the unique extension with kernel T v and quotient C 2 , which has periodic cohomology[46, Lemma 4.4].Note that O v is 4-periodic by[46, Corollary 5.6] and, by Proposition 2.2, it also has type IV since it has a quotient O.