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Revista de la Unión Matemática Argentina

versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637

Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005

 

Eta series and eta invariants of Z4-manifolds

Ricardo A. Podestá,

Universidad Nacional de Córdoba, Argentina.

Supported by Conicet and Secyt-UNC

Abstract
In this paper, for each n =  4r + 3  , we construct a family of compact flat spin n  -manifolds with holonomy group isomorphic to ℤ4   such that the spectrum of the Dirac operator D  is asymmetric. For these manifolds we will obtain explicit expressions for the eta series, η(s)  , in terms of Hurwitz zeta functions, and for the eta-invariant, η  , associated to D  . The explicit expressions will show the meromorphic continuation of η(s)  to ℂ  is in fact everywhere holomorphic.

Keywords and phrases: Dirac operator; Eta series; Eta invariant; Flat manifolds

2000 Mathematics Subject Classification. Primary 58J53; Secondary 58C22, 20H15.

Introduction

If A  is a positive self-adjoint elliptic differential operator on a compact n  -manifold M  , then it has a discrete spectrum, denoted by SpecA(M   )  , consisting of positive eigenvalues λ  with finite multiplicity dλ  . This spectrum can be properly studied by the zeta function         ∑   - s  ζA(s) =    λ  , where the sum is taken over the non zero eigenvalues of A  and Re(s) >  nd  , with d  the order of A  .
If A  is no longer positive, then the eigenvalues can now be positive or negative. In this case, the spectrum is said to be asymmetric if for some λ ∈  SpecA(M  )  we have dλ ⁄= d -λ  . To study this phenomenon, Atiyah, Patodi and Singer introduced in [APS] the "signed" version of the zeta function, namely, the so called eta series defined by

           ∑  ηA(s) =          sign(λ) ∣λ ∣- s.          0⁄= λ∈SpecA

This series converges for         n  Re(s) >  d  and defines a holomorphic function ηA(s)  which has a meromorphic continuation to ℂ  . It is a non trivial fact that this function is really finite at s = 0  (See [APS2] for n  odd, and [Gi], [Gi2], for n  even). The number ηA(0)  is a spectral invariant, called the eta invariant, which gives a measure of the spectral asymmetry of A  .

In this paper, we will take A  to be the Dirac operator D  . It is a first order elliptic essentially self-adjoint operator defined on spin manifolds, that is, manifolds admitting a spin structure. Consider the eta function associated to D  , denoted simply by η(s)  . The determination of η(s)  and η(0)  is in general a difficult task and explicit computations of these objects are not easy to find in the literature.
For compact flat spin manifolds (see Preliminaries) we have the following picture. Pfäffle computed the η  -invariants in the 3-dimensional case ([Pf]). A general expression for η(s)  for an arbitrary n  -manifold M  with a spin structure ɛ  is given in [MP2]. There, in the particular case of manifolds with holonomy group       k  F ≃ ℤ 2   , 1 ≤ k ≤  n - 1  , the authors obtained a very explicit expression for η(s)  , in terms of differences of Hurwitz zeta functions          ∑  ζ(s,α) =    j≥0(j + α)-s  , for Re(s) > 1  and α ∈ (0,1]  . This allowed to compute the η  -invariant simply by evaluation at s = 0  . Also in [MP2], similar results were obtained for a family of compact flat spin p  -manifolds with holonomy group F  ≃ ℤp  , with p  prime of the form 4r + 3  .
These results led us to expect that the eta series of any compact flat spin manifold with abelian holonomy group should be expressible in terms of differences of Hurwitz zeta functions ζ(s,α)  for α ∈ (0,1] ∩ ℚ  .
The goal of the present paper is to deal with the simplest case not covered in [MP2], that is, when F ≃  ℤ4   . More precisely, we consider a rather large family of compact flat spin manifolds M  with holonomy group ℤ4   having asymmetric Dirac spectrum and we compute the corresponding eta series and eta invariant for every manifold in the family (in the case of symmetric spectrum one has that η(s) ≡ 0  , see (4.1)). The general expression for η(s)  is either of the form

       CM,ɛ(     1        3 )  η(s) = (8π)s ζ(s, 4) - ζ(s, 4)

or

       C    (  (                )    (                ))  η(s) = (8Mπ,ɛ)s  an ζ(s, 18) - ζ(s, 78) + bn ζ(s, 38) - ζ(s, 58)

where CM,ɛ,an,bn  are explicit constants depending on M  , the spin structure ɛ  of M  , and the dimension n  of M  .

Acknowledgements. I would like to thank Professor Roberto Miatello for several useful comments on a preliminary version that have contributed to improve the exposition.

1. Preliminaries

Flat manifolds. We refer to [Ch]. A Bieberbach group is a discrete, cocompact, torsion-free subgroup Γ  of the isometry group I(ℝn)  of ℝn  . Such Γ  acts properly discontinuously on ℝn  , thus M Γ = Γ \ ℝn  is a compact flat Riemannian manifold with fundamental group Γ  . By the Killing-Hopf theorem any such manifold arises in this way. Any element γ ∈ I(ℝn)  = O(n) ⋉  ℝn  decomposes uniquely as γ = BL         b  , with B  ∈ O(n)  ,       n  b ∈ ℝ  and Lb  denotes translation by b  . The translations in Γ  form a normal maximal abelian subgroup L Λ   of finite index, with Λ  a lattice in   n  ℝ  which is B  -stable for each BLb  ∈ Γ  . As usual, one identifies LΛ   with Λ  . The restriction to Γ  of the canonical projection r : I(ℝn) → O(n)  given by BLb  ↦→  B  is a group homomorphism with kernel LΛ   and F := r(Γ )  is a finite subgroup of O(n)  . The group F  ≃ Λ \Γ  is called the holonomy group of Γ  and is isomorphic to the linear holonomy group of the Riemannian manifold M Γ   . The action of F  on Λ  by conjugation is usually called the integral holonomy representation of Γ  . By an F  -manifold we understand a Riemannian manifold with holonomy group F  . In this paper we shall consider ℤ4   -manifolds which, by the Cartan-Ambrose-Singer theorem, are necessarily flat.

Spin groups. For standard results on spin geometry we refer to [LM] or [Fr]. Let Cl(n)  denote the Clifford algebra of ℝn  with respect to the standard inner product 〈,〉 on ℝn  and let ℂl(n) = Cl(n) ⊗  ℂ  be its complexification. If {e ,...,e }    1      n is the canonical basis of   n  ℝ  then a basis for Cl(n)  is given by the set {ei1 ⋅⋅⋅eik : 1 ≤ i1 < ⋅⋅⋅ < ik ≤ n} . One has that vw +  wv + 2〈v,w 〉 = 0  holds for all         n  v,w ∈ ℝ  , thus eiej = - ejei  and   2  ei =  - 1  for i,j =  1,...,n  . Inside the group of units of Cl(n)  we have the compact Lie group

Spin(n) = {v1 ⋅⋅⋅v2k : ∥vj∥ = 1, 1 ≤ j ≤ 2k},

which is connected if n ≥ 2  and simply connected if n ≥  3  . There is a Lie group epimorphism μ : Spin(n) → SO(n)  with kernel {±1} given by v ↦→  (x ↦→  vxv-1)  .

If Bj  is a matrix for 1 ≤ j ≤ m  , we will denote by diag(B1, ...,Bm)  the block matrix having the block B    j  in the diagonal position j  . For t ∈ ℝ  let B(t) =  [cost - sint]           sint cost  and, fort1,...,tm ∈ ℝ  , define
                {                     diag(B(t1),...,B(tm))       if   n = 2m,  x0(t1,...,tm) :=    diag(B(t ),...,B(t  ),1)    if   n = 2m +  1,                             1          m                 ∏m  (                       )  x(t1,...,tm) :=      cos(tj) + sin(tj)e2j- 1e2j ∈ Spin(n).                 j=1
(1.1)

It is easy to check that, for any k ∈ ℤ  , the elements x(t1,...,tm)  satisfy
   x(t1,...,tm)k = x(kt1,...,ktm)    - x(t1,t2,...,tm) = x(t1 + π,t2,...,tm).
(1.2)

We will also need to fix maximal tori in Spin(n)  and SO(n)  . They are respectively given by T  = {x(t1,...,tm) : tj ∈ ℝ, 1 ≤ j ≤ m} and by T0 =  {x0(t1,...,tm) :  tj ∈ ℝ,j = 1, ...,m} (see [LM] or [Fr]). The restriction μ : T →  T0   is a 2-fold cover and
μ(x(t1,...,tm)) = x0(2t1,...,2tm).
(1.3)

Spin representations. We consider (L,S)  an irreducible complex representation of the Clifford algebra ℂl(n)  , restricted to Spin(n)  . The complex vector space S  has dimension 2m  with m  := [n2]  . If n  is odd, then (L,S)  is irreducible for Spin(n)  and is called the spin representation. If n  is even, then S =  S+ ⊕ S- where each S± is irreducible of dimension 2m-1   . If L ± denote the restricted action of L  on S± then    ±  ±  (L  ,S  )  are called the half-spin representations of Spin(n)  . We shall write (Ln, Sn)  and   ±   ±  (L n,S n)  for (L, S)  and   ±   ±  (L  ,S )  when we wish to specify the dimension.

We will make repeatedly use of the following result (see [MP2]) which gives the values of the characters χLn   and χL±n   of the spin and half spin representations on T  . If n =  2m  , then

                         ( m             m       )                       m -1  ∏           m ∏  χL±n (x(t1,...,tm)) = 2        costj ± i     sin tj .                             j=1           j=1
(1.4)

Furthermore,                        ∏  χL (x(t1,...,tm)) = 2m   m   costj    n                      j=1  for n =  2m  or n = 2m  + 1  .

Spin structures. If (M,  g)  is an orientable Riemannian manifold, let B(M  )  be the bundle of oriented frames on M  and π : B(M ) → M  the canonical projection. B(M  )  is a principal SO(n)  -bundle over M  . A spin structure on M  is an equivariant 2-fold cover    ˜  p :B(M  ) → B(M  )  where    ˜  ˜π :B(M  ) → M  is a principal Spin(n)  -bundle and π ∘ p = ˜π  . Such M  endowed with a spin structure is called a spin manifold.

On compact flat manifolds M Γ =  Γ \ℝn  , Γ  a Bieberbach group, the spin structures are in a one to one correspondence with group homomorphisms ɛ  commuting the diagram

                 Spin(n)                /     |          ɛ / / /     |μ        / /           |     /-/-----------  Γ        r       SO(n)
(1.5)

that is, satisfying μ∘ ɛ = r  where r(γ) =  B  if γ = BL   ∈ Γ         b  (see [Fr], [LM]). We shall denote by (M  Γ ,ɛ)  the spin manifold M Γ   endowed with the spin structure ɛ  as in (1.5).

The spectrum of the Dirac operator. If (L, S)  is the spin representation, the vector bundle S(M  Γ ,ɛ) := Γ \(ℝn × S) → Γ \ ℝn = M Γ   with action γ ⋅ (x, w) = (γx,L( ɛ(γ))(w))  , where γ ∈ Γ  , w  ∈ S  , is called the spinor bundle of M  Γ   . The space Γ ∞(S(M Γ ,ɛ))  of smooth sections of the spinor bundle can be identified with the set {f  : ℝn → S smooth   : f(γx) = L( ɛ(γ))f(x)} . One can consider the Dirac operator D  acting on smooth sections f  of S(M   ,ɛ)       Γ  by           ∑n       ∂f-  Df  (x) =    i=1ei ⋅ ∂xi (x)  , where ei ⋅ w = L(ei)(w)  for w ∈ S  . One has that D  is an elliptic first-order differential operator, which is symmetric and essentially self-adjoint. Furthermore, over compact manifolds, D  has a discrete spectrum consisting of real eigenvalues ±  2πμ  , μ ≥  0  , of finite multiplicity d±μ  . If f ∈ kerD  , f  is called a harmonic spinor.

In [MP2] explicit expressions for the multiplicities  ±  dμ  for any compact flat spin manifold (M Γ ,ɛ)  with translation lattice Λ  and holonomy group F  were obtained. We now recall the ingredients for these expressions.

Let F1 = {B  ∈ F =  r(Γ ) : nB = 1} where nB :=  dim ker(B  - Id)  . Put   *         *          2πiλ⋅u  Λ ɛ = {u ∈ Λ  : ɛ(λ) = e    ,λ ∈ Λ} , with   *  Λ the dual lattice of Λ  , and
  *           *  Λ ɛ,μ = {u ∈ Λ ɛ : ∥u ∥ = μ}.
(1.6)

Now, for each γ =  BLb ∈  Γ  , let (Λ*ɛ,μ)B  denotes the set of elements fixed by B  in Λ *ɛμ  , that is
  *   B          *  (Λɛ,μ)  = {u ∈ Λ ɛ,μ : Bv = v}.
(1.7)

Furthermore, for γ ∈ Γ  , let xγ  be a fixed, though arbitrary, element in the maximal torus of Spin(n - 1)  , conjugate in Spin(n)  to ɛ(γ)  . Finally, define a sign σ(u,x γ)  , depending on u  and on the conjugacy class of x   γ  in Spin(n - 1)  , in the following way. If γ =  BLb ∈ Λ \Γ  and        *B  u ∈ (Λ ɛ) \{0} , let hu ∈ Spin(n)  such that      - 1  huu hu  = ∥u ∥en  . Hence,         -1  hu ɛ(γ)hu  ∈ Spin(n - 1)  . Take σɛ(u,xγ) = 1  if         -1  hu ɛ(γ)hu  is conjugate to xγ  in Spin(n -  1)  and σɛ(u,x γ) = - 1  otherwise. As a consequence, σ( - u, xγ) = - σ(u,xγ)  and σ(αu, xγ) = σ(u, xγ)  for every α > 0  (see Definition 2.3, Remark 2.4 and Lemma 6.2 in [MP2] for details).

For n  odd, the multiplicity of the eigenvalue ± 2πμ  , for μ > 0  , is given by
              (   ±         -1-     ∑        ∑     - 2πiu⋅b  dμ (Γ ,ɛ) = ∣F ∣                   e       ⋅ χL±n-1(xγ) +                   γ ∈Λ\Γ  u∈(Λ*ɛ,μ)B                   B ⁄∈ F1                     ∑       ∑                          )                                   e-2πiu⋅b ⋅ χ ±σ(u,x )(xγ) ,                           u∈(Λ*  )B           Ln- 1 γ                   γB ∈∈Λ\FΓ     ɛ,μ                        1
(1.8)

while for n  even, it is given by the first term in (1.8) (i.e., the sum over B ⁄∈ F1   ) with L ±    n-1   replaced by L    n-1   . For μ = 0  , with n  even or odd, we have that            1--∑                         F  d0(Γ ,ɛ) = ∣F∣   γ∈Λ\Γ χLn (ɛ(γ)) = dim S  , if ɛ∣Λ = 1  , and d0(Γ ,ɛ) = 0  , otherwise.

2. A family of spin ℤ4   -manifolds

In this section, for each n ≥ 3  , we shall construct a family of n  -dimensional pairwise non-homeomorphic spin ℤ4   -manifolds, having asymmetric Dirac spectrum. In this way, we will obtain non trivial eta series and eta invariants, to be computed in the next sections. PutJ˜:= [01 -01]  . For each j,l ≥ 1  and k ≥ 0  , we set
B    :=  diag( ˜J,...,J˜,- 1,...,- 1,1,...,1 ),    j,k         ◟--◝◜-◞  ◟---◝◜---◞ ◟--◝◜--◞                  j         k          l
(2.1)

where 2j + k + l = n ≥ 3  .

Then B    ∈ O(n)    j,k  , B 4 = Id   j,k  and B    ∈ SO(n)    j,k  if and only if k  is even. Let Λ =  ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤen  be the canonical lattice of   n  ℝ  and for j,k,l  as before define the groups
Γ j,k := 〈Bj,kL en,Λ〉.                4
(2.2)

We have that Λ  is B    j,k  -stable. Since (Bm  -  Id) me  = 0 ∈ Λ     j,k      4 n  , for 0 ≤  m  ≤ 3  , and  ∑3      m  e              ∑3       m  (   m=0 Bj,k)-n4 = en ∈ Λ \(    m=0 B j,k)Λ  , by Proposition 2.1 in [MR], each Γ j,k  is a Bieberbach group. In this way, we have a family
Fn = {M    := Γ   \ℝn  : 1 ≤ j ≤ [n-1], 0 ≤ k ≤ n - 2j - 1, l ≥ 1}           j,k     j,k                 2
(2.3)

of compact flat manifolds with holonomy group F ≃ ℤ4   . It is easy to see that the cardinality of   n  F  is given by    n    n- 1(     n- 1    )      2  #F   =  [-2-]n - [-2--] - 1 = o(n )  .

Lemma 2.1. For Mj,k ∈ Fn   we have
                l    j+k  H1(Mj,k, ℤ) ≃ ℤ  ⊕ ℤ 2  .
(2.4)

Hence the manifolds in Fn   are non-homeomorphic to each other.

Proof. We compute H1(Mj,k, ℤ) ≃ Γ j,k∕[Γ j,k,Γ j,k]  . For           e  γ =  Bj,kL -n4   , we have [Γ   ,Γ  ]  =   〈[γ,L  ] = L          : 1 ≤ i ≤ n〉    j,k  j,k            ei     (Bj,k- Id)ei             =   〈L -e1±e2,...,L- e2j-1±e2j,L2e2j+1,...,L2e2j+k〉.  Using this, and the fact that  4  γ  = Len   , it is easy to see that                 l    j+k  H1(Mj,k, ℤ) ≃ ℤ  ⊕ ℤ 2  .  Thus, if Mj,k  and Mj ′,k′ are homeomorphic then l = l′ and j + k =  j′ + k′ . Since n =  2j + k + l = 2j′ + k′ + l′ we have that j = j′ and k = k ′ . Therefore, the manifolds in Fn  are non-homeomorphic to each other.

We now study the existence of spin structures on n  -dimensional ℤd  -manifolds following [MP], where the existence of spin structures on ℤk2   -manifolds was considered. Let Γ  be a Bieberbach group with holonomy group F ≃  ℤd  and translation lattice Λ  . Then M   =  Γ \ℝn    Γ  with Γ =  〈γ, Λ〉 where γ = BL          b  , B  ∈ O(n)  , b ∈ ℝn  , B Λ = Λ  and   d  B   = Id  .

Assume there is a spin structure defined on M Γ   , that is, a group homomorphism ɛ : Γ → Spin(n)  such that μ∘ɛ = r  . Then, necessarily ɛ(L ) ∈ {±1}     λ , for λ ∈ Λ  . Thus, if λ1,...,λn  is a ℤ  -basis of Λ  and we set δi := ɛ(Lλi)  , for every     ∑  λ =    imiλi ∈ Λ  with mi ∈  ℤ  , we have          ∏   mi   ∏  ɛ(L λ) =   iδi  =   mi odd δi  .

For any γ = BLb  ∈ Γ  we will fix a distinguished (though arbitrary) element in μ- 1(B)  , denoted by uB  . Thus, ɛ(γ) =  σuB,  where σ ∈ { ±1} depends on γ  and on the choice of uB  .

The morphism ɛ  is determined by its action on the generators of Γ  . Hence, we will identify this morphism with the (n + 1)  -tuple
ɛ ≡ (δ1,...,δn,σ uB)
(2.5)

where δi = ɛ(L λi)  and σ  is defined by the equation ɛ(γ) =  σuB  .

Now, since ɛ  is a morphism and γ = BLb  ∈ Γ  , for any λ ∈ Λ  we have
ɛ(LB λ) = ɛ(γL λγ-1) = ɛ(γ)ɛ(L λ)ɛ(γ-1) = ɛ(Lλ).
(2.6)

Therefore, if ɛ  is a spin structure on M    Γ   , since γd ∈ L         Λ   , then the character ɛ∣Λ : Λ → {±1} must satisfy the following conditions for any γ = BLb  ∈ Γ  :
             d          d  (ɛ1)      ɛ(γ ) = (σuB)  (ɛ )      ɛ(L       ) = 1,  λ ∈ Λ.    2          (B- Id)λ
(2.7)

We thus set
ˆΛ(Γ ) := {χ ∈ Hom( Λ,{±1})  : χ satisfies (ɛ ) and (ɛ )}.                                            1        2
(2.8)

The next result says that these necessary conditions for the existence of spin manifolds are also sufficient in the case of manifolds with cyclic holonomy groups. We adapt the proof of Theorem 2.1 in [MP] to our case.

Proposition 2.2. If Γ =  〈BLb, Λ〉 is a Bieberbach group with holonomy group F  = 〈B 〉 ≃ ℤd   and σ  is as in (2.5), then the map ɛ ↦→  (ɛ∣Λ, σ)  defines a bijective correspondence between the spin structures on M Γ   and the set ˆΛ(Γ ) × {±1} . Hence, the number of spin structures on M Γ   is either 0  or 2r   for some 1 ≤  r ≤ n  .

Proof. It suffices to prove that, given ɛ ∈ Λˆ( Γ )  , we can extend it into a group homomorphism from Γ  to Spin(n)  , also called ɛ  , satisfying (1.5).

Let γ = BL         b  . Since Λ  is normal in Γ  and Bd = Id  , we see that any γ  ∈ Γ   0  can be written uniquely as       k  γ0 = γ L λ  , with 0 ≤ k ≤ d - 1  , λ ∈  Λ  . For any choice of        -1  uB  ∈ μ  (B)  , we set ɛ(γ) = σuB  , with σ ∈  {±1} , and, for a general element in Γ  , we define

   k          k  ɛ(γ Lλ) = ɛ(γ) ɛ(L λ)

for 0 ≤ k ≤ d - 1  . Thus, we get a well defined map ɛ : Γ → Spin(n)  such that μ ∘ ɛ = r  , and we claim it is a group homomorphism.

In fact, note that if γ = BLb  then γk = BkLb(k)   where         ∑k -1  b(k) :=    i=0 B - ib  . We have that γkγl = Bk+lLB  -lb(k)+b(l)   and also γkγl = γk+l = Bk+lLb(k+l)   . Hence, by using these relations and condition (ɛ )   2  we get

ɛ(γkL λγlLλ′)  =  ɛ(BkLb(k)+ λBlLb(l)+λ′) = ɛ(Bk+lLB -l(b(k)+λ)+b(l)+λ′)                       k+l                k+l         ′       k       l                 =  ɛ(γ   LB -lλ+λ′) = ɛ(γ)   ɛ(L λ)ɛ(L λ) = ɛ(γ L λ)ɛ(γ L λ′)

for any λ, λ′ ∈ Λ  .

It is clear that the number of spin structures is either   r  2  , for some r ≥ 1  , or 0, in case the equations given by conditions (ɛ1   ) and (ɛ2   ) are incompatible ones. Since each of these equations divides by 2 the number of spin structures and the covering torus has exactly 2n  such structures we have that r ≤ n  . This completes the proof.

Since spin manifolds are orientable, we need to restrict ourselves to the manifolds Mj,k  with k =  2k         0   even. We have the following result

Corollary 2.3. Every orientable ℤ4   -manifold Mj,k   , k = 2k0   , has   n- j  2   spin structures ɛ  parametrized by the (n + 1)  -tuples (δ1,...,δn,σuBj,k)  as in (2.5) satisfying:
δ1 = δ2, ⋅⋅⋅ , δ2j-1 = δ2j   and      δn = (- 1)j
(2.9)

where u    = (√2-)j(1 + e e )⋅⋅ ⋅(1 + e    e  )e    ⋅⋅⋅e   Bj,k     2        1 2          2j-1 2j  2j+1    2j+k   .

Proof. We first note that            π     π  Bj,k = x0(◟2,.◝.◜.,2◞,π◟,-.◝.◜.,π◞,0◟,..◝.◜,0◞) ∈ T0                j       k0       [l]                                 2   in the notation of (1.1). Hence,  - 1            π      π π      π  μ   (Bj,k) = ±x( 4,..., 4,2,..., 2,0...,0) ∈ T  , by (1.3) and we take           π      π π     π  uBj,k = x(4,..., 4,2,..., 2,0...,0)  .

Let γj,k = Bj,kL en4   . Since ɛ(γj,k) = σuBj,k   and γ4j,k = Len   for every j,k  , condition (ɛ1)  gives

δ  = ɛ(γ   )4 = u4    = x(π,...,π ,2π,...,2π ,0,...,0) = (- 1)j,   n       j,k      Bj,k     ◟--◝◜--◞ ◟---◝◜---◞                               j        k0

where we have used (1.2) in the third equality. Moreover, condition (ɛ2)  gives 1 = ɛ(L(Bj,k-Id)e2i-1) = δ2i- 1δ2i  for 1 ≤ i ≤ j  , hence (2.9) holds. Since there are no more relations imposed on the δ   i  's the result follows.

Remark 2.4. It would be natural to consider the larger family   n  F  with matrices in (2.1) replaced by
Bj,h,k = diag( ˜J,...,J˜,J,...,J,- 1,...,- 1,1,...,1),               ◟--◝◜--◞ ◟-◝◜--◞ ◟----◝◜---◞ ◟--◝◜-◞                 j≥1      h≥0       k≥0       l≥0
(2.10)

where J := [01 1 0 ]  , 2(j + h) + k + l = n  and h + l ⁄= 0  . That is, consider Gn =  {Mj,h,k := Γ j,h,k\ℝn} where Γ j,h,k = 〈Bj,h,kL en,Λ 〉                  4 if l ≥ 1  and Γ j,h,k = 〈Bj,h,kL e2j+1,Λ 〉                   2 if l = 0  .

However, this larger family of ℤ4   -manifolds has trivial eta series unless h = 0  and l = 1  , i.e. the case previously considered. In fact, we have nBj,h,k ≥ 2  for every j,h,k  with h + l ≥ 2  and, by Corollary 2.6 in [MP2], the spectrum of D  is symmetric. The remaining case (h = 1  and l = 0  ) is more involved, but one checks it by proceeding similarly as in the next section.

3. The spectrum of the Dirac operator

Since we are looking for spectral asymmetry of D  , by Corollary 2.6 in [MP2], we will restrict ourselves to orientable odd dimensional manifolds in Fn  with F  ⁄= ∅   1 (recall that                     n B  F1 =  {B ∈ F  : dim( ℝ ) =  1} ). Hence, we fix n =  2m +  1  and take
Fn1 :=  {Mj,k ∈ Fn  : k = 2k0, l = 1}.
(3.1)

Also, for each          n  Mj,k ∈ F 1   we choose, in the notation of (1.1) and Corollary 2.3, the spin structure
       (  ɛ σj,k =  1,...,1,(- 1)j,σx( π4,..., π4, π2,..., π2)),   σ ∈ {±1}.                            ◟--◝◜--◞ ◟--◝k◜--◞                               j        2
(3.2)

For simplicity, we will use the following shorter notation
x     (θ1,...,θt) := x( θ1,...,θ1,...,θt,...,θt).   k1,...,kt                ◟--◝◜--◞      ◟--◝◜--◞                            k1            kt
(3.3)

By using expression (1.8), we will explicitly compute the multiplicity of the eigenvalues of the Dirac operator D  of the spin manifolds        σ  (Mj,k, ɛj,k)  .

We shall need the following auxiliary function. Let              j  ω(j) :=  3-(-21)-   , i.e.
       {  ω(j) =   1            if j is even           2            if j is odd.
(3.4)

Theorem 3.1. Let n = 2m  + 1 = 4r + 3  . The ℤ   4   -manifolds M    ∈ Fn    j,k    1   with spin structures  σ  ɛj,k   as in (3.2) have asymmetric Dirac spectrum and, in the notation of (1.6), the multiplicity of the non zero eigenvalue ± 2π μ  of D  is given by

          {                      r+[-t-]         j   ±  σ       4r- 1∣Λ ɛσj,k,μ∣ ± σ (- 1) ω(j)2m -ω(j)-[2]  μ =  2tω+(1j)-  dμ (ɛj,k) =    r- 1              4    ∣Λ ɛσj,k,μ∣                            otherwise
(3.5)

for k ⁄= 0  , and by

           {                         (                  t)               4r-1∣Λɛσm,0,μ∣ ± (- 1)r 2r-1 (- 1)t2r + σ (- 1)[2] μ = 2t+21  d±μ(ɛσm,0) =    r-1               4   ∣Λɛσm,0,μ∣                                   otherwise

for k = 0  , where t ∈ ℕ0   in both cases and ω(j)  is as in (3.4).

Furthermore, for every         n  Mj,k ∈ F1   there are no non-trivial harmonic spinors, that is, d±0 (ɛσj,k) = 0  .

Proof. For fixed j,k  , let γ = Bj,kL en4-   be the generator of Γ j,k  . For 1 ≤ h ≤ 3  , letb  ∈ ℝn   h  be defined by γh = Bh  L         j,k bh   and we put
             ∑  S ±(μ) :=             e-2πiu⋅bh χ ± σ(u,x  )(x  h).   h                 h          Ln-1  γh   γ            u∈(Λɛσj,k,μ)B
(3.6)

Since F1(Γ j,k) ⁄= ∅ , because nB =  nB3 = 1  , and since nB2 = 1  if and only if k = 0  , the formula in (1.8) now reads
            (                                           )  d±μ (ɛσj,k) = 14  2m-1 ∣Λ ɛσj,k,μ∣ + S ±1 (μ) + δk,0S ±2 (μ) + S ±3 (μ) ,
(3.7)

where δ   k,0   is Kronecker's delta function.

Now, since  σ  ɛj,k(γ) ∈ T  , the maximal torus in Spin(n - 1)  , we can take x γ = ɛσj,k(γ) = σxj,k0(π4, π2)  , xγ2 = xj,k0(π2,π)  and xγ3 = σxj,k0(3π4 , 3π2-)  since

xγh = ɛ(γh) = ɛ(γ)h = (σx γ)h

and x(θ ,...,θ  )h = x(hθ ,...,h θ )     1      m          1        m  for h ∈ ℕ  (see (1.2)). Furthermore, since B   e  = e    j,k n    n  and for u = en  we can take hu = 1  , then σ(en,x γh) = 1  for each 1 ≤ h ≤  3  . Moreover, σ(μen, xγh) = 1  and σ(- μen,xγh) = - 1  , because μ >  0  (see Preliminaries). On the other hand bh = he4n   .

Since Λ  is the canonical lattice  n  ℤ  then  *  Λ  = Λ  and we can write  *  Λɛ = Λ + u ɛ  where      ∑  uɛ =   {i:δi= -1}ei  . Furthermore, since δi = 1  for 1 ≤  i ≤ n - 1  andδn = (- 1)j  (see (3.2)), we have that Λ *ɛσ  = ℤe1 ⊕  ⋅⋅⋅ ⊕ ℤen    j,k  if j  is even and  *                              1  Λ ɛσj,k = ℤe1 ⊕ ⋅⋅⋅ ⊕ ℤen -1 ⊕ (ℤ + 2)en  if j  is odd. Thus,
       h    {  ℤen             j even  (Λ *ɛσ )Bj,k =         1     j,k          (ℤ + 2)en       j odd.
(3.8)

Clearly, if μ  is such that   *    Bhj,k  (Λ ɛσj,k,μ)    = ∅ for every 1 ≤ h ≤ 3  , only the identity of Γ  can give a non-zero contribution to (1.8) and the multiplicity formula now reads d±μ (ɛσj,k) = 4r- 1∣Λɛσ ,μ∣                    j,k . Thus, from now on, we will assume that μ  satisfies    *    Bh  (Λ ɛσj,k,μ) j,k ⁄= ∅ for some 1 ≤ h ≤  3  . Then, we have   *    Bh  (Λ ɛσj,k,μ) j,k = {± μen} with μ ∈ ℕ  for j  even and           1  μ ∈  ℕ0 + 2   for j  odd and hence we have

           π                   π  S±h (μ) =  e-2ihμ χL±  (xγh) +  e2ihμ χL∓  (xγh).                    n-1                 n-1
(3.9)

If, furthermore, χL -n- 1(x γh) = - χL+n-1(x γh)  holds, then
            π       π  Sh±(μ) = (e- 2ihμ - e2ihμ)χL ±n-1(xγh) = - 2 isin(πh2μ)χL ±n-1(xγh).
(3.10)

By (1.4), the characters   ±  χLn-1(xγh)  have the expression

              h  m-1(     hπ  j     hπ  k0   m      hπ  j     hπ  k0)  χL±n-1(xγh) = σ 2     (cos(-4 )) (cos(-2 )) ±  i (sin( 4-))(sin(-2 ))  .

The explicit values, for 1 ≤  h ≤ 3  , are given in the following table:
|-----------|----------------------|-------------------------|  -χL±n-1(xγh)----------k->-0--------------------k =-0-----------  |-----------|-------m--1m--√2-j----|-----m-1--√2-m------m----|  |--h-=-1----|---±σ-2----i-(-2-)----|--σ-2----(2-)-(1-±-i-)---|  |  h = 2    |          0           |        ±2m -1im         |  |-----------|----m-1-m--√2--j----k0|---m--1-√2-m---------m---|  ---h-=-3----±-σ-2----i-(-2-)(--1)---σ-2----(-2 )-((- 1)-±-i-)-
(3.11)

Case 1: k >  0  . Let h = 1  or 3  . Suppose that j  is even, then μ ∈ ℕ  . If μ  is even, S ±(μ) = 0   h  because sin(πhμ) = 0       2  . Let μ = 2t + 1  with t ∈ ℕ0   . Since    (2t+1)hπ         t+[h]  sin(   2   ) = (- 1)  2   , by (3.10) we have

  ±               μ-21+[h2]  S h (μ) = - 2i (- 1)      χL ±n-1(x γh).

Replacing these values in (3.7) we get

              (                                                )  d± (ɛσ ) =   1  2m- 1∣Λ σ   ∣ - 2 i(- 1)t(χ ±  (x ) - χ ±  (x 3))   μ   j,k      4(        ɛj,k,μ              Ln-1  γ     Ln-1  γ     )               1   m- 1  σ             t+1  m  m+1 √2- j(         k0)           =   4  2   ∣Λ ɛj,k,μ∣ ± σ (- 1)  2  i   ( 2 ) 1 - (- 1)        (3.12)                r-1                t+r  m-1-[j]           =   4   ∣Λɛσj,k,μ ∣ ± σ (- 1) 2      2

where we have used that m  = 2r + 1  and that k0   is odd, because j  is even and n =  2(j + k ) + 1 ≡ 3 (4)              0  .

Now, if j  is odd,     2t+1  μ =   2   with t ∈ ℕ0   . One has that
    (π(2t+1))      (3π(2t+1))        [t2] √2-  sin    4    =  sin     4     = (- 1)  ( 2 ).
(3.13)

Then, by using that k0   is even because j  is odd, we get that

               (                         √-                          )  d± (ɛσ )  =  1  2m- 1∣Λ  σ  ∣ - 2 i(- 1)[t2](-2)(χ ±  (x  ) + χ ±  (x 3))   μ   j,k      4 (       ɛj,k,μ               2    Ln-1  γ     Ln-1  γ  )               1   m- 1                [t2]+1 m  m+1  √2-j+1(         k0)            =  4  2    ∣Λ ɛσj,k,μ∣ ± σ (- 1)   2  i    ( 2 )   1 + (- 1)      (3.14)                r-1                r+[t] m -2-[j]            =  4   ∣Λɛσj,k,μ∣ ± σ (- 1) 2 2      2.

Taking      2t+1-  μ =  ω(j)   , with         3-(--1)j-  ω(j) =    2   , from (3.12)  and (3.14)  we finally obtain expression (3.5).

Case 2: k =  0  . Then j  is odd and      2t+1-  μ =   2   with t ∈ ℕ0   . We have that
S ±2 (μ) = ±( - 1)t+1 2m im+1 = ±( - 1)t+r 22r+1,
(3.15)

by (3.10); and, for h = 1,3  , by (3.11) we obtain

                   √-   (  π                     π                 )  S± (μ)   =  σ 2m -1(-2)m  e--2ihμ((- 1)[h2] ± im)+ e2 ihμ((- 1)[h2] ∓ im)   h                 2    (                                   )                m -1 √2-m       [h2]     πhμ     m          πhμ          =  σ 2    (2 )   (- 1)  2cos( 2  ) ± i (- 2i)sin( 2 ) .

Then, since    (      )      (       )  cos π(2t+1)  - cos  3π(2t+1)- = 0         4              4  , by using (3.13) we have
                           √-  S±1 (μ) + S ±3 (μ) = ∓ σ 2m+1 (-22 )m+1 sin(π(2t4+1))                           r+[t] r+1                 = ± σ (- 1)  2 2   .
(3.16)

By introducing the values of (3.15) and (3.16) in (3.7), we get to the expressions in the statement of the proposition.

Finally, the claim concerning to the multiplicity of the 0-eigenvalue follows directly from the expressions just after (1.4) and (1.8), respectively.

Remark 3.2. From the multiplicity formulae obtained in Proposition 3.1 we see that, generically, there are no Dirac isospectrality between manifolds in Fn    1   .

On the other hand, if m  = 2r  , that is, if n =  4r + 1  , then the spectrum of D  is symmetric with multiplicities given by  ±  σ      r-1  dμ(ɛj,k) = 4   ∣Λɛσj,k,μ∣ for every pair j,k  . This follows by (3.12)  and (3.14)  in the proof of Theorem 3.1, in the case k > 0  , and by (3.7) for k = 0  , since both (3.15) and (3.16) vanish in this case. Hence, for a fixed n  , {(M2j,k,ɛσ  ) : 1 ≤ j ≤ r}           2j,k , is a set of m  mutually D  -isospectral ℤ4   -manifolds, and similarly for            σ  {(M2j+1,k,ɛ2j,k) : 1 ≤ j ≤ r} .

4. Eta series and eta invariants of ℤ4   -manifolds

In general, for a differential operator A  having positive and negative eigenvalues we can decompose the spectrum SpecA(M  ) = S ˙∪ A  where S  and A  are the symmetric and the asymmetric components of the spectrum, respectively. That is, if λ = 2π μ  , λ ∈ S  if and only if d+ (M ) = d- (M  )    μ        μ  . We say that Spec  (M  )       A  is symmetric if A  = ∅ .

As a measure of the spectral asymmetry of a differential operator A  on a manifold M  , Atiyah, Patodi and Singer introduced the eta series
             ∑        sign(-λ)   --1--- ∑    d+μ-(M-) --d-μ(M-)-  ηA(s) =                 ∣λ ∣s   = (2π)s             ∣μ∣s           λ∈ SpecD(M)                  μ∈-1A               λ⁄= 0                       2π
(4.1)

generalizing the zeta functions for the Laplacian. It is known that this series converges absolutely for          n  Re(s) >  d  , where d  is the order of A  , and defines a holomorphic function ηA(s)  in this region, having a meromorphic continuation to ℂ  that is holomorphic at s = 0  ([APS2], [Gi2]). The eta invariant is defined by ηA := ηA(0)  . It is known that if n ⁄≡  3 mod  (4)  then η(s) ≡ 0  for every Riemannian manifold M  (see [Fr]).

Now, we let A =  D  , the Dirac operator. By using the results obtained in the previous section we shall compute the expression for the eta series and the values of the η  -invariants for the spin ℤ4   -manifolds considered. We have the following result

Theorem 4.1. Let n = 2m  + 1 = 4r + 3  . The eta series for the ℤ   4   -manifolds M    ∈ Fn    j,k    1   with spin structures  σ  ɛj,k   as in (3.2) are given for k > 0  by
                2ω(j)-1            Cj,σ-- ∑        [h+ω2(j)] (  2h+1 )  ηɛσj,k(s) = (8π)s       (- 1)      ζ s,4ω(j)                   h=0
(4.2)

with                          j  Cj,σ = σ (- 1)r 2m+1-ω(j)- [2]   , and by
          (- 1)r 2r ∑ 3 (        h+1   )  (      )  ηɛσm,0(s) = ------s-      σ + (- 1)[2 ]2r  ζ s, 2h+81              (8π)   h=0
(4.3)

where          ∑ ∞         - s  ζ(s,α) =   j=0(j + α)   is the Hurwitz zeta function for Re(s) >  1  , α ∈ (0,1]  , and ω(j)  is as defined in (3.4).

Furthermore, the meromorphic continuation of ηɛσj,k(s)  to ℂ  is everywhere holomorphic for all manifolds Mj,k ∈ Fn           1   .

Note. Observe that, for k > 0  , all the eta functions {ηɛσ  (s)}     2j,k are mutually proportional and the same happens with    σ  {ηɛ2j+1,k(s)} .

Proof. By Proposition 3.1 we have that
                    {        r+[-t-] m+1- ω(j)-[j]   +  σ      -  σ        σ (- 1)   ω(j) 2          2        k > 0  dμ(ɛj,k) - dμ(ɛj,k) =    (- 1)r 2r(σ (- 1)[t2] + (- 1)t2r)   k = 0.
(4.4)

Now, by (4.1) and (4.4), in the case k > 0  , we have
                          t            Cj,σ  ∞∑   (- 1)[ω(j)]  ηɛσj,k(s) =  ----s-    --2t+1-s--            (2 π)  t=0  (ω(j))
(4.5)

where C   =  σ(- 1)r 2m+1 -ω(j)-[j2]   j,σ   .

If j  is even, the series in (4.5) equals

 ∞                 (  ∞             ∞          )  ∑   --(--1)t--   1-   ∑   ---1----  ∑   ---1----     1-(     1        3)      (2t + 1)s = 4s       (t + 1)s -     (t + 3)s  =  4s ζ(s, 4) - ζ(s, 4) .  t=0                  t=0     4     t=0      4

where we have separated the contributions of 2t  and 2t + 1  , and hence in this case
          σ (- 1)r 2m-[j2](              )  ηɛσj,k(s) = --------s----- ζ(s, 14) - ζ(s, 34) .                (8π)
(4.6)

For j  odd, the series in (4.5) now equals

          t  ∑∞  (- 1)[2]      1  ∞∑      1         1          1          1      -----1-s  =   4s    -----1-s + -----3-s - ----5-s-- -----7-s   t=0 (t + 2)          t=0 (t + 8)    (t + 8)    (t + 8)   (t + 8)                    1- (    1        3        5        7 )                =   4s  ζ(s,8) + ζ(s,8) - ζ(s,8) - ζ(s,8) .

where we have separated the contributions of 4t + h  , with 0 ≤ h ≤  3  , and hence

          Cj,σ  ∑3      h  ηɛσj,k(s) = ----s-   (- 1)[2]ζ(s, 2h8+1).            (8π)  h=0
(4.7)

By putting together expressions (4.6) and (4.7) we get formula (4.2).

On the other hand, for k = 0  , by (4.1) and (4.4), we have

               r r ∑∞         [t2]       t r  ηɛσ (s) = (--1)-2-     σ-(--1)--+-(--1)-2-.    m,0        (2 π)s   t=0       (t + 12)s

The series above equals

∑∞  (σ + 2r)    (σ - 2r)    (σ - 2r)    (σ + 2r)      ------1-s + ------3s--  -----5-s-- ------7-s  t=0 (4t + 2)    (4t + 2)    (4t + 2)    (4t + 2)

where again we have separated the cases 4t + h  , 0 ≤ h ≤  3  . Now, proceeding as before, we obtain
          (-1)r2r(       r (    1        7 )         r(     3        5 ))  ηɛσm,0(s) = -(8π)s-  (σ + 2 ) ζ(s,8) - ζ(s,8) +  (σ -  2 ) ζ(s,8) - ζ(s,8)  .
(4.8)

From here it is clear that (4.3) holds.

The last assertion clearly follows from the explicit expressions for eta series obtained in (4.6), (4.7) and (4.8) since the Hurwitz zeta function ζ(s,α)  has a simple pole at s = 1  with residue 1 (see [Ap]). □

Corollary 4.2. The eta invariants of the spin ℤ4   -manifolds Mj,k ∈ Fn          1   with spin structures ɛσ   j,k   are given by

          {           ω(j)       j               σ (- 1)r+[-2-]2m- 1- [2]    k > 0, 1 ≤ j < 2r + 1  ηɛσj,k(0) =        r  r      r-1               (- 1) 2 (σ + 2   )       k = 0, j = 2r + 1.
(4.9)

where ω(j)  is as in (3.4)  . In particular ηɛσj,k(0) ∈ ℚ \{0} .

Proof. This is a consequence of the expressions given in Theorem 4.1 and the fact that ζ(0,α) =  1-  α            2  for every α ∈ (0,1]  . □

We now illustrate the results in the lowest dimensions considered, that is n = 3  and n =  7  .

Example 4.3. For n = 3  there is only one manifold in  3  F1   , namely M1,0   where        [ ˜ ]  B1,0 =  J 1 . Since r = 0  and k = 0  , by (4.3) we have

          -2--(    1        7 )                --2-(     3        5)  ηɛ+1,0(s) =  (8π)s ζ(s,8) - ζ(s,8) ,     ηɛ-1,0(s) = (8π)s ζ(s, 8) - ζ(s, 8)

and by (4.9)

ηɛ+ (0) = 32,         ηɛ- (0) = - 12.    1,0                   1,0

This is in agreement with the values obtained in [Pf].

Example 4.4. For n = 7  there are 3 manifolds in F71   . They are M1,4   , M2,2   and M3,0   where        [J˜      ]  B1,4 =    -I - I                 1 ,        [ ˜J      ]  B2,2 =     ˜J -I                 1 ,        [       ]           ˜J  B3,0 =    J˜˜              J 1 and        [     ]  - I =   -1 -1 . Now, by (4.2) and (4.3) we get

                 (                                     )    σ         --4σ  (    1        7 )   (    3        5 )  ηɛ1,4(s)  =   (8π)s   ζ(s,8) - ζ(s,8)  +  ζ(s,8) - ζ(s,8)  η σ (s)  =   --4σ (ζ(s, 1) - ζ(s, 3))   ɛ2,2        (8π)s (    4        4                        )  η + (s)  =   --2-  3(ζ(s, 1) - ζ(s, 7)) - (ζ(s, 3 ) - ζ(s, 5))   ɛ3,0        (8π)s        8        8          8         8              --2- (   (    1        7 )    (    3        5 ))  ηɛ-3,0(s)  =   (8π)s  -   ζ(s,8) - ζ(s,8)  - 3 ζ(s,8) - ζ(s,8)

and, again by (4.9), also

ηɛσ1,4(0) = - 4σ,    ηɛσ2,2(0) = - 2σ,     ηɛ+ (0) = - 4,     ηɛ- (0) = 3.                                           3,0                3,0

Remark 4.5. To conclude, we conjecture that for a compact flat manifold of dimension 4r + 3  , with a "nice" integral holonomy representation, the eta series η(s)  can be put in terms of differences of Riemann-Hurwitz zeta functions ζ(s,α)  , where α ∈ (0,1] ∩ ℚ  , and that the meromorphic continuation to ℂ  is holomorphic everywhere. Hence, from this expression, the η  -invariant is easily computed simply by evaluation at s = 0  . More precisely, we claim that the eta series has the expression

               ∑N          (                     )  ηΓ ,ɛ(s) = C-Γ ,ɛ-   fj,Γ ,ɛ(s) ζ(s,αj) - ζ(s,1 - αj)            (2 π)s j=1

where N <  ∣F ∣ , C Γ ,ɛ  is a constant depending on M  and on the spin structure ɛ  and each fj,Γ ,ɛ(s)  is an entire function (trigonometric or constant). The results in this paper, together with those in [MP2] bring support to this conjecture. We plan to get deeper into this question in the future.

References

[Ap]   Apostol T., Introduction to analytic number theory, Springer Verlag, NY, 1976.         [ Links ]

[APS]   Atiyah, M.F., Patodi V.K., Singer, I.M., Spectral asymmetry and Riemannian geometry, Bull. Lond. Math. Soc. 5, (229–234) 1973.         [ Links ]

[APS2]   Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry I, II, III, Math. Proc. Cambridge Philos. Soc. 77, (43–69) 1975, 78 (405–432) 1975, 79, (71–99) 1976.         [ Links ]

[Ch]   Charlap L., Bieberbach groups and flat manifolds, Springer Verlag, Universitext, 1988.         [ Links ]

[Fr]   Friedrich T., Dirac operator in Riemannian geometry, Amer. Math. Soc. GSM 25, 1997.         [ Links ]

[Gi]   Gilkey P., The Residue of the Local Eta Function at the Origin, Math. Ann. 240, (183–189) 1979.         [ Links ]

[Gi2]   Gilkey P., The Residue of the Global η  Function at the Origin, Adv. in Math. 40, (290–307) 1981.         [ Links ]

[LM]   Lawson H.B., Michelsohn M.L., Spin geometry, Princeton University Press, NJ, 1989.         [ Links ]

[MP]   Miatello R.J., Podestá R.A., Spin structures and spectra of Z2k-manifolds, Math. Zeitschrift 247, (319–335) 2004. arXiv:math.DG/0311354.         [ Links ]

[MP2]   Miatello R.J., Podestá R.A., The spectrum of twisted Dirac operators on compact flat manifolds, TAMS, to appear. arXiv:math.DG/0312004.         [ Links ]

[MR]   Miatello R., Rossetti J.P., Flat manifolds isospectral on p  -forms, Jour. Geom. Anal. 11, (647–665) 2001.         [ Links ]

[Pf]   Pfäffle F., The Dirac spectrum of Bieberbach manifolds, J. Geom. Phys. 35, (367–385) 2000.         [ Links ]

Ricardo A. Podestá
FaMAF–CIEM
Universidad Nacional de Córdoba
Córdoba, Argentina.
podesta@mate.uncor.edu

Recibido: 10 de agosto de 2004
Aceptado: 17 de junio de 2005

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