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Positive Scalar Curvature and crystallographic fundamental groups.

Noé Bárcenas barcenas@matmor.unam.mx http://www.matmor.unam.mx /  barcenas Centro de Ciencias Matemáticas. UNAM
Ap.Postal 61-3 Xangari. Morelia, Michoacán MEXICO 58089  and  Mario Velásquez Universidad Nacional de Colombia Sede Bogotá Departamento de Matemáticas, Facultad de Ciencias Cra. 30 cll 45 - Ciudad Universitaria, Bogotá, Colombia mavelasquezme@unal.edu.co https://sites.google.com/site/mavelasquezm/home
Abstract.

We examine positive and negative results for the Gromov-Lawson-Rosenberg Conjecture within the class of crystallographic groups. We give necessary conditions within the class of split extensions of free abelian by cyclic groups to satisfy the unstable Gromov-Lawson-Rosenberg Conjecture. We also give necessary conditions within the same class of groups which are counterexamples for the conjecture.

Key words and phrases:
Gromov-Lawson-Rosenberg Conjecture, Group homology, (connective) topological K-theory, extensions of ℤn\mathbb{Z}^{n} by ℤ/m\mathbb{Z}/m
2020 Mathematics Subject Classification:
46L80, 53C27

1. Introduction

The (unstable) Gromov-Lawson-Rosenberg Conjecture for a discrete group Γ\Gamma predicts that a closed spin nn-dimensional manifold MnM^{n} where n≥5n\geq 5, with fundamental group Γ\Gamma and classifying map for the fundamental group f:M→B​Γf:M\to B\Gamma, the vanishing of the group homomorphism

α​(M)=A∘pB​Γ​D​(fM):Ωnspin​(B​Γ)⟶K​On​(C∗∗​(Γ))\alpha(M)=A\circ p_{B\Gamma}D(f_{M}):\Omega_{n}^{\rm spin}(B\Gamma)\longrightarrow KO_{n}(C_{*}^{*}(\Gamma))

given as the composition

Ωnspin​(B​Γ)​⟶D​(fM)​k​on​(B​Γ)​⟶pB​Γ​K​On​(B​Γ)​⟶𝐴​K​On​(Cr∗​(Γ)),\Omega_{n}^{\rm spin}(B\Gamma)\overset{D(f_{M})}{\longrightarrow}ko_{n}(B\Gamma)\overset{p_{B\Gamma}}{\longrightarrow}KO_{n}(B\Gamma)\overset{A}{\longrightarrow}KO_{n}(C_{r}^{*}(\Gamma)),

decides about the existence of a metric of positive scalar curvature on MM. Some explanations are due. DD is the map which sends a spin bordism class fM:M→B​Γf_{M}:M\to B\Gamma to the image of the k​oko-fundamental class f∗​([M])∈k​on​(B​Γ)f_{*}([M])\in ko_{n}(B\Gamma). The map pB​Γ:k​on​(B​Γ)→K​On​(B​Γ)p_{B\Gamma}:ko_{n}(B\Gamma)\to KO_{n}(B\Gamma) is the natural transformation of periodicity, and AA denotes the real assembly map A:K​On​(B​Γ)→K​On​(Cr∗​(Γ))A:KO_{n}(B\Gamma)\to KO_{n}(C_{r}^{*}(\Gamma)).

There exist counterexamples to this conjecture. T. Schick in [Sch98] showed that for the group Γ=ℤ4×ℤ/3\Gamma=\mathbb{Z}^{4}\times\mathbb{Z}/3, there exists a five dimensional manifold MM with fundamental group Γ=π1​(M)=ℤ4×ℤ/3\Gamma=\pi_{1}(M)=\mathbb{Z}^{4}\times\mathbb{Z}/3 for which

α​(M)=0∈K​O5​(Cr∗​(Γ)),\alpha(M)=0\in KO_{5}(C_{r}^{*}(\Gamma)),

but MM admits no metric of positive scalar curvature.

The result initiated a series of subsecuent articles stating group cohomological conditions which produce counterexamples for the (unstable) Gromov-Lawson-Rosenberg Conjecture, including [DP03a], and specially [DSS03], where the techniques are used to construct a torsionfree example, which is even a fundamental group of a compact manifold admitting a CAT​(0){\rm CAT}(0)-cubical complex structure.

We would like to mention [DP03b], [DL13] and [Hug21] as some sources for positive results on the stable Gromov-Lawson-Rosenberg-Conjecture. The positive results therein concern groups satisfying the Baum Connes Isomorphism conjecture, which satisfy condition 1.1, and some other conditions about the maximal finite subgroups.

We will consider in this article for a group homomorphism ρ:ℤ/m→G​ln​(ℤ)\rho:\mathbb{Z}/m\to Gl_{n}(\mathbb{Z}), split extensions of the type

1⟶ℤn⟶Γ=ℤn⋊ρℤ/m⟶ℤ/m⟶1.1\longrightarrow\mathbb{Z}^{n}\longrightarrow\Gamma=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/m\longrightarrow\mathbb{Z}/m\longrightarrow 1.

The integral cohomology of such groups Γ\Gamma has been computed in a series of articles under several additional sets of hypothesis including:

  • •

    The group Γ\Gamma is torsionfree and mm is a prime number [CV65].

  • •

    The action is compatible in the sense of [AGPP08], which allows for a specific resolution of the trivial ℤ​[Γ]\mathbb{Z}[\Gamma]-module ℤ\mathbb{Z}, and the collapse of the Lyndon-Hochschild-Serre spectral sequence computing the integral cohomology of Γ\Gamma without extension problems at the E2E_{2}-term.

  • •

    The action of Γ\Gamma on ℝn−{0}\mathbb{R}^{n}-\{0\} is free outside of the origin. [LL12].

  • •

    The action of Γ\Gamma on ℝn−{0}\mathbb{R}^{n}-\{0\} is free outside of the origin, and mm is a prime number [DL13].

  • •

    The natural number mm is free of squares, without further assumption on the action [SV24a].

With the exception of [SV24a] and [AGPP08], these conditions are used because they imply the following maximality properties within the family of finite subgroups of Γ\Gamma.

Condition 1.1.

[Conditions MM and N​MNM]

  • •

    Each finite subgroup HH of Γ\Gamma is a subgroup of a unique maximal finite subgroup MM, and there exists a finite collection (up to conjugacy) ℳ\mathcal{M} of them.

  • •

    The normalizer in Γ\Gamma of MM is MM itself.

Lück and Davis in [DL13] used the results of these computations together with the construction of specific models for the classifying space for proper actions [LW12] to derive computations of complex, real, and real connective KK-homology of both the classifying space B​ΓB\Gamma, and the classifying space for proper actions E¯​Γ\underline{E}\Gamma.

Extending these results, the second named author and Sánchez performed computations of both the complex K​UKU- homology of the classifying spaces B​ΓB\Gamma, denoted by K​U∗​(B​Γ)KU_{*}(B\Gamma), and the equivariant K​UKU-homology groups of the classifying spaces for proper actions, denoted by K​U∗Γ​(E¯​Γ)KU_{*}^{\Gamma}(\underline{E}\Gamma).

In this work we will make structural statements about the algebraic structure of real connective k​oko-homology groups of B​ΓB\Gamma, denoted by k​o∗​(B​Γ)ko_{*}(B\Gamma), which will be the base for positive and negative results for the (unstable) Gromov-Lawson-Rosenberg conjecture for high dimensional smooth spin manifolds with fundamental group Γ\Gamma.

The hypothesis that we will impose on the group Γ\Gamma is the following

Condition 1.2.

[Condition for positive results] Let mm be and odd natural number and assume that ρ:ℤ/m→G​ln​(ℤ)\rho:\mathbb{Z}/m\to Gl_{n}(\mathbb{Z}) is a group homomorphism such that the group action of ℤ/M\mathbb{Z}/M on ℝn−{0}\mathbb{R}^{n}-\{0\} is free.

The following is our main positive result on the Gromov-Lawson-Rosenberg conjecture.

Theorem 1.3.

Let MnM^{n} be an nn-dimensional smooth spin manifold, where n≥5n\geq 5 is even, and with fundamental group isomorphic to Γ\Gamma, where Γ\Gamma satisfies condition 1.2. Denote by fM:M→B​Γf_{M}:M\to B\Gamma the classifying map for the fundamental group. Assume that α​(M)=0\alpha(M)=0. Then MM admits a metric of positive scalar curvature.

Theorem 1.3 will be proved for nn even as Theorem 4.1, and nn odd as Theorem 4.5. The main structural statements for their proof, namely Lemmas 2.2 and 2.6 for the even case, and Lemma 4.2 are of different nature and therefore stated and proved separately.

Within the class of crystallographic groups

1→ℤn→Γ=ℤn⋊ρℤ/m→ℤ/m→1,1\to\mathbb{Z}^{n}\to\Gamma=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/m\to\mathbb{Z}/m\to 1,

there exist groups for which the unstable Gromov-Lawson-Rosenberg conjecture is known to be true, namely

  • •

    The number mm is prime and the action is free outside of the origin, according to [DL13].

  • •

    The groups adressed in section 2.

On the other hand side, the group ℤ4×ℤ/3\mathbb{Z}^{4}\times\mathbb{Z}/3 belongs to the family of central extensions

1→ℤn→Γ=ℤn⋊ρℤ/m→ℤ/m→1,1\to\mathbb{Z}^{n}\to\Gamma=\mathbb{Z}^{n}\rtimes_{\rho}\mathbb{Z}/m\to\mathbb{Z}/m\to 1,

for a representation ρ:ℤ/3→G​L5​(ℤ)\rho:\mathbb{Z}/3\to GL_{5}(\mathbb{Z}) whose action on ℝ5\mathbb{R}^{5} is trivial. When we localize at a prime number pp, we have a complete determination of the group cohomology of Γ\Gamma in terms of the decomposition of a finite index submodule of ℤn\mathbb{Z}^{n} as a ℤ​[ℤ/m]\mathbb{Z}[\mathbb{Z}/m]-module where the summands are the irreducible representations ℤ\mathbb{Z} (trivial representation), 𝕀​[ℤ/n]\mathbb{I}[\mathbb{Z}/n] (augmentation ideal), and ℤ​[ℤ/m]\mathbb{Z}[\mathbb{Z}/m]. The information related to these decompositions is one of the ingredients for the negative results on the Gromov-Lawson-Rosenberg conjecture.

The second ingredient will be the verification, using the computations of group cohomology by [SV24a], that there exists a family of crystallographic groups for which the method introduced by Schick in [Sch98] applies. The following is our main negative result, which will be proved as Theorem 5.1.

Theorem.

Suppose mm is square-free. Let ℤn\mathbb{Z}^{n} be a ℤ/m\mathbb{Z}/m-module, and suppose that there exists a prime p∣mp\mid m such that if we consider the (r,s,t)(r,s,t) decomposition of MM viewed as a ℤ/p\mathbb{Z}/p-module, where r≥4r\geq 4, and s+t≥1s+t\geq 1 then ℤn⋊ℤ/m\mathbb{Z}^{n}\rtimes\mathbb{Z}/m is a counter-example for the unstable Gromov-Lawson-Rosenberg conjecture.

1.1. Aknowledgements

The first named author aknowledges support of DGAPA-PAPIIT Grant IN101423. Both authors thank the exchange program CIC-UNAM.

2. Connective k​oko-homology of B​ΓB\Gamma.

Recall that Γ\Gamma fits in an extension

(1) 1→ℤn→Γ→𝜋ℤ/m→1,1\to\mathbb{Z}^{n}\to\Gamma\xrightarrow{\pi}\mathbb{Z}/m\to 1,

Assume that condition 1.2 holds. Then, as a consequence of [LW12], we obtain the following result.

Theorem 2.1.

There is a commutative diagram with exact rows

⨁(N)∈𝒩k​o~r​(B​N)\textstyle{\bigoplus_{(N)\in\mathcal{N}}\widetilde{ko}_{r}(BN)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}k​or​(B​Γ)\textstyle{ko_{r}(B\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}β\scriptstyle{\beta}A∘pB​Γ\scriptstyle{A\circ p_{B\Gamma}}k​or​(B¯​Γ)\textstyle{ko_{r}(\underline{B}\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pB¯​Γ\scriptstyle{p_{\underline{B}\Gamma}}K​Or​(Cr∗​(Γ;ℝ))\textstyle{KO_{r}(C_{r}^{*}(\Gamma;\mathbb{R}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}K​Or​(B¯​Γ)\textstyle{KO_{r}(\underline{B}\Gamma)}

where the bottom map is the composite of the inverse of the Baum–Connes map and the map K​OmΓ​(E¯​Γ)→K​Om​(B​Γ)KO_{m}^{\Gamma}(\underline{E}\Gamma)\to KO_{m}(B\Gamma) is the induction map with respect to the homomorphism Γ→1\Gamma\to 1.

Proof.

It is a consequence of the cellular pushout relating B​ΓB\Gamma and B¯​Γ\underline{B}\Gamma when the group Γ\Gamma satisfies conditions M and NM. ∎

Now suppose that fM:M→B​Γf_{M}:M\to B\Gamma is a classifying map of MM and

α​(M)=A∘pB​Γ​(D​[fM])=0.\alpha(M)=A\circ p_{B\Gamma}(D[f_{M}])=0.

First, by the commutativity of the above diagram, we have pB¯​Γ∘β​(D​[fM])=0p_{\underline{B}\Gamma}\circ\beta(D[f_{M}])=0. Now we will analyze ker⁡(pB¯​Γ).\ker(p_{\underline{B}\Gamma}).

Lemma 2.2.

ker⁡(pB¯​Γ)\ker(p_{\underline{B}\Gamma}) only contains mm-torsion.

Proof.

Let pp be a prime dividing mm such that m=ps​m′m=p^{s}m^{\prime} with (p,m′)=1(p,m^{\prime})=1 and let rr be the product of the primes dividing mm. By Lemma 2.3 the quotient map B¯​Γ→B¯​(Γ/ℤ/ps)\underline{B}\Gamma\to\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}) induces an isomorphism

k​o∗​(B¯​Γ)ℤ/ps⊗ℤ​[1/p]→k​o∗​(B¯​(Γ/ℤ/ps))⊗ℤ​[1/p].ko_{*}(\underline{B}\Gamma)_{\mathbb{Z}/{p^{s}}}\otimes\mathbb{Z}[1/p]\to ko_{*}(\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}))\otimes\mathbb{Z}[1/p].

We have a commutative diagram

k​o∗​(B¯​Γ)ℤ/ps⊗ℤ​[1/p]\textstyle{ko_{*}(\underline{B}\Gamma)_{\mathbb{Z}/{p^{s}}}\otimes\mathbb{Z}[1/p]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}pB¯​Γ\scriptstyle{p_{\underline{B}\Gamma}}k​o∗​(B¯​(Γ/ℤ/ps))⊗ℤ​[1/p]\textstyle{ko_{*}(\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}))\otimes\mathbb{Z}[1/p]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pB¯​(Γ/ℤ/ps)\scriptstyle{p_{\underline{B}(\Gamma/\mathbb{Z}/{p^{s}})}}K​O∗​(B¯​Γ)ℤ/ps⊗ℤ​[1/p]\textstyle{KO_{*}(\underline{B}\Gamma)_{\mathbb{Z}/p^{s}}\otimes\mathbb{Z}[1/p]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}K​O∗​(B¯​(Γ/ℤ/ps))⊗ℤ​[1/p]\textstyle{KO_{*}(\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}))\otimes\mathbb{Z}[1/p]}

Then it is enough to prove that

k​o∗​(B¯​(Γ/ℤ/ps))ℤ/ps⊗ℤ​[1/p]→K​O∗​(B¯​(Γ/ℤ/ps))ℤ/ps⊗ℤ​[1/p]ko_{*}(\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}))_{\mathbb{Z}/p^{s}}\otimes\mathbb{Z}[1/p]\to KO_{*}(\underline{B}(\Gamma/\mathbb{Z}/{p^{s}}))_{\mathbb{Z}/p^{s}}\otimes\mathbb{Z}[1/p]

only contains m/psm/p^{s}-torsion.

The commutativity of the previous diagram follows from the fact that periodicity is a transformation of homology theories, and that the composition of a homology theory with the formation of coinvariants is a homology theory.

Proceeding inductively over all primes dividing mm it is enough to prove that

k​o∗​(B​ℤn)ℤ/m⊗ℤ​[1/m]→pB​ℤnK​O∗​(B​ℤn)ℤ/m⊗ℤ​[1/m]ko_{*}(B\mathbb{Z}^{n})_{\mathbb{Z}/m}\otimes\mathbb{Z}[1/m]\xrightarrow{p_{B\mathbb{Z}^{n}}}KO_{*}(B\mathbb{Z}^{n})_{\mathbb{Z}/m}\otimes\mathbb{Z}[1/m]

is injective. But it is a consequence of the injectivity of the maps on coefficients from connective real K-theory and periodic real K-theory as in noted by Davis and Lück in the proof of Thm. 0.7 in [DL13]. Then we find that ker⁡(pB¯​Γ)\ker(p_{\underline{B}\Gamma}) only contains mm-torsion. ∎

Now we need to recall a couple of results that we will need in the following.

Lemma 2.3.

Let pp be a prime number and let GG be a pp-group. For any GG-CW complex XX with quotient map π:X→G∖X\pi:X\to G\setminus X and any homology theory ℋ∗​(−){\mathcal{H}}_{*}(-), the induced map πr:(ℋr​(X)⊗ℤ​[1/p])G→ℋr​(G∖X)⊗ℤ​[1/p]\pi_{r}:\left({\mathcal{H}}_{r}(X)\otimes\mathbb{Z}[1/p]\right)_{G}\to{\mathcal{H}}_{r}(G\setminus X)\otimes\mathbb{Z}[1/p] is an isomorphism for all r∈ℤr\in\mathbb{Z}.

Proof.

The argument is similar to the one given for Prop. A.4. in [DL13]. Given a GG-CW-complex XX, we get a natural transformation

j∗:(ℋm​(X)⊗ℤ​[1/p])G→ℋm​(G∖X)⊗ℤ​[1/p].j_{*}:\left({\mathcal{H}}_{m}(X)\otimes\mathbb{Z}[1/p]\right)_{G}\to{\mathcal{H}}_{m}(G\setminus X)\otimes\mathbb{Z}[1/p].

Both sides are GG-homology theories and moreover j∗j_{*} is an isomorphism when X=G/HX=G/H we have that j∗j_{*} is an isomorphism for ever XX. ∎

We aso need the following result of Lück-Weiermann.

Theorem 2.4 (Corollary 2.8 in [LW12]).

Let ℱ⊆𝒢{\mathcal{F}}\subseteq{\mathcal{G}} be families of subgroups of a group Γ\Gamma such that every element in 𝒢−ℱ{\mathcal{G}}-{\mathcal{F}} is contained in a unique maximal element in 𝒢−ℱ{\mathcal{G}}-{\mathcal{F}}. Let ℳ{\mathcal{M}} be a complete system of representatives of the conjugacy classes of subgroups in 𝒢−ℱ{\mathcal{G}}-{\mathcal{F}} which are maximal in 𝒢−ℱ{\mathcal{G}}-{\mathcal{F}}. Let 𝒮​𝒰​ℬ​(M)\mathcal{SUB}(M) be the family of subgroups of MM. Then, there is a cellular Γ\Gamma-pushout

(2) ⨆M∈ℳBℱ∩NΓ​M​(NΓ​M)\textstyle{\bigsqcup_{M\in{\mathcal{M}}}B_{{\mathcal{F}}\cap N_{\Gamma}M}(N_{\Gamma}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}i\scriptstyle{i}λ\scriptstyle{\lambda}Bℱ​(Γ)\textstyle{B_{{\mathcal{F}}}(\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}⨆M∈ℳB𝒮​𝒰​ℬ​(M)∪(ℱ∩NΓ​M)​(NΓ​M)\textstyle{\bigsqcup_{M\in{\mathcal{M}}}B_{\mathcal{SUB}(M)\cup({\mathcal{F}}\cap N_{\Gamma}M)}(N_{\Gamma}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X\textstyle{X}

such that XX is a model for B𝒢​(Γ)B_{{\mathcal{G}}}(\Gamma).

For a group Γ\Gamma given by (1) let us denote by Γp=π−1​(ℤ/ps)\Gamma_{p}=\pi^{-1}(\mathbb{Z}/p^{s}), where psp^{s} is the highest power of pp which divides mm. For any subgroup G⊆ℤ/mG\subseteq\mathbb{Z}/m we denote by GpG_{p} the subgroup of Γ\Gamma defined as Gp=G/(ℤ/ps∩G).G_{p}=G/(\mathbb{Z}/p^{s}\cap G).

For a homology theory ℋ∗​(−){\mathcal{H}}_{*}(-) denote by ℋ∗​(−)(p){\mathcal{H}}_{*}(-)_{(p)} to the localization of ℋ∗{\mathcal{H}}_{*} at pp, that is, ℋ∗​(−)⊗ℤ(p){\mathcal{H}}_{*}(-)\otimes\mathbb{Z}_{(p)}, where ℤ(p)\mathbb{Z}_{(p)} is the ring of integers localized at the prime ideal (p)(p).

Lemma 2.5.

For any (reduced or unreduced) homology theory ℋ∗​(−){\mathcal{H}}_{*}(-) and p∣mp\mid m, we have that

  1. (i)

    ℋ∗​(B​Γ)(p)≅(ℋ∗​(B​(Γp))(p))Gp{\mathcal{H}}_{*}(B\Gamma)_{(p)}\cong\left({\mathcal{H}}_{*}(B(\Gamma_{p}))_{(p)}\right)_{G_{p}}

  2. (ii)

    ℋ∗​(B¯​Γ)(p)≅(ℋ∗​(B¯​(Γp))(p))Gp.{\mathcal{H}}_{*}(\underline{B}\Gamma)_{(p)}\cong\left({\mathcal{H}}_{*}(\underline{B}(\Gamma_{p}))_{(p)}\right)_{G_{p}}.

Proof.
  1. (i)

    The extension induces a fibration of classifying spaces.

    B​Γp→B​Γ→B​GpB\Gamma_{p}\to B\Gamma\to BG_{p}

    Consider the Leray-Serre spectral sequence associated to this fibration, for the homology theory ℋ∗​(−)(p){\mathcal{H}}_{*}(-)_{(p)}. By [Mil62] this spectral sequence converges to ℋ∗​(B​Γ)(p){\mathcal{H}}_{*}(B\Gamma)_{(p)} with second page:

    Eα,β2=Hα​(Gp;ℋβ​(B​Γp)(p))={(ℋβ​(B​Γp)(p))Gpα=00α≠0.E^{2}_{\alpha,\beta}=H_{\alpha}\left(G_{p};{\mathcal{H}}_{\beta}(B\Gamma_{p})_{(p)}\right)=\begin{cases}\left({\mathcal{H}}_{\beta}(B\Gamma_{p})_{(p)}\right)_{G_{p}}&\alpha=0\\ 0&\alpha\neq 0.\end{cases}

    This implies that the sequence collapses without extension problems, so we have proved the first statement.

  2. (ii)

    The idea of the proof is similar to that of Lemma 3.1 in [SV24b]. Let ℱ{\mathcal{F}} be a family of subgroups of Γ\Gamma, we say that Γ\Gamma satifies the (ps,ℱ)(p^{s},{\mathcal{F}})-condition if for every homology theory ℋ∗​(−){\mathcal{H}}_{*}(-), the induction

    (ℋ∗​(Bℱ∩Γp​Γp)(p))Gp→I​n​dΓpΓℋ∗​(Bℱ​Γ)(p),\left({\mathcal{H}}_{*}(B_{{\mathcal{F}}\cap\Gamma_{p}}\Gamma_{p})_{(p)}\right)_{G_{p}}\xrightarrow{Ind_{\Gamma_{p}}^{\Gamma}}{\mathcal{H}}_{*}(B_{\mathcal{F}}\Gamma)_{(p)},

    is an isomorphism. Here, Bℱ​ΓB_{\mathcal{F}}\Gamma denotes the orbit space of the classifying space of Γ\Gamma with respect to ℱ{\mathcal{F}}.

    Suppose that mm is the product of powers of distinct primes p1s1​⋯​prsrp_{1}^{s_{1}}\cdots p_{r}^{s_{r}}. Let

    ℱi={H⊂Γ∣H​ is finite and ​|H|​ divides ​p1s1​⋯​pisi}.{\mathcal{F}}_{i}=\{H\subset\Gamma\mid H\text{ is finite and }|H|\text{ divides }p_{1}^{s_{1}}\cdots p_{i}^{s_{i}}\}.

    Now we will show that by induction on ii that Γ\Gamma satisfies the (ps,ℱi)(p^{s},{\mathcal{F}}_{i})-condition, for every ii.

    For i=0i=0, it is the statement (i).

Now we will construct a model for B𝒮​𝒰​ℬ​(M)∪(ℱi∩NΓ​M)​(NΓ​M)B_{\mathcal{SUB}(M)\cup({\mathcal{F}}_{i}\cap N_{\Gamma}M)}(N_{\Gamma}M).

We have a pushout

(3) B𝒮​𝒰​ℬ​(M)∩(ℱi∩NΓ​M)​(NΓ​M)\textstyle{B_{\mathcal{SUB}(M)\cap({\mathcal{F}}_{i}\cap N_{\Gamma}M)}(N_{\Gamma}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}i\scriptstyle{i}λ\scriptstyle{\lambda}Bℱi∩NΓ​M​(NΓ​M)\textstyle{B_{{\mathcal{F}}_{i}\cap N_{\Gamma}M}(N_{\Gamma}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}B𝒮​𝒰​ℬ​(M)​(NΓ​M)\textstyle{B_{\mathcal{SUB}(M)}(N_{\Gamma}M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}B𝒮​𝒰​ℬ​(M)∪(ℱi∩NΓ​M)​(NΓ​M)\textstyle{B_{\mathcal{SUB}(M)\cup({\mathcal{F}}_{i}\cap N_{\Gamma}M)}(N_{\Gamma}M)}

In both cases, families 𝒮​𝒰​ℬ​(M)\mathcal{SUB}(M) and 𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M have a maximal element, MM and Mi+1=M/(ℤ/pi+1si+1)M_{i+1}=M/(\mathbb{Z}/p_{i+1}^{s_{i+1}}) respectively, then E​(WΓ​M)E(W_{\Gamma}M) with the action induced by the quotient map is a model for E𝒮​𝒰​ℬ​(M)​NΓ​ME_{\mathcal{SUB}(M)}N_{\Gamma}M, then

B𝒮​𝒰​ℬ​(M)​NΓ​M=B​WΓ​MB_{\mathcal{SUB}(M)}N_{\Gamma}M=BW_{\Gamma}M

and E​(NΓ​M/(Mi+1))E(N_{\Gamma}M/(M_{i+1})) with the action induced by the quotient map is a model for
E𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M​NΓ​ME_{\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M}N_{\Gamma}M, and hence

B𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M​NΓ​M=B​(NΓ​M/Mi+1).B_{\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M}N_{\Gamma}M=B(N_{\Gamma}M/M_{i+1}).

Notice that both groups WΓ​MW_{\Gamma}M and NΓ​M/Mi+1N_{\Gamma}M/M_{i+1} are finite subgroups of ℤ/m\mathbb{Z}/m. We have

ℋ∗​(B𝒮​𝒰​ℬ​(M)​NΓ​M)(p)\displaystyle{\mathcal{H}}_{*}(B_{\mathcal{SUB}(M)}N_{\Gamma}M)_{(p)} ≅ℋ∗​(B​WΓ​M)(p)\displaystyle\cong{\mathcal{H}}_{*}(BW_{\Gamma}M)_{(p)}
≅(ℋ∗​(B​(WΓ​M∩Γp/M))(p))Gp​ by (i)\displaystyle\cong\left({\mathcal{H}}_{*}(B(W_{\Gamma}M\cap\Gamma_{p}/M))_{(p)}\right)_{G_{p}}\text{ by (i) }
≅(ℋ∗​(B𝒮​𝒰​ℬ​(M)​(NΓ​M∩Γp))(p))Gp.\displaystyle\cong\left({\mathcal{H}}_{*}(B_{\mathcal{SUB}(M)}(N_{\Gamma}M\cap\Gamma_{p}))_{(p)}\right)_{G_{p}}.

and in a similar way

ℋ∗​(B𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M​NΓ​M)(p)\displaystyle{\mathcal{H}}_{*}(B_{\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M}N_{\Gamma}M)_{(p)} ≅(ℋ∗​(B𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M​(NΓ​M∩Γp))(p))Gp.\displaystyle\cong\left({\mathcal{H}}_{*}(B_{\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M}(N_{\Gamma}M\cap\Gamma_{p}))_{(p)}\right)_{G_{p}}.

In others words, NΓ​MN_{\Gamma}M satisfies (ps,𝒮​𝒰​ℬ​(M))(p^{s},\mathcal{SUB}(M))-condition and (ps,𝒮​𝒰​ℬ​(M)∩ℱi∩NΓ​M)(p^{s},\mathcal{SUB}(M)\cap{\mathcal{F}}_{i}\cap N_{\Gamma}M)-condition. By the five lemma applied to the morphism of Mayer-Vietoris sequences given by restrict the pushout (3) to Γp\Gamma_{p} we get that NΓ​MN_{\Gamma}M satisfies the (ps,𝒮​𝒰​ℬ​(M)∪(ℱi∩NΓ​M))(p^{s},\mathcal{SUB}(M)\cup({\mathcal{F}}_{i}\cap N_{\Gamma}M))-condition.

Finally again by the five lemma applied to the morphism of Mayer-Vietoris sequences given by restriction to Γp\Gamma_{p} of the pushout (2) associated to families ℱi⊆ℱi+1{\mathcal{F}}_{i}\subseteq{\mathcal{F}}_{i+1} we get Γ\Gamma satisfies the (ps,ℱi)(p^{s},{\mathcal{F}}_{i})-condition for every ii and for i=ri=r we obtain (ii).

∎

The following result was proved as Lemma 4.4 in [LL12].

Lemma 2.6.

In the exact sequence

0⟶H2​r​(B¯​(ℤn⋊ℤ/ps))​⟶f¯∗​H2​r​(ℤn⋊ℤ/ps)​⟶φ2​r​⨁P∈𝒫​H2​r​(B​P),0\longrightarrow H^{2r}(\underline{B}(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s}))\overset{\bar{f}_{*}}{\longrightarrow}H^{2r}(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s})\overset{\varphi^{2r}}{\longrightarrow}\underset{P\in\mathcal{P}}{\bigoplus}H^{2r}(BP),
  • •

    The homomorphism φ2​r\varphi^{2r} has torsionfree kernel.

  • •

    The abelian group H2​r​(B¯​(ℤn⋊ℤ/ps))H^{2r}(\underline{B}(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s})) is finitely generated and torsionfree.

Applying Lemma 2.5 and the Universal Coefficient Theorem, we obtain the following result.

Corollary 2.7.

The abelian groups

H2​r+1​(B¯​Γ,ℤ)H_{2r+1}(\underline{B}\Gamma,\mathbb{Z})

are finitely generated and torsion-free.

3. Proof of the positive result in the even dimensional case.

In this section we will prove the even version of Theorem 1.3.

Theorem 3.1.

Let MnM^{n} be an nn-dimensional smooth spin manifold, where n≥5n\geq 5 is even, and with fundamental group isomorphic to Γ\Gamma. Denote by fM:M→B​Γf_{M}:M\to B\Gamma the classifying map for the fundamental group. Assume that α​(M)=0\alpha(M)=0. Then MM admits a metric of positive scalar curvature.

The proof of this theorem requires some work.

Lemma 3.2.

For ∗* even, k​o∗​(B​Γ)ko_{*}(B\Gamma) does not contain mm-torsion

Proof.

By Lemma 2.5 it is enough to prove that for any odd prime pp dividing mm, we have that k​o∗​(B​(ℤn⋊ℤ/ps))Gpko_{*}(B(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s}))_{G_{p}} does not contain psp^{s}-torsion.

Consider the Atiyah-Hirzebruch- Leray-Serre spectral sequence associated to the extension

0→ℤn→ℤn⋊ℤ/ps→ℤ/ps→0,0\to\mathbb{Z}^{n}\to\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s}\to\mathbb{Z}/p^{s}\to 0,

coverging to k​o∗​(B​(ℤn⋊ℤ/ps))ko_{*}(B(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s})). The second page is given by

Ep,q2=Hi​(ℤ/ps;k​oj​(B​ℤn)).E^{2}_{p,q}=H_{i}(\mathbb{Z}/p^{s};ko_{j}(B\mathbb{Z}^{n})).

Let us first compute koj(Bℤn))ko_{j}(B\mathbb{Z}^{n})). By Lemma 5.3 in [DL13] we have isomorphisms of abelian groups

(4) k​oj​(B​ℤn)⊗ℤ​[1/2]\displaystyle ko_{j}(B\mathbb{Z}^{n})\otimes\mathbb{Z}[1/2] ≅⨁l=0nHl​(ℤn)⊗k​oj​(∗)⊗ℤ​[1/2]\displaystyle\cong\bigoplus_{l=0}^{n}H_{l}(\mathbb{Z}^{n})\otimes ko_{j}(*)\otimes\mathbb{Z}[1/2]
(5) k​oj​(B​ℤn)⊗ℤ(2)\displaystyle ko_{j}(B\mathbb{Z}^{n})\otimes\mathbb{Z}_{(2)} ≅⨁l=0nHl​(ℤn)⊗k​oj​(∗)(2).\displaystyle\cong\bigoplus_{l=0}^{n}H_{l}(\mathbb{Z}^{n})\otimes ko_{j}(*)_{(2)}.

Now we will prove that both are actually isomorphisms of ℤ​[ℤ/ps]\mathbb{Z}[\mathbb{Z}/p^{s}]-modules. Note that k​oj​(∗)⊗ℤ​[1/2]ko_{j}(*)\otimes\mathbb{Z}[1/2] is torsion free, then the Chern character shows that the map (4) is an isomorphism of ℤ​[ℤ/ps]\mathbb{Z}[\mathbb{Z}/p^{s}]-modules. The isomorphism (5) implies that the Atiyah-Hirzebruch spectral sequence with second page

Ei,j2=Hi​(ℤn;k​oj​(∗)(2))E^{2}_{i,j}=H_{i}(\mathbb{Z}^{n};ko_{j}(*)_{(2)})

converging to k​oi+j​(B​ℤn)(2)ko_{i+j}(B\mathbb{Z}^{n})_{(2)} collapses. On the other hand, this spectral sequence is natural with respect to ℤ/ps\mathbb{Z}/p^{s}-module structure of ℤn\mathbb{Z}^{n}. We have a filtration of ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-modules

k​or​(B​ℤn)(2)=F0,r⊇…⊇Fr,0⊇Fr+1,−1=0ko_{r}(B\mathbb{Z}^{n})_{(2)}=F_{0,r}\supseteq\ldots\supseteq F_{r,0}\supseteq F_{r+1,-1}=0

and exact sequences of ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-modules

0→Fi+1,r−i−1→Fi,r−i→Hi​(ℤn)⊗k​or−i​(∗)(2)→00\to F_{i+1,r-i-1}\to F_{i,r-i}\to H_{i}(\mathbb{Z}^{n})\otimes ko_{r-i}(*)_{(2)}\to 0

It is enough to prove that the above exact sequence splits as ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-modules.

Suppose first that r−i≡3,5,6,7r-i\equiv 3,5,6,7 (mod 8), then k​or−i​(∗)=0ko_{r-i}(*)=0, and so the exact sequence splits trivially.

If r−i≡0,4r-i\equiv 0,4 (mod 8), then k​or−i​(∗)≅ℤko_{r-i}(*)\cong\mathbb{Z}. On the other hand, Hi​(ℤn)⊗k​or−i​(∗)(2)H_{i}(\mathbb{Z}^{n})\otimes ko_{r-i}(*)_{(2)} is a finite generated ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-module that is torsion free as ℤ(2)\mathbb{Z}_{(2)}-module, as the action is free outside the origin, the norm element xx of ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}] acts by zero, then Hi​(ℤn)⊗k​om−i​(∗)(2)H_{i}(\mathbb{Z}^{n})\otimes ko_{m-i}(*)_{(2)} can be considered as a ℤ(2)​[ℤ/ps]/⟨x⟩\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]/\langle x\rangle-module. But ℤ(2)​[ℤ/ps]/x\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]/x is a Dedekind domain (being isomorphic to the psp^{s}-ciclotomic ring), then Hi​(ℤn)⊗k​om−i​(∗)(2)H_{i}(\mathbb{Z}^{n})\otimes ko_{m-i}(*)_{(2)} is ℤ(2)​[ℤ/ps]/x\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]/x-projective, and hence it is ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-projective, and hence the sequence splits.

Finally, if r−i≡1,2r-i\equiv 1,2 (mod 8). As the Atiyah-Hirzebruch spectral sequence collapses, we have an splitting of abelian groups

s:Hi​(ℤn)⊗k​or−i​(∗)(2)→Fi,r−is\colon H_{i}(\mathbb{Z}^{n})\otimes ko_{r-i}(*)_{(2)}\to F_{i,r-i}

such that π∘s=i​d\pi\circ s=id. Define

s~:Hi​(ℤn)⊗k​or−i​(∗)(2)\displaystyle\widetilde{s}:H_{i}(\mathbb{Z}^{n})\otimes ko_{r-i}(*)_{(2)} →Fi,r−i\displaystyle\to F_{i,r-i}
y\displaystyle y ↦∑g∈ℤ/psg⋅s​(g−1​x).\displaystyle\mapsto\sum_{g\in\mathbb{Z}/p^{s}}g\cdot s(g^{-1}x).

s~\widetilde{s} is a homomorphism of ℤ(2)​[ℤ/ps]\mathbb{Z}_{(2)}[\mathbb{Z}/p^{s}]-modules and as pp is odd π∘s~\pi\circ\widetilde{s} is multiplication by psp^{s}, but it is the identity because k​om−i​(∗)ko_{m-i}(*) is isomorphic to ℤ/2\mathbb{Z}/2.

Then the maps (4) and (5) are isomorphisms of ℤ​[ℤ/ps]\mathbb{Z}[\mathbb{Z}/p^{s}]-modules. Using that we have

H^i+1​(ℤ/ps;k​oj​(B​ℤn))≅⨁lH^i+1​(ℤ/ps;Hj−4​l​(ℤn)).\widehat{H}^{i+1}(\mathbb{Z}/p^{s};ko_{j}(B\mathbb{Z}^{n}))\cong\bigoplus_{l}\widehat{H}^{i+1}(\mathbb{Z}/p^{s};H_{j-4l}(\mathbb{Z}^{n})).

Then using the universal coefficient theorem and Theorem 3.2 in [LL12] we obtain

H^i+1​(ℤ/ps;k​oj​(B​ℤn))=0​ if ​i+j​ is even .\widehat{H}^{i+1}(\mathbb{Z}/p^{s};ko_{j}(B\mathbb{Z}^{n}))=0\text{ if }i+j\text{ is even .}

In particular this implies that the canonical map

E0,2​r2=k​o2​r​(B​ℤn)ℤ/ps→k​o2​r​(B​ℤn)ℤ/psE^{2}_{0,2r}=ko_{2r}(B\mathbb{Z}^{n})_{\mathbb{Z}/p^{s}}\to ko_{2r}(B\mathbb{Z}^{n})^{\mathbb{Z}/p^{s}}

is injective, because H^−1​(ℤ/ps;k​o2​r​(B​ℤn))=0\widehat{H}^{-1}(\mathbb{Z}/p^{s};ko_{2r}(B\mathbb{Z}^{n}))=0. Then k​o2​r​(B​ℤn)ℤ/psko_{2r}(B\mathbb{Z}^{n})_{\mathbb{Z}/p^{s}} does not contain psp^{s}-torsion. On the other hand, Ei,j2E^{2}_{i,j} is zero if i+ji+j is even and positive, then

k​o2​r​(B​ℤn)ℤ/ps=E0,2​r2=E0,2​r∞≅k​o2​r​(B​(ℤn⋊ℤ/ps)),ko_{2r}(B\mathbb{Z}^{n})_{\mathbb{Z}/p^{s}}=E^{2}_{0,2r}=E^{\infty}_{0,2r}\cong ko_{2r}(B(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s})),

then k​o2​r​(B​(ℤn⋊ℤ/ps))ko_{2r}(B(\mathbb{Z}^{n}\rtimes\mathbb{Z}/p^{s})) does not contain psp^{s}-torsion and we conclude that k​o∗​(B​Γ)ko_{*}(B\Gamma) does not contains mm-torsion if ∗* is even. ∎

Proof.

(Theorem 4.1) Notice that from the Atiyah-Hirzebruch spectral sequence we obtain, for even indices, k​o~∗​(B​ℤ/ps)=0\widetilde{ko}_{*}(B\mathbb{Z}/p^{s})=0. Then from Theorem 2.1 we conclude that for even degrees β\beta is injective and from Lemma 2.2 we get D​[fM]=0D[f_{M}]=0, by Prop. 12.1 in [DL13] we get that MM admits a metric with positive scalar curvature.

∎

4. Proof of the positive result in the odd dimensional case.

In this section we will prove the odd version of Theorem 1.3

Theorem 4.1.

Let MnM^{n} be an nn-dimensional smooth spin manifold, where n≥5n\geq 5 is odd, and with fundamental group isomorphic to Γ\Gamma. Denote by fM:M→B​Γf_{M}:M\to B\Gamma the classifying map for the fundamental group. Assume that α​(M)=0\alpha(M)=0. Then MM admits a metric of positive scalar curvature.

As in the previous section the proof of this theorem requires some lemmas.

Lemma 4.2.

Let rr be a natural number. Then, the k​oko-homology group

k​o2​r+1​(B¯​Γ)ko_{2r+1}(\underline{B}\Gamma)

does not contain pp-torsion for p≠2p\neq 2.

Proof.

Recall the Atiyah-Hirzebruch spectral sequence converging to the k​oko-homology groups localized at pp, k​o∗​(B¯​Γ)(p)ko_{*}(\underline{B}\Gamma)_{(p)} with E2E^{2}-term

Ei,j2=Hi​(B¯​Γ,k​oj​(∗)(p)).E^{2}_{i,j}=H_{i}(\underline{B}\Gamma,{ko_{j}(\ast)}_{(p)}).

The relevant elements on the E2E^{2}-term of the Atiyah-Hirzebruch spectral sequence for the computation of k​o2​r+1​(B¯​Γ)ko_{2r+1}(\underline{B}\Gamma) are those for which i+ji+j is odd. Let us distinguish the following cases:

  • •

    Assume ii is odd, and notice that because of 2.7, the homology groups Hi​(B¯​Γ,ℤ)H_{i}(\underline{B}\Gamma,\mathbb{Z}) are torsionfree abelian. Since i+ji+j is odd, it follows that jj is even. Let us analyze first the case where j≡0​or≡4j\equiv 0\,{\rm or}\,\equiv 4 modulo 8. In both cases, k​oj​(∗)ko_{j}(\ast) is free abelian of rank one, and k​oj​(∗)(p){ko_{j}(\ast)}_{(p)} is a free ℤ(p)\mathbb{Z}_{(p)}-module of rank one. For j≡2j\equiv 2 modulo 88, the group k​oj​(∗)(p)ko_{j}(\ast)_{(p)} is zero, and k​oj​(∗)≅0ko_{j}(\ast)\cong 0 for j≡6j\equiv 6 modulo 8.

  • •

    Assume that ii is even. Then jj is odd. If j≡3, 5,or​ 7j\equiv 3,\,5,\,\,{\rm or}\,7 modulo 88, then k​oj​(∗)≅0ko_{j}(\ast)\cong 0. If j≡1j\equiv 1 modulo 88, then k​oj(p)=0{ko_{j}}_{(p)}=0.

In either case, we see that Ei,j2E_{i,j}^{2} is either zero or a free ℤ(p)\mathbb{Z}_{(p)}-module of finite rank. Moreover, because of the rational triviality of the differentials of the Atiyah-Hirzebruch spectral sequence for k​o2​r+1​(B¯​Γ)(p)ko_{2r+1}(\underline{B}\Gamma)_{(p)}, the spectral sequence collapses without differentials and extension problems, converging to free ℤ(p)\mathbb{Z}_{(p)}-modules.

We analyze now the Atiyah-Hirzebruch spectral sequence for k​o2​r+1​(B¯​Γ)​[1p]ko_{2r+1}(\underline{B}\Gamma)[\frac{1}{p}]. By Lemma 2.3, these groups are isomorphic to

(k​o2​r+1​(B​ℤn)​[1p])ℤ/ps,\left(ko_{2r+1}(B\mathbb{Z}^{n})\left[\frac{1}{p}\right]\right)_{\mathbb{Z}/p^{s}},

By (5), there are positive integers rir_{i}, such that, these groups are isomorphic to

⨁𝑖​k​o2​r+1−i​(∗)ri​[1p].\underset{i}{\bigoplus}ko_{2r+1-i}(\ast)^{r_{i}}\left[\frac{1}{p}\right].

If ii is even, the groups k​o2​r+1−iko_{2r+1-i} are either zero or two-torsion, and hence k​o2​r+1−i​(∗)ri​[1p]=0ko_{2r+1-i}(\ast)^{r_{i}}\left[\frac{1}{p}\right]=0.

If ii is odd, the sum 2​r+1−i2r+1-i is even. Put i=2​li+1i=2l_{i}+1 for lil_{i} a natural number or zero. if r−lir-l_{i} is even, then 2​(r−li)+2≡ 2​mod​ 42(r-l_{i})+2\equiv\,2\,{\rm mod}\,4, and k​o2​r+1−i​(∗)​[1p]=0ko_{2r+1-i}(*)\left[\frac{1}{p}\right]=0. If r−lir-l_{i} is odd, 2​(r−li)+22(r-l_{i})+2 is divisible by four, and hence k​o2​r+1−i​(∗)​[1p]=k​o2​(r−li)+2​(∗)​[1p]=ℤ​[1p]ko_{2r+1-i}(*)\left[\frac{1}{p}\right]=ko_{2(r-l_{i})+2}(*)\left[\frac{1}{p}\right]=\mathbb{Z}\left[\frac{1}{p}\right].

In either case, we conclude that the groups are zero or free ℤ​[1p]\mathbb{Z}\left[\frac{1}{p}\right]-modules. ∎

The following two results concern computations of the spin bordism groups of the classifying space, and the real KK-theory of the real group C∗C^{*}-algebra of the finite group ℤ/ps\mathbb{Z}/p^{s}.

Lemma 4.3.

For any mm odd, the map

D~:Ω~∗S​p​i​n​(B​ℤ/m)→k​o~∗​(B​ℤ/m)\widetilde{D}:\widetilde{\Omega}_{*}^{Spin}(B\mathbb{Z}/m)\to\widetilde{ko}_{*}(B\mathbb{Z}/m)

is surjective.

Proof.

By Lemma 2.5 (i), it is enough to prove that

D~:Ω~∗S​p​i​n​(B​ℤ/ps)→k​o~∗​(B​ℤ/ps)\widetilde{D}:\widetilde{\Omega}^{Spin}_{*}(B\mathbb{Z}/p^{s})\to\widetilde{ko}_{*}(B\mathbb{Z}/p^{s})

is surjective for any odd prime pp.

Let MM be a ℤ/ps\mathbb{Z}/p^{s}-module. By the standard resolution of ℤ\mathbb{Z} as a trivial ℤ​[ℤ/ps]\mathbb{Z}[\mathbb{Z}/p^{s}]- module, for any i≥1i\geq 1, the localization of the homology groups with coefficients in MM,

Hi​(ℤ/ps,M)​[1p]=0H_{i}(\mathbb{Z}/p^{s},M)\left[\frac{1}{p}\right]=0

holds.

It follows that for i≥1i\geq 1 the maps

(6) Hi​(ℤ/ps,M)→Hi​(ℤ/ps,M)(p)→Hi​(B​ℤ/ps;M(p))H_{i}(\mathbb{Z}/p^{s},M)\to H_{i}(\mathbb{Z}/p^{s},M)_{(p)}\to H_{i}(B\mathbb{Z}/p^{s};M_{(p)})

are all isomorphisms. Consider the Atiyah-Hirzebruch spectral sequences converging to Ω∗S​p​i​n​(B​ℤ/ps)\Omega_{*}^{Spin}(B\mathbb{Z}/p^{s}), Ω∗S​p​i​n​(B​ℤ/ps)(p)\Omega_{*}^{Spin}(B\mathbb{Z}/p^{s})_{(p)}, k​o∗​(B​ℤ/ps)ko_{*}(B\mathbb{Z}/p^{s}) and k​o∗​(B​ℤ/ps)(p)ko_{*}(B\mathbb{Z}/p^{s})_{(p)}. By the comparison lemma for spectral sequences [Wei94, Theorem 5.2.12] and isomorphism (6), we have a commutative diagram

Ω~m​(B​ℤ/ps)\textstyle{\widetilde{\Omega}_{m}(B\mathbb{Z}/p^{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}D~\scriptstyle{\widetilde{D}}k​o~m​(B​ℤ/ps)\textstyle{\widetilde{ko}_{m}(B\mathbb{Z}/p^{s})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}≅\scriptstyle{\cong}Ω~m​(B​ℤ/ps)(p)\textstyle{\widetilde{\Omega}_{m}(B\mathbb{Z}/p^{s})_{(p)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}D~(p)\scriptstyle{\widetilde{D}_{(p)}}k​o~m​(B​ℤ/ps)(p)\textstyle{\widetilde{ko}_{m}(B\mathbb{Z}/p^{s})_{(p)}}

Then it is enough to prove the surjectivity of D~(p)\widetilde{D}_{(p)}

The Atiyah-Hirzebruch spectral sequence for computing pp-local Spin bordism and k​oko-homology collapse at the E2E^{2} term, yielding isomorphisms

Ei,j∞=H~i​(ℤ/ps)⊗(ΩjSpin)(p)E_{i,j}^{\infty}=\tilde{H}_{i}(\mathbb{Z}/p^{s})\otimes\left(\Omega_{j}^{\rm Spin}\right)_{(p)}
Ei,j∞=H~i​(ℤ/ps)⊗(koj)(p)E_{i,j}^{\infty}=\tilde{H}_{i}(\mathbb{Z}/p^{s})\otimes{\left({\rm ko}_{j}\right)}_{(p)}

Taking a look at the map on the (p)(p)-localized coefficients

D(p):(ΩjSpin)(p)→(koj)(p),D_{(p)}:\left(\Omega_{j}^{\rm Spin}\right)_{(p)}\to\left({\rm ko}_{j}\right)_{(p)},

which are non -zero only for jj a multiple of four, we see that the map is surjective at the level of coefficients, and thus surjective at the E∞E^{\infty}- term.

∎

We now recall the following consequence of Theorem 9.4 in page 415 of [DL13].

Lemma 4.4.

Let pp be an odd prime number. The real KK-theory of the real group C∗C^{*}-algebra for the cyclic group ℤ/ps\mathbb{Z}/p^{s} is as follows:

K​O0​(ℝ​[ℤ/ps])KO_{0}(\mathbb{R}[\mathbb{Z}/p^{s}]) ℤ1+ps−12\mathbb{Z}^{1+\frac{p^{s}-1}{2}}
K​O1​(ℝ​[ℤ/ps])KO_{1}(\mathbb{R}[\mathbb{Z}/p^{s}]) ℤ/2\mathbb{Z}/2
K​O2​(ℝ​[ℤ/ps])KO_{2}(\mathbb{R}[\mathbb{Z}/p^{s}]) ℤ/2⊕ℤps−12\mathbb{Z}/2\oplus\mathbb{Z}^{\frac{p^{s}-1}{2}}
K​O3​(ℝ​[ℤ/ps])KO_{3}(\mathbb{R}[\mathbb{Z}/p^{s}]) 0
K​O4​(ℝ​[ℤ/ps])KO_{4}(\mathbb{R}[\mathbb{Z}/p^{s}]) ℤ1+ps−12\mathbb{Z}^{1+\frac{p^{s}-1}{2}}
K​O5​(ℝ​[ℤ/ps])KO_{5}(\mathbb{R}[\mathbb{Z}/p^{s}]) 0
K​O6​(ℝ​[ℤ/ps])KO_{6}(\mathbb{R}[\mathbb{Z}/p^{s}]) ℤps−12\mathbb{Z}^{\frac{p^{s}-1}{2}}
K​O7​(ℝ​[ℤ/ps])KO_{7}(\mathbb{R}[\mathbb{Z}/p^{s}]) 0
Theorem 4.5.

Let M2​r+1M^{2r+1} be a 2​r+12r+1-dimensional smooth spin manifold, where r≥2r\geq 2 is odd, and with fundamental group isomorphic to Γ\Gamma. Denote by fM:M→B​Γf_{M}:M\to B\Gamma the classifying map for the fundamental group. Assume that α​(M)=0\alpha(M)=0. Then MM admits a metric of positive scalar curvature.

Proof.

Consider the diagram from Theorem 2.1 together with the additional left column given by A∘pB​NA\circ p_{BN}.

⨁(N)∈𝒩k​o~2​r+1​(B​N)\textstyle{\bigoplus_{(N)\in\mathcal{N}}\widetilde{ko}_{2r+1}(BN)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}φ2​r+1\scriptstyle{\varphi_{2r+1}}A∘pB​N\scriptstyle{A\circ p_{BN}}k​o2​r+1​(B​Γ)\textstyle{ko_{2r+1}(B\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}β\scriptstyle{\beta}A∘pB​Γ\scriptstyle{A\circ p_{B\Gamma}}k​o2​r+1​(B¯​Γ)\textstyle{ko_{2r+1}(\underline{B}\Gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}pB¯​Γ\scriptstyle{p_{\underline{B}\Gamma}}⨁N∈(𝒩)​K​O~2​r+1​(ℝ​[ℤ/ps])\textstyle{\underset{N\in\mathcal{(N})}{\bigoplus}\widetilde{KO}_{2r+1}(\mathbb{R}[\mathbb{Z}/p^{s}])}K​O2​r+1​(Cr∗​(Γ;ℝ))\textstyle{KO_{2r+1}(C_{r}^{*}(\Gamma;\mathbb{R}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}K​O2​r+1​(B¯​Γ)\textstyle{KO_{2r+1}(\underline{B}\Gamma)}

Recall that from 4.2, the group k​o2​r+1​(B¯​Γ)ko_{2r+1}(\underline{B}\Gamma) does not contain mm-torsion, and from 2.2, ker⁡pB¯​Γ\ker p_{\underline{B}\Gamma} only consists of mm-torsion. It follows that β​(D[FM])=0\beta(D_{[F_{M}]})=0. By the exactness of the upper line, we can find XN∈k~​o2​r+1​(B​N)X_{N}\in\widetilde{k}o_{2r+1}(BN) in the preimage under the map φ2​r+1\varphi_{2r+1}. By the surjectivity lemma 4.3, we can find classes [MN,FN]∈Ω2​r+1S​p​i​n​(B​N)[M_{N},F_{N}]\in\Omega_{2r+1}^{Spin}(BN) such that D​[MN​→FN​B​N]=XND[M_{N}\overset{F_{N}}{\to}BN]=X_{N}. By surgery, we can assume that FMF_{M} is 2-connected.

Now, recall that by the computation of the real KK-theory of the group C∗C^{*} algebra of ℤ/ps\mathbb{Z}/p^{s} 4.4, the reduced K​OKO-theory groups are zero in odd degrees. By the proof of the Gromov-Lawson -Rosenberg Conjecture for groups with periodic cohomology [BGS97], the classes [MN,FN][M_{N},F_{N}] admit representatives [MN+,FM][M_{N}^{+},F_{M}], where MN+M_{N}^{+} has positive scalar curvature.

We consider now the class

D[FM:M→BΓ]=D[⊔N∈𝒩MN+⟶⊔N∈𝒩BN⟶BΓ]D[F_{M}:M\to B\Gamma]=D[\underset{N\in\mathcal{N}}{\sqcup}M_{N}^{+}\longrightarrow\underset{N\in\mathcal{N}}{\sqcup}BN\longrightarrow B\Gamma]

and notice that it admits a representative of positive scalar curvature. This finishes the proof of theorem 4.5.

∎

5. Construction of Counterexamples

In this section, we will give a condition on the action of ℤ/m\mathbb{Z}/m on ℤn\mathbb{Z}^{n} that implies that the group ℤn⋊ℤ/m\mathbb{Z}^{n}\rtimes\mathbb{Z}/m is a counterexample for the GLR conjecture.The assumption with be that mm is odd and free of squares to use results from [SV24a].

Let pp be a prime that divides mm, then by Prop. 4.5 in [SV24a] there is ℤ/m\mathbb{Z}/m-submodule N⊆ℤnN\subseteq\mathbb{Z}^{n} of finite index such that this index is coprime with pp and such that there is a decomposition of ℤ/m\mathbb{Z}/m-modules

N≅ℤr⊕(ℤ​[ℤ/p])s⊕It.N\cong\mathbb{Z}^{r}\oplus(\mathbb{Z}[\mathbb{Z}/p])^{s}\oplus I^{t}.

Where ℤ\mathbb{Z} has the trivial ℤ/p\mathbb{Z}/p-action, I⊆ℤ​[ℤ/p]I\subseteq\mathbb{Z}[\mathbb{Z}/p] is the augmentation ideal and both II and ℤ​[ℤ/p]\mathbb{Z}[\mathbb{Z}/p] are endowed with the canonical ℤ/p\mathbb{Z}/p-action and moreover ℤr\mathbb{Z}^{r}, (ℤ​[ℤ/p])s(\mathbb{Z}[\mathbb{Z}/p])^{s} and ItI^{t} are itself ℤ/m\mathbb{Z}/m-modules. We say the NN is a ℤ/p\mathbb{Z}/p-module of type (r,s,t)(r,s,t). By Lemma 5.3 in [SV24a] we can suppose that ℤn\mathbb{Z}^{n} is a ℤ/m\mathbb{Z}/m-module is of type (r,s,t)(r,s,t). We have the following result.

Theorem 5.1.

Suppose mm is square-free. Let ℤn\mathbb{Z}^{n} be a ℤ/m\mathbb{Z}/m-module, and suppose that there exists a prime p∣mp\mid m such that if we consider the (r,s,t)(r,s,t) decomposition of MM viewed as a ℤ/p\mathbb{Z}/p-module, where r≥4r\geq 4, and s+t≥1s+t\geq 1 then ℤn⋊ℤ/m\mathbb{Z}^{n}\rtimes\mathbb{Z}/m is a counter-example for the GLR conjecture.

Proof.

Notice that, by Corollary. 4.2 in [AGPP08] the Lyndon-Hoschild-Serre spectral sequence for (co-)homology associated to the extension (1) collapses. In particular, H1​(Γ)H^{1}(\Gamma) contains as a subgroup H1​(ℤr×ℤ/m)H^{1}(\mathbb{Z}^{r}\times\mathbb{Z}/m), then let a1​…,a4a_{1}\ldots,a_{4} be some generators of the torsion-free part of H1​(ℤn×ℤ/m)H^{1}(\mathbb{Z}^{n}\times\mathbb{Z}/m), viewed as elements in H1​(Γ)H^{1}(\Gamma). For each aia_{i}, we have the dual elements x^i∈H1​(ℤn×ℤ/m)\widehat{x}_{i}\in H_{1}(\mathbb{Z}^{n}\times\mathbb{Z}/m), let xi=ι∗​(x^i)x_{i}=\iota_{*}(\widehat{x}_{i}), where ι:ℤr×ℤ/m→Γ\iota:\mathbb{Z}^{r}\times\mathbb{Z}/m\to\Gamma is the inclusion

and let y^\widehat{y} be an element of pp-torsion in H1​(ℤr×ℤ/m)H_{1}(\mathbb{Z}^{r}\times\mathbb{Z}/m). Now we follow the argument in [Sch98]. Let w=ι∗​(x^1×…​x^4×y^)∈H5​(Γ)w=\iota_{*}(\widehat{x}_{1}\times\ldots\widehat{x}_{4}\times\widehat{y})\in H_{5}(\Gamma). Let us prove that a1∩(a2∩(a3∩w))≠0a_{1}\cap(a_{2}\cap(a_{3}\cap w))\neq 0.

First note that by the Kunneth formula applied to the decomposition as (r,s,t)(r,s,t)-modules, the map ι∗:H5​(ℤr×ℤ/m)→H5​(Γ)\iota_{*}:H_{5}(\mathbb{Z}^{r}\times\mathbb{Z}/m)\to H_{5}(\Gamma) is injective. On the other hand, by the naturality of the cap product we have

ι∗​(a3∩(x^1×…​x^4×y^))=ι∗​(ι∗​(a3)∩(x^1×…​x^4×y^))=a3∩w.\iota_{*}(a_{3}\cap(\widehat{x}_{1}\times\ldots\widehat{x}_{4}\times\widehat{y}))=\iota_{*}(\iota^{*}(a_{3})\cap(\widehat{x}_{1}\times\ldots\widehat{x}_{4}\times\widehat{y}))=a_{3}\cap w.

But a3∩(x^1×…​x^4×y^)≠0,a_{3}\cap(\widehat{x}_{1}\times\ldots\widehat{x}_{4}\times\widehat{y})\neq 0, and similarly a1∩(a2∩(a3∩(w)))≠0a_{1}\cap(a_{2}\cap(a_{3}\cap(w)))\neq 0 and now the same argument in Example 2.2 in [Sch98] applies. Then the group Γ\Gamma is a counterexample for GLR conjecture. ∎

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