Torus bundles over lens spaces

Let $p$ be an odd prime and let $\rho:\mathbb{Z}/p\rightarrow\operatorname{GL}_n(\mathbb{Z})$ be an action of $\mathbb{Z}/p$ on a lattice and let $\Gamma:=\mathbb{Z}^n\rtimes_{\rho}\mathbb{Z}/p$ be the corresponding semidirect product. The torus bundle $M:=T^n_{\rho}\times_{\mathbb{Z}/p}S^{\ell}$ over the lens space $S^{\ell}/\mathbb{Z}/p$ has fundamental group $\Gamma$. When $\mathbb{Z}/p$ fixes only the origin of $\mathbb{Z}^n$, Davis and L\"uck \cite{DavisLuckTorusBundles} compute the $L$-groups $L^{\langle j\rangle}_m(\mathbb{Z}[\Gamma])$ and the structure set $\mathcal{S}^{geo,s}(M)$. In this paper, we extend these computations to all actions of $\mathbb{Z}/p$ on $\mathbb{Z}^n$. In particular, we compute $L^{\langle j\rangle}_m(\mathbb{Z}[\Gamma])$ and $\mathcal{S}^{geo,s}(M)$ in a case where $\underline{E}\Gamma$ has a non-discrete singular set.


Introduction
In [9] and [10], Davis and Lück study groups of the form where p is an odd prime and ρ : ℤ/p → GL n (ℤ) has no nonzero fixed points.They compute the topological K-theory of the real and complex group C * -algebras of Γ in [9].Along the way, they compute K * (BΓ) and several other K-theory groups.Letting T n ρ denote the torus with ℤ/p-action determined by ρ and letting S ℓ denote a sphere with a free ℤ/p-action, define M := T n ρ × ℤ/p S ℓ .The manifold M is a torus bundle over a lens space and the assumption that ℤ/p acts freely on ℤ n \ {0} implies that the action of ℤ/p on T n ρ has discrete fixed points.In [10], Davis and Lück use the computations from [9] to determine the L-groups of ℤ[Γ] and the structure set of M in the sense of surgery theory.
For the L-theory computation, Davis and Lück use the Farrell-Jones Conjecture for Γ to conclude ).
Then they compute the homology group by inverting p and inverting 2. After inverting p, H Γ m (EΓ; L ⟨−∞⟩ ℤ ) becomes part of a split short exact sequence.After inverting 2, there is an isomorphism and one applies the computations of [9].
In this paper, we study the case where the action of ℤ/p on ℤ n is not necessarily free on ℤ n \ {0}.A free abelian group with a ℤ/p-action can be written as , where M ⊗ ℚ ≅ ℚ(ζ) a and N ⊗ ℚ ≅ ℚ[ℤ/p] b .Here, we use ζ to denote a primitive p-th root of unity.We say that such a module is of type (a, b, c) in which case n = a(p − 1) + bp + c.The ℤ[ℤ/p]-module ℤ n will be denoted L. As in [10] define Γ := L ⋊ ρ ℤ/p and define M := T n ρ × ℤ/p S ℓ .The fixed points of the corresponding ℤ/p-action on T n ρ is a disjoint union of a(p − 1) many (b + c)-dimensional tori rather than a discrete set when L is of type (a, b, c).Proving that L Theorem 4.1.Suppose Γ is of type (a, b, c).Then: (1) The differentials d i,j r in the Atiyah-Hirzebruch-Serre spectral sequence for the fibration BΓ → Bℤ/p vanish for r ≥ 2.
Using the techniques in [10], we are then able to compute the L-groups of ℤ[Γ] and the structure sets of M. The description of the simple structure sets in [10] are nice because the torsion comes from L(ℤ).In our case, we will inevitably encounter 2-torsion coming from L h 2m (ℤ[ℤ/p]).Fortunately, we are still able to obtain the integral results below (as opposed to results that only hold after inverting 2).This is essentially due to the splitting after inverting p of the maps in Proposition 3.2 and our understanding of the Whitehead groups (in particular, [12,Theorem 1.10]).
In the theorems below P will denote the set of conjugacy classes of nontrivial finite subgroups of Γ (all of which are isomorphic to ℤ/p).We let N Γ P denote the normalizer of P in Γ and we let W Γ P := N Γ P/P denote the Weyl group.The groups L ⟨j⟩ m (ℤ[Γ]) are Ranicki's surgery groups with decoration and S per,⟨j⟩ n+ℓ+1 (M) is Ranicki's algebraic structure set.The set S geo,⟨j⟩ n+ℓ+1 (M) is the geometric structure set obtained from the surgery exact sequence by using connective L-theory.We refer to Section 6 for more details.When j = 2 (resp.1), Theorem 6.12 specializes to a computation of the usual simple structure set (resp.homotopy structure set) in the sense of surgery theory.
The L-groups appearing in these computations are computable; the normalizers are isomorphic to ℤ b+c × ℤ/p and the Weyl groups are isomorphic to ℤ b+c so Shaneson splitting allows us describe these groups in terms of the L-groups of ℤ and ℤ[ℤ/p].
Remark.The isomorphisms in the theorems above do not come from "natural" maps.For instance, the map N Γ P → Γ induces a map L Remark.To prove Theorem 5.5, it suffices to prove the case that Γ is of type (a, b, 0), i.e., Γ = L ⋊ ℤ/p, where L is a free abelian group with no ℤ-summands with trivial ℤ/p action.Indeed, one can inductively apply Shaneson splitting to obtain the more general case.It would be convenient to do this with the structure sets as well.However, we are not aware of a reference that gives Shaneson splitting for the structure sets.

Geometric interpretations of the structure set
In the situation of [10], the computation of the structure set is interpreted as follows.Suppose f : N → M is a structure and let f : N → M ≅ T n × S ℓ denote the ℤ/p-cover.The proof of [10,Theorem 10.6] applies in our case so we have the following interpretation of the H n (T n ρ ; L(ℤ)⟨1⟩) ℤ/p summand of the structure set.
Understanding the ).These groups have 2-torsion which involves the ideal class groups.Rationally, there is an isomorphism In the above expression, R ± ℂ (ℤ/p) denotes the group of virtual complex ℤ/p-representations whose characters are of the form χ ± χ −1  and reg denotes the regular representation.We give a heuristic description of this summand in the case Γ = ℤ n × ℤ/p (i.e., when M is the product of a torus and a lens space L ℓ ).Note that the inner sum on the right-hand side is indexed by the standard subtori ) has a ℤ/p-cover so we may take its ρ-invariant to obtain an element of (R (−1) k+ℓ+1/2 ℂ (ℤ/p)/⟨reg⟩) ⊗ ℚ.This gives an element in the summand corresponding to the torus T k .This description is not technically correct; without some modifications, we do not know how to show it is well-defined.A rigorous interpretation of these ρ-invariants will be the subject of future work.

Outline
In Section 2, we review properties of the ℤ[ℤ/p]-module L and we state relevant properties of the group Γ.In Section 3, we introduce some machinery from [10]; our L-theory computations rely on the Farrell-Jones conjecture and a description of EΓ as a homotopy pushout.Section 4 is the main computational part of the paper.It is devoted to computing the topological K-theory of BΓ and recording some consequences of the computation.The main computational tool we use is the Atiyah-Hirzebruch-Serre spectral sequence.In Sections 5 and 6, we compute the L-groups of ℤ[Γ] and the structure set of M.These computations follow the computations in [10] very closely.For Section 6, in particular, our results follow from the proofs of [10] with only slight modifications.

Group theoretic preliminaries
We are interested in groups Γ of the form L ⋊ ℤ/p, where L is a finitely generated free abelian group.Throughout this paper, p will always be an odd prime.Curtis and Reiner classified these groups in [7,Theorem 74.3].

Theorem 2.1. Let ζ be a primitive p-th root of unity. If L is an indecomposable integral ℤ/p representation, then L is either of the following:
(1) B ⊆ ℚ(ζ) a fractional ideal with action given by multiplication by ζ .The group ring ℤ[ℤ/p] is an example of a module of the form (2); it is of the form ℤ[ζ] ⊕ 1 ℤ.When p = 3, for instance, the assignment defines an isomorphism of ℤ[ℤ/p]-modules.
The group Γ determines a torus bundle M over a lens space with fundamental group ℤ/p.Note that the manifold corresponding to (L ⊕ ℤ) ⋊ ℤ/p is a product M × S 1 .As taking products with S 1 is a well understood operation in topology, we will sometimes make the simplification that L does not have any ℤ summands.b, c).Then the following hold: (1) The virtually cyclic subgroups of Γ are isomorphic to either the trivial group, ℤ/p, ℤ or ℤ × ℤ/p.
(2) Let P denote the set of conjugacy classes of maximal finite subgroups of Γ.Then |P| = p a .
(3) If P is a finite subgroup of order p, then the normalizer N Γ P is isomorphic to ℤ b+c × P and the Weyl group Proof.The elements of our group can be written as xy i , where x ∈ L and y is a generator of ℤ/p.Let ρ(−) : L → L denote the action of ℤ/p on L.
To show the first statement, if xy and x  y are in a subgroup H, then x(x  ) −1 must be in H. Hence, if H is finite, x = x   .It follows that the nontrivial finite subgroups are isomorphic to ℤ/p.Suppose V is an infinite virtually cyclic subgroup that is not infinite cyclic.Then V must surject onto ℤ with kernel ℤ/p and V ∩ L is an infinite cyclic group.Let xy be a torsion element of V and let v ∈ V ∩ L. Then

Equivariant homology and the Farrell-Jones conjecture
In this section, we introduce some preliminary material on equivariant homology following [8] and on the Farrell-Jones conjecture, which will allow us to compute L and K-groups.

Equivariant homology
Let G be a discrete group.Given a covariant functor E : Grpd → Sp from the category of small groupoids to spectra, define the equivariant homology groups of a G-CW-complex X to be where Or(G) is the orbit category of G and G/H is the groupoid associated to the G-set G/H and − ∧ Or(G) − denotes a coend.The functor X − + sends an orbit G/H to the fixed point set In this paper, we will take E to be K −∞ ℤ and L ⟨j⟩ ℤ for j = 2, 1, 0, . . ., −∞.The corresponding homology theories have the property that ) for all m ∈ ℤ.We also use equivariant topological K-theory, which sends G/H to the representation ring R ℂ (H) when G is finite.
Equivariant cohomology is defined analogously.We refer to [8] for more details.

Remark.
The notation H G m (X; L ⟨j⟩ ℤ ) denotes the Davis-Lück equivariant homology as mentioned above whereas the notation H m (X; L ⟨j⟩ (ℤ)) denotes the generalized homology of X with coefficients in the spectrum L ⟨j⟩ (ℤ).

Classifying spaces
Definition 3.1.Let G be a group.A family of subgroups is a nonempty set F of subgroups closed under taking subgroups and conjugation.A classifying space for F, denoted E F G, is a G-CW-complex satisfying If {e} is the family consisting of only the trivial group, then E {e} G = EG.The primary families we will consider are Vcyc, the collection of virtually cyclic subgroups, and Fin, the collection of finite subgroups.We will use the following notation: Specifying to the case where Γ = L ⋊ ℤ/p, [13, Corollary 2.10] shows that there is the following homotopy pushout diagram: Proceeding as in [9, Lemma 7.2] and using that EW Γ P is a model of EN Γ P as an N Γ P-space, we obtain the following long exact sequences.

The Farrell-Jones conjecture
One of the primary computational tools that we use is the Farrell-Jones conjecture, which has been proved in many cases.[4] proves the conjecture for our group Γ.

Topological K-theory
This is the main computation section of the paper.The goal of this section is to prove the following theorem.
(2) If b ̸ = 0 or c ̸ = 0, then there is an isomorphism of abelian groups In Theorem 4.1 we use Ẑp to denote the p-adic integers.We will reduce to the case c = 0 and proceed by induction on b.The case where L is type (a, 0, 0) is [9, Theorem 3.1].

Group cohomology
We collect some important facts about group cohomology.For a finite group G and a There is an exact sequence

Lemma 4.2. Let G be a finite group and let M be a finitely generated ℤ[G] module with no p-torsion for all primes p dividing the order of G. Then, for all i ∈ ℤ, there is an isomorphism
Proposition 4.3.Suppose L is a module of type (a, b, 0).Then there is an isomorphism of Tate cohomology groups Proof.We follow the proof of [9, Lemma 1.10 (i)] where the case b = 0 is done.As Ĥ(ℤ/p; M) ≅ Ĥ(ℤ/p; M ⊗ ℤ (p) ), We may assume b = 1 so Here, the second line is obtained by the isomorphism (b, m)  → (α 0 b, m) and the fourth line follows from a similar isomorphism.The third line follows from the fact that where a j is the number of partitions of j.
We will also need the following lemma, which appears in the proof of [9,Lemma 1.10].
The following result computes the fixed sets of the ℤ/p-action on the torus corresponding to a module of type (a, b, c).
Proof.In the case L is of type (a, 0, 0), this is [9, Lemma 1.9(v)].The case where L is of type (0, 0, c) is straightforward.Since T n ρ is equivariantly a product, it suffices to show that, when L is of type (0, 1, 0), the fixed set is a circle.
Suppose L is of type (0, 1, 0).Consider the following short exact sequence of ℤ/p-modules.
This gives rise to the top exact sequence in the diagram below: ≅ By the proof of Proposition 4.3, there is an isomorphism of ℤ/p-modules L ⊗ ℝ ≅ ℝ[ℤ/p] which identifies L with a finite index ℤ/p-submodule of ℤ[ℤ/p].This gives the vertical maps.This also implies that H 1 (ℤ/p; L) = 0.It follows that (T p ρ ) ℤ/p ≅ S 1 .

Cohomology of T n ρ
In order to relate these algebraic results to the problem of computing topological K-theory, we record some results on the cohomology of T n ρ as a ℤ[ℤ/p]-module.Let L be a module of type (a, b, c) determining the representation ρ.Then, as is also a module of type (a, b, c).
We will need to use the topological K-theory of T n ρ considered as a ℤ[ℤ/p]-module.The Atiyah-Hirzebruch spectral sequence collapses for tori so, as an abelian group, K m (T n ) ≅ ⨁ ℓ∈ℤ H m+2ℓ (T n ).The proof of [9, Lemma 3.3] shows this is also true as ℤ[ℤ/p]-modules.Lemma 4.7.Let T n ρ be a torus with a ℤ/p-action as above.Then as a ℤ[ℤ/p]-module,

Facts about spectral sequences
Suppose E → B is a fibration with connected base space and with fiber F. Let H * be a generalized cohomology theory and let H * be a generalized homology theory.There are Atiyah-Hirzebruch-Serre spectral sequences In the cohomology spectral sequence above, we have when B is path connected with fundamental group G. Thus, E 0,j ∞ is a subgroup of H j (F) G .In the homology spectral sequence, The following is in the appendix of [9].

equal to the map on cohomology induced by the inclusion
The fibrations we use will come from group extensions N → G → G/N, where G/N is finite.We have the inclusion of BN as the fiber of BG → B(G/N).This map induces the inclusion N → G so, up to homotopy we may think of BN → BG as a covering space with fibers G/N.If G/N is finite, then there is a transfer map τ * : H m (BN) → H m (BG) such that the composition is the norm map.For a generalized homology theory there is a transfer map τ * : is the norm map.To summarize, we have the following result.
Proposition 4.9.Suppose N → G → G/N is a group extension with G/N finite.In the cohomological Atiyah-Hirzebruch-Serre spectral sequence for the fibration BN if it is in the image of the norm map.In the homological spectral sequence, an element x ∈ E 2 0,j represents a nonzero element in E ∞ 0,j if the norm of x is nonzero.
We specialize [5,Theorem 13.2] to the following statement.
Theorem 4.10.If there is an N > 0 such that the differentials d i,j r in the spectral sequence (4.1) vanish for r > N, then this converges strongly in the following sense.For the filtration the following hold:

Vanishing of differentials
We now prove Theorem 4.1 (1).First, we reduce to the case where Γ is of type (a, b, 0).Proof.If Γ is of type (a, b, c + 1), then it is isomorphic to Γ  × ℤ, where Γ  is of type (a, b, c).Since BΓ ≅ S 1 × BΓ  , there is a morphism of spectral sequences coming from the map of fibrations BΓ  → BΓ over Bℤ/p.Let Bℤ/p (s)  denote the s-skeleton of Bℤ/p.Let Y s denote the preimage of Bℤ/p (s)  under the map BΓ  → Bℤ/p.The exact couple giving rise to the spectral sequence for BΓ   is given by the abelian groups The exact couple giving rise to the spectral sequence for BΓ is given by abelian groups K m (Y s × S 1 ) and K m (Y s × S 1  , Y s−1 × S 1 ).We can write the groups in this exact couple as a A s,m−s . The maps between these groups are sums of the maps between the exact couple for BΓ   .Our inductive hypothesis therefore implies that the differentials vanish after the first page of the spectral sequence.
Remark.When c > 0, we do not need to know Theorem 4.1 (1) in order to prove 4.1 (2) provided we have the c = 0 case.Suppose Theorem 4.1 (1) is true for groups of type (a, b − 1, 0).We will show this is true for L a representation of type (a, b, 0).First we introduce some notation.

Notation. • Given
we make the following abbreviations: • For d = 1, . . ., b, define L d to be the representation of type (a, b − 1, 0), where the d-th summand of N L is removed.We have group homomorphisms Clearly, the composition ψ d ∘ ϕ d is the identity.We will henceforth denote L d ⋊ ℤ/p by Γ d .It follows that there is the retraction of bundles below: The maps Bϕ d and Bψ d induce maps on cohomology, which we denote by ϕ * d and ψ * d .The terms of the Atiyah-Hirzebruch-Serre spectral sequence for BΓ d will be denoted E i,j r,d .
• By abuse of notation, we will define ϕ d : • For a module M, let Λ even M be the sum of all Λ 2r M and define Λ odd M similarly.
• When one of the r i is neither 0 nor p, then Ĥ * (ℤ/p; A r ) = 0. Indeed, Lemma 4. Recall we are considering the Atiyah-Hirzebruch-Serre spectral sequence We first check that, if i > 0, then d i,j is trivial.Suppose that j is even.The term can be decomposed as a sum and the term E i+2,j−1 2 can be decomposed as a sum Moreover, we may identify the image of By the inductive hypothesis and the fact that r,d is the identity, these terms are in E i,j ∞ .It therefore remains to consider the effect of d i,j 2 on the subgroup corresponding to r = (p, . . ., p).Thus we consider either a map depending on the parity of i and b.In either case, Proposition 4.4 implies that these maps are 0.This completes the proof that d i,j 2 = 0 when i > 0 and when j is even.The case that j is odd follows identically.Now, suppose that i = 0.It follows from [9, Lemma 1.10 (i)] that H 2 (ℤ/p; Λ odd M * L ⊗ A (p,...,p) ) = 0 so we just need to show that the restriction of the differential must be trivial.But by Proposition 4.9 and the fact that Ĥ0 (ℤ/p; Λ odd M * L ⊗ A (p,...,p) ) = 0, the left-hand side must be in E 0,j ∞ .The cases r > 2 follows from a similar analysis.

Convergence
In this subsection, we prove the second part of Theorem 4.1 in the case Γ is of type (a, b, 0).The general case follows from this computation.We induct by assuming that the second part is true for groups of type (a, b  , 0), where b  < b.
Define the filtration for K m (BΓ) given by the spectral sequence by is a free abelian group, it suffices to compute F 1 .We first make a simplification.Suppose that N L has rank bp as an abelian group.Let Γ  be the group ) denote the filtration on K m (BΓ  ).

Lemma 4.12.
There is an isomorphism .This is a map of bundles over Bℤ/p so this induces a map on the spectral sequences.The induced maps H s (ℤ/p; For the remainder of the section, we will assume In the computation of .Using this, we construct a map of bundles over Bℤ/p.Let E denote the sphere bundle S bp × ℤ/p (T a(p−1) × Eℤ/p).

Lemma 4.13.
There is an isomorphism Moreover,the spectral sequence Proof.There is a section B(M L ⋊ ℤ/p) → E such that Th 0 := E/B(M L ⋊ ℤ/p) is the Thom space of the real vector bundle × Eℤ/p).This gives us an exact sequence where ℝ b has trivial ℤ/p action and V is a real ℤ/p-representation obtained by forgetting the complex structure of a complex ℤ/p-representation.Writing Th 1 as the Thom space of V × ℤ/p (T a(p−1) × Eℤ/p), we have Th 0 = Σ b Th 1 .We need to show that Km (B(M L ⋊ℤ/p)) ≅ Km (Th 1 ).Define E n := V × ℤ/p (T a(p−1) × Eℤ/p (n) ), where Eℤ/p (n)  denotes the n-skeleton of Eℤ/p and let Th n 1 denote the Thom space of E n considered as a vector bundle over T a(p−1) × ℤ/p Eℤ/p (n)  .The Thom isomorphism for K-theory implies K m (Th n 1 ) ≅ K m (T a(p−1) × ℤ/p Eℤ/p (n) ).
This induces an isomorphism on inverse systems indexed by n.In particular, K m (Th n 1 ) is Mittag-Leffler.Thus, The proof of the second part is similar to the proof of the first part of Theorem 4.1.
Proof of Theorem 4.1 (2), case b = 1.Denote the filtration on K m (E) coming from the fibration E → Bℤ/p by It follows from Lemma 4.13 and the b = 0 case of Theorem 4.1 that G 1 ≅ Ẑ(p−1)p a p .Suppose i ≥ 1 and j is odd.The E i,j 2 term for the Atiyah-Hirzebruch-Serre spectral sequence for K-theory of the fibration BΓ → Bℤ/p is On the other hand, the corresponding term for the fibration E → Bℤ/p is The decompositions of the coefficients above follow from the Künneth formula for K-theory [1].Naturality of the Künneth formula shows that these decompositions respect the ℤ/p-module structures and that BΓ → E induces isomorphisms on E i,j 2 .A similar argument shows that this is an isomorphism when j is even as well.Therefore, Theorem 4.10 implies that G 1 ≅ F 1  .

The b > 1 case
Assume now that b > 1 and that Theorem 4.1 is true for groups of type (a, b  , 0), where b  < b.We will need to make some more observations.Lemma 4.14.Suppose we have filtered abelian groups with a filtration preserving split injection G 1 → F 1 whose splitting also preserves the filtration.Then is a filtered abelian group with slices Proof.The splitting implies G 1 ∩ F s = G s , where we take intersections in F 1 and identify the G s with their images in F 1  .The result follows by considering the filtration Denote the filtration on K m (BΓ d ) corresponding to the spectral sequence for BΓ d by The maps F s d → F s are split injections.In particular, F s 1 → F s → F s /F 1 is a split short exact sequence.Now consider the square The right vertical map and the top map split which gives a splitting for the bottom map.Therefore, there is a split short exact sequence where F s 1 + F s 2 denotes the subgroup of F s generated by F s 1 and F s 2 .Continuing this way, we obtain split short exact sequences This shows that ) is a filtration for the subgroup of K m (BΓ) with slices H i (ℤ/p; (ΛN L ) ⊗ A r ), where at least one of r 1 , r 2 , . . ., r d is 0.

Lemma 4.15.
There is an isomorphism where Here, κ b,m = (−1) b+1 when m is even and κ b,m = 0 when m is odd.
Proof.Let us abbreviate One checks that there is the following resolution: Note that the F 1 α 1 ,...,α d are finitely generated free Ẑp -modules.Moreover, any abelian group homomorphism of such modules is a Ẑp -module homomorphism.¹As an abelian group, ∑ b α=1 F 1 α is torsion-free; the retractions Since Ẑp is a principal ideal domain, we see that ∑ b α=1 F 1 α is a finitely generated free Ẑp -module.The result follows from tensoring with Qp and counting dimensions.For d = 1, . . ., b − 1, the induction hypothesis implies which accounts for the term κ b,m .
1 One examines the bijections where the second bijection follows from the fact that Ẑp has a unique subgroup of index p n .Proof.This follows from expanding 1 H s (ℤ/p; Λ even M L ), s odd, H s (ℤ/p; Λ odd M L ), s even.
Recall that we have assumed the submodule N L is isomorphic to ℤ[ℤ/p] b so there is the sphere bundle quotient E discussed in Lemma 4.13.As before, we let {G s } denote the filtration on K m (E).Note that there is a split injection of filtered abelian groups F 1 1,...,b → G 1 . One checks that and that the map of filtered abelian groups ) induces an isomorphism on slices.Using that The isomorphism follows similarly.Therefore, we obtain an isomorphism Using Lemma 4.15 and Lemma 4.16, we obtain F 1 ≅ Ẑ(p−1)p a 2 b−1 p as desired.

Corollaries of the K-theory computation
We record some consequences of Theorem 4.1 that we will need for computing the L-groups and the structure sets.These results are proven for groups of type (a, 0, 0) in [9] so we will assume that either b ̸ = 0 or c ̸ = 0 in this section.We need to import the following results, which can be found in [9].
Lemma 4.17.For a finite group G, there is an isomorphism Theorem 4.18 (Universal Coefficients Theorem).For any CW-complex and all m ∈ ℤ, there is an short exact sequence Furthermore, when X is finite, there is a an exact sequence These sequences are natural in X.

Theorem 4.19 (Equivariant Universal Coefficients Theorem).
Suppose that H is a finite group and that X is an H-CW-complex.For m ∈ ℤ, there is a short exact sequence of R ℂ (H)-modules Furthermore, when X is finite, there is an exact sequence Corollary 4.20.The differentials in the following Atiyah-Hirzebruch-Serre spectral sequence vanish: Proof.The proof of this result is similar to the proof of Theorem 4.1 so we only sketch it.First, note that we may reduce to the case that Γ is type (a, b, 0) as in Lemma 4.11.
As a ℤ[ℤ/p]-module, K j (T n ρ ) is isomorphic to the dual K j (T n ρ ) * .Since dualization commutes with taking direct sums and dualization sends modules of type (a, b, c) to modules of type (a, b, c), the induction argument in the proof of Theorem 4.1 proves that the differentials d 2 i,j vanish when i > 2. Now, we check that the differentials mapping to the left column vanish.By the induction hypothesis, it suffices to show that the restriction of the differential summands of the form we only need to check the differentials vanish in the first case.The left column consists of terms K j (T n ρ ) ℤ/p .In order to show the differentials vanish, it suffices to show that the transgression K j (T n ρ ) ℤ/p → K j (BΓ) is injective.The norm map K j (T n ρ ) ℤ/p → K j (T n ρ ) ℤ/p factors through the transgression.Since Ĥ−1 (ℤ/p; (Λ even M * L ⊗ A (p,...,p) ) * ) ≅ Ĥ1 (ℤ/p; Λ even M * L ⊗ A (p,...,p) ) ≅ 0 the norm map is injective on the summand H 0 (ℤ/p; (Λ even M * L ⊗ A (p,...,p) ) * ).Hence, this term is in E ∞ 0,j .The proof that d r i,j vanishes for r > 2 is similar.

Corollary 4.21.
There is an isomorphism where Hom ℤ (K m (T n ρ ) ℤ/p , ℤ) is the image of the map induced by the inclusion of the fiber T n ρ → BΓ.
Proof.Define B s := T n ρ × ℤ/p Eℤ/p (s)  .We have the following direct system of short exact sequences: Taking the colimit, we obtain By considering the Atiyah-Hirzebruch-Serre spectral sequence for the fibration B s → Bℤ/p (s)  and comparing it to that of the fibration BΓ → Bℤ/p, we see that , where A s is some p-group and C s is a (possibly trivial) finitely generated free abelian group.Indeed, the limit of the A s is exactly F 1 in the filtration of K m (BΓ).Moreover, by considering morphisms of spectral sequences, C s is not in the image of K m (B s+1 ).Therefore, the right hand term is isomorphic to Hom , ℤ) (we abuse notation here and let A s denote the p-group in K m+1 (B s )).We obtain isomorphisms , where Â denotes the Pontryagin dual of a locally compact abelian group A. We refer to the proof of [9, Theorem 4.1] and [16] for details regarding Pontryagin duality.
It remains to check that the subgroup Hom ℤ (K m (T n ρ ) ℤ/p , ℤ) is the image of the map induced by T n ρ → BΓ.The inclusion induces the composition By the commutativity of the diagram the result follows.
In the future, we will write K m (T n ρ ) ℤ/p rather than Hom ℤ (K m (T n ρ ) ℤ/p , ℤ).
Corollary 4.22.After inverting 2, KO m (BΓ) is the sum of a finitely generated free ℤ[ 1  2 ]-module and a p-torsion group.Moreover, the inclusion T n ρ → BΓ induces a surjection on the finitely generated free ℤ[ 1  2 ]-module.Proof.Consider the following diagram: The horizontal composites are multiplication by 2. Thus, after inverting 2, i * is injective.Applying Corollary 4.21 proves the first part.
For the second part, let x ∈ KO m (BΓ) be an element in the finitely generated free ℤ[ 1  2 ]-submodule of KO m (BΓ).Then i * x is in the image of the middle vertical map by Corollary 4.21.It pulls back to an element y ∈ K m (T n ρ ).But then x is the image of 1 2 r * y under the outer vertical maps.Corollary 4.23.The groups K m Γ (EΓ), KO m Γ (EΓ) and KO Γ m (EΓ) are p-torsion free.
Proof.First, we show that K m Γ (EΓ) is p-torsion free.Let Rℂ (ℤ/p) denote the reduced complex representation ring of the group ℤ/p.Proposition 3.2 gives us the top row of the following diagram: The bottom row comes from applying K-theory to the homotopy pushout diagram obtained from the quotient of diagram 3.1 by Γ.The vertical map ⨁ (P)∈P Rℂ (ℤ/p) Γ (EΓ) be an element of order p.Then x must pull back to an element in K m (BΓ) which then pulls back to and element x ∈ ⨁ (P)∈P Ẑ2 b+c−1 p (here we use that K m (BΓ) is torsion free).Using the transfer, one can check that K m (BΓ) is the sum of a finitely generated free abelian group with a finite p-group.The image of φ must be contained in the p-group (it is the entire p-group as K m (BΓ) is torsion free).Thus φ factors through for some N.But every element in this quotient can be represented by an element in the image of ⨁ (P)∈P Rℂ (ℤ/p) 2 b+c−1 . Let x ∈ ⨁ (P)∈P Rℂ (ℤ/p) 2 b+c−1 be an element lifting the projection of x.Then we obtain that x maps to x ∈ K m Γ (EΓ) but exactness implies that x = 0.This shows that K m Γ (EΓ) has no p-torsion.Lemma 4.17 and Theorem 4.19 imply that K Γ m (EΓ) has no p-torsion.Since multiplication by 2 in KO Γ m (EΓ) factors through K Γ m (EΓ), it follows that KO Γ m (EΓ) has no p-torsion.

L-theory computations
For geometric applications, one is typically interested in the groups There are maps and we define This theory is developed in [14].The group L ) for all j.One of the primary L-groups that appear in our computations are the groups L ⟨j⟩ m (ℤ[N Γ P]) and L ⟨j⟩ m (ℤ[W Γ P]) so we take some time to discuss these groups here.Recall that N Γ P ≅ ℤ b+c × ℤ/p and W Γ P ≅ ℤ b+c .Shaneson splitting [14,Theorem 17.2] gives isomorphisms The quotient is So, these groups can be computed in terms of the L-groups of the group ℤ/p.Finally, we record the L-groups of ℤ/p.The following theorem can be found in [3] and [2].
is not an isomorphism.

5.1
The L ⟨−∞⟩ computation Theorem 5.2.There is an isomorphism Proof.We show that the isomorphism holds when p is inverted and when 2 is inverted.This suffices since the groups are finitely generated.Using Proposition 3.4 and Proposition 3.2, we have the following long exact sequence: By the remark after Proposition 3.2, we can make the identification After inverting p, the sequence splits into short exact sequences But when p is inverted, the right-hand term is isomorphic to H m (T n ρ ; L(ℤ)) ℤ/p by a transfer argument [9, Proposition A.4].This is free so the sequence splits.Note that K m is a free abelian group since free abelian.Now it remains to show that the groups are isomorphic after inverting 2. By [10, Theorem 4.2], equivariant L-theory homology and equivariant KO-homology agree after inverting 2. We obtain the resulting long exact sequence.
We can write KO m (BΓ)[ 1  2 ] ≅ F ⊕ A, where F is a free ℤ[ 1  2 ]-module and A is a p-torsion group.The map KO Γ m (EΓ)[ 1  2 ] → KO m (BΓ)[ 1  2 ] is invertible after inverting p so there is a partial section defined on a p-power index subgroup of KO m (BΓ)[ 1  2 ].Since KO Γ m (EΓ) has no p-torsion, this subgroup must be a p-power index subgroup of F, hence isomorphic to F. This partial splitting gives a subgroup K m [ 1  2 ] ⊕ F, which is p-power index in KO Γ m (EΓ)[ 1  2 ].Therefore, there is an isomorphism ] is a free ℤ[ 1  2 ]-module isomorphic to a p-power index subgroup of H m (BΓ; L(ℤ))[ 1  2 ].

Arbitrary decorations
For the groups studied in [10], the assembly map ) is an isomorphism for all decorations j.This is essentially because the normalizers of finite subgroups are isomorphic to ℤ/p and because the analogous result holds for ℤ/p.In our case, the normalizers are of the form ℤ b+c × ℤ/p so the situation becomes more complicated.
In order to study L-theory with arbitrary decorations, we need to use Whitehead groups.Theorem 5.5.For j = 2, 1, 0, . . ., −∞, there is an isomorphism Proof.The homology group H m ((T n ρ ) P ; L ⟨−∞⟩ (ℤ[ℤ/p])) fits into the exact sequence where the map L(ℤ) → L ⟨−∞⟩ (ℤ[ℤ/p]) splits.By the Farrell-Jones conjecture (or Shaneson splitting), we may identify It follows from Theorem 5.2 and the resulting identification that the result is true when 2 is inverted.Now, we need to check that the result is true when p is inverted.Since Wh j (N Γ P), Wh j (Γ) = 0 for j ≤ −1, the Farrell-Jones conjecture and Proposition 3.2 imply that is a split short exact sequence when p is inverted.We claim the same is true for j > −1.To verify this claim, we induct on j, using j = −1 as the base case.Consider the following diagram: . . . . . . . . .
Remark.The isomorphism in Theorem 6.2 can be rewritten as

The periodic structure set of M
In order to compute the periodic structure sets S Proof.The proof is similar to the proof of [10, Lemma 8.3] so we give an outline.
We first show that the differentials in the first spectral sequence vanish.It suffices to show that the differentials vanish after inverting p and after localizing p.After inverting p, the only nonzero terms are in the column E 2 0,j so the differentials must vanish.After localizing at p and applying [10, Theorem 4.2], it suffices to show that the differentials for the homology theory KO * (−) (p) vanish.But multiplication by 2 in the homology theory KO * factors through the map K * .This exhibits a KO * (BΓ) (p) as a retract of K * (BΓ) (p) .The result follows from Corollary 4.20, which asserts that the differentials in K * (BΓ) (p) vanish.
To show that the differentials in the second spectral sequence vanishes, we consider the map f : M → BΓ.This induces a map from the second spectral sequence to the first which is bijective on terms E 2 i,j for i < ℓ and surjective on terms E 2 ℓ,j .Since the terms E 2 i,j (M) = 0 for i > ℓ, this implies that the differentials vanish as desired.
Let F ℓ,n (−) and E r ℓ,n (−) denote the filtration terms and E r terms of the spectral sequences in Proposition 6.3.Note that F ℓ,n (M) = H n+ℓ (M; L(ℤ)) because the base space is ℓ-dimensional.Note also that there is always a quotient pr : The maps η n+ℓ+1 are from the surgery exact sequence and A n (T n ρ ) ℤ/p is the assembly map.We define μ to be the composite of the left vertical maps and we define σ to be the composite of μ with the isomorphism L n (ℤ[ℤ n ρ ]) ℤ/p ≅ H n (T n ρ ; L(ℤ)) ℤ/p .Our goal now is to show the following.Therefore, the right vertical map is surjective, which completes the proof.Proof.The proof is the same as the proof of [10,Lemma 8.5].
From Lemma 6.4 and Lemma 6.5, we conclude that σ × S

( 2 )( 3 )Example 1 .
B ⊕ ℤ, where B ⊆ ℚ(ζ) is a fractional ideal, and t ⋅ (b, m) = (ζb + mb 0 , m), where b 0 ∈ B \ (1 − ζ)B.We will denote this by B ⊕ b 0 ℤ.Two such representations of this form are isomorphic if the fractional ideals represent the same element in the ideal class group.ℤ with a trivial action.Let ζ be a p-th root of unity.Then ℤ[ζ] is an example of a module of the form (1) in Theorem 2.1.

3 , [ 11 ,Proposition 3 . 4 .
Theorem 65] and [11, Proposition 75], we obtain the following.The map EΓ → pt induces isomorphisms on L ⟨−∞⟩ ℤ module structures of L and ℤ[ℤ/p] do not change if we change the choice of b 0 and b 1 so long as they remain outside (1 − ζ)B i .By multiplying b 0 with some integer prime to p, we may assume b 0

Lemma 4 . 11 .
Suppose 4.1 (1) is true for groups Γ of type (a, b, c).Then it is true for groups of type (a, b, c + 1).
5 shows that Λ m ℤ[ℤ/p] is free when 1 ≤ m ≤ p − 1 and the vanishing of the Tate cohomology follows from Proposition 4.3.As we are not interested in all r ∈ ℤ b , we define the following: R b,m := {(r 1 , . . ., r b ) ∈ {0, p} b |r 1 + ⋅ ⋅ ⋅ + r b ≡ m mod 2}.For a subset d ⊆ {1, . . ., b}, define R b,m,d ⊆ R b,m to be those b-tuples such that r d = 0 if d ∈ d.Proof of Theorem 4.1 (1).We proceed by induction on b with the case b = 0 having been done in [9, Lemma 3.3].
] denote the torus T bp with a ℤ/p-action corresponding to the action of ℤ/p on the lattice ℤ[ℤ/p] b .Similarly, let T bp N L denote the torus with ℤ/p-action corresponding to the lattice N L .Since (N L ) (p) ≅ ℤ[ℤ/p] (p) , there is some N prime to p such that T bp N L is an N-sheeted regular cover of T bp ℤ[ℤ/p] .From this, we obtain an N-sheeted regular cover BΓ → BΓ

F 1 ,
induction on b will address the terms H i (ℤ/p; ΛM * L ⊗ A r ) when r ̸ = (p, . . ., p).In order to deal with the terms with r = (p, . . ., p), we will need to consider a sphere bundle quotient of BΓ.Recall we have assumed that N L ≅ ℤ[ℤ/p] b so T bp ≅ ℝ[ℤ/p] b /N L .Let x 0 ∈ T bp denote the image of 0 ∈ ℝ[ℤ/p] b and let D denote a ℤ/p-invariant disk neighborhood of x 0 .Then the quotient T bp /(T bp \ D) is the representation sphere of the regular representation ℝ[ℤ/p] b where α 1 , . . ., α d are distinct integers in {1, . . ., b} and where the intersection occurs in F 1 ⊆ K m (BΓ).For d < b, each F 1 α 1 ,...,α d is the image of the p-adic part of K m (B((M L ⊕ ℤ[ℤ/p] b−d ) ⋊ ℤ/p)) under an appropriate retraction BΓ → B((M L ⊕ ℤ[ℤ/p] b−d ) ⋊ ℤ/p).For d = b, the group F 1 1,...,b is the image of the p-adic part of K m (B(M L ⋊ ℤ/p)) under the projection BΓ → B(M L ⋊ ℤ/p).

xy is torsion if and only if x is in the kernel of the norm map Norm : L → L. Moreover, if x and z are in the kernel of the norm map, one checks that the group generated by xy is conjugate to the group generated by zy if an only if x − z is in the image of 1 − y : L → L. Therefore, P is in bijection with H 1 (ℤ/p; L). It follows from Proposition 4.3, which we prove later
. Finally, Proposition 6.3 implies that E ∞ ℓ,n ≅ E 2 ℓ,n for both spectral sequences.This explains the second, third, fourth and fifth rows of the following diagram: ℓ (Eℤ/p; H n (T ρ ; L(ℤ)) H n (T n ρ ; L(ℤ)) ℤ/p H ℓ (ℤ/p; H n (T n ρ ; L(ℤ))) L n (ℤ[ℤ n ρ ]) ℤ/p H ℓ (ℤ/p; H n (T n ρ ; L(ℤ))).