aa r X i v : . [ m a t h . DG ] M a y Eta invariants for flat manifolds
A. Szczepa´nski ∗ Institute of Mathematics, University of Gda´nskul. Wita Stwosza 57, 80-952 Gda´nsk, PolandE-mail: [email protected] 27, 2018
Abstract
Using a formula from H. Donnelly [5], we prove that for a familyof seven dimensional flat manifolds with cyclic holonomy groups the η invariant of the signature operator is an integer number. We alsopresent an infinite family of flat manifolds with integral η invariant.Our main motivation is a paper of D. D. Long and A. Reid ”On thegeometric boundaries of hyperbolic 4-manifolds”, see [9]. Key words. η invariant, flat manifold, cusp cross-section Mathematics Subject Classification : 58J28, 20H15, 53C25
Let M n be a closed Riemannian manifold of dimension n. We shall call M n flat if, at any point, the sectional curvature is equal to zero. Equivalently, M n is isometric to the orbit space R n / Γ , where Γ is a discrete, torsion-free and co-compact subgroup of O ( n ) ⋉ R n = Isom( R n ). From the Bieberbach theorem(see [4], [18], [19]) Γ defines the short exact sequence of groups0 → Z n → Γ p → G → , (1) ∗ The author was supported by Max Planck Institute in Bonn G is a finite group. Γ is called a Bieberbach group and G its holonomygroup. We can define a holonomy representation φ : G → GL ( n, Z ) by theformula: ∀ g ∈ G, φ ( g )( e i ) = ˜ ge i (˜ g ) − , (2)where e i ∈ Γ are generators of Z n for i = 1 , , ..., n, and ˜ g ∈ Γ such that p (˜ g ) = g. Our main motivation is the paper of D. D. Long and A. W. Reid [9].Using the methods from [2], the authors of [9] proved that an obstruction,for the flat 4 n − n -manifold, isthe non-integrality of the η invariant of the signature operator. They gave(see [9]) an example of a 3-dimensional flat manifold M with η ( M ) / ∈ Z , see Example 1.H. Donnelly in [5] formulated a general formula for the above η invariantfor some special class of flat manifolds. From (1) it is easy to see that anyflat manifold M n is diffeomorphic to T n /G, where T n = R n / Z n is the n -dimensional torus. Hence, we can say that a map T n → T n /G is regular covering of T n /G with covering group G . The above formulaexpresses the η invariant of the quotient space T n /G with the η invariantof T n , and some properties of the covering map (or a group action). Suchapproach was already considered in an original Atiyah, Patodi, Singer paper[2, pp. 408-413].Let T n − be any flat (4 n − S be the unit circle and G be a finite group which acts on S × T n − = T n − , such that T n − /G is anoriented flat manifold with holonomy group G. Let Γ = π ( T n − /G ) , g ∈ G and g = p (˜ g ) = ¯ A, where ˜ g = ( ¯ A, b ) ∈ Γ ⊂ SL (4 n − , Z ) ⋉ R n − . We assumethat g acts on T n − in the following way g ( x, ¯ x ) = ( x + a, A ¯ x + ¯ a ) , (3)where b = ( a, ¯ a ) ∈ S × T n − and A ∈ SL (4 n − , Z ) . Equivalently, itmeans that ¯ A = [ A ] . Here an element b ∈ R n − also denotes its image in R n − / Z n − . Since Γ is torsion free we can assume that 0 = a ∈ R / Z for g = 1 . Our main result (Theorem 1) is the following:2 f M is a seven dimensional, oriented flat manifold with cyclic holonomygroup, which satisfies condition (3), then η ( M ) ∈ Z . There exists a classification of flat manifolds up to dimension six, (see [13]). Itwas computed with the support of computer system CARAT. This algorithmalso gives a method for the classification of seven dimensional Bieberbachgroups with a cyclic holonomy group. It was done by R. Lutowski (see [10]and [11]). In the proof of the above result we were concentrated only onthose oriented flat manifolds which satisfy condition (3). Summing up, ifthere exists a seven dimensional flat manifold with the η invariant / ∈ Z , theneither it has a noncyclic holonomy group or it has a cyclic holonomy with aspecial holonomy representation.There is already some literature on the η invariant of flat manifolds. Forexample see [6],[12],[14],[16]. However, in all these articles the authors mainlyare concentrated on the η invariant of the Dirac operator.Let us present a structure of the paper. We prove Theorem 1 in section 3. Forthe proof, we use a generalized formula from H. Donnelly [5]. It is recalled insection 2, see Propositions 1, 2 and Remark 1. In the last section we presenttwo families of flat manifolds which exactly satisfy assumptions from [5]. Forthe first family of oriented n -dimensional flat manifolds with the holonomygroup ( Z ) n − (see [15]), we prove that the η invariant is always equal tozero. In the case of the second family we give an exact formula (17) for the η invariant. However, we do not know how to prove that the values of the η invariant are in Z . We do it only in a very special case.For the proof of the main result we use the computer package CARAT, see[13], [10] and [11]. We thank R. Lutowski for his assistance in the use ofCARAT and checking our calculations in the proof of Theorem 1. Moreover,the author would like to thank W. Miklaszewski, B. Putrycz for their help inthe use of ”MATHEMATICA” version 7 and M. Mroczkowski for improvingEnglish. Finally, we would like thank the referee(s) for a careful reading andmany constructive remarks. So called Hantzsche-Wendt manifolds. Donnelly’s formula
Since the result of H. Donnelly [5] is more then 30 years old let us recall it withsome details and comments. We keep the notations from the introduction.Let X n = X be a compact oriented Riemannian manifold of dimension 4 n with non-empty boundary Y n − = Y. Assume that the metric of X is aproduct near the boundary Y. Let Λ( Y ) be the exterior algebra of Y (see [1],[5]), and let B : Λ( Y ) → Λ( Y ) be a first order self-adjoint elliptic differentialoperator defined by Bφ = ( − n + p +1 ( ǫ ∗ d − d ∗ ) φ, where ∗ is the duality operator on Y and φ is either a 2 p -form ( ǫ = 1) or a(2 p − ǫ = − B preserves the parity of forms on Y and commuteswith φ ( − p ∗ φ, so that B = B ev ⊕ B odd and B ev is isomorphic to B odd .B ev has pure point spectrum consisting of eigenvalues λ with multiplicitydim( λ ). The spectral function η ( s, Y ) = Σ λ =0 (sign λ )(dim λ ) | λ | − s converges for Re( s ) sufficiently large and has a meromorphic continuation tothe entire complex s -plane. Moreover η (0 , Y ) is finite, see [1].Consider the finite group G acting isometrically on a manifold Y andsuppose g ∈ G. Then the map defined by g on sections of Λ ev ( Y ) commuteswith B ev . This induces linear maps g ∗ λ on each eigenspace, with eigenvalue λ, of B ev . The spectral function η g ( x, Y ) = Σ λ =0 (sign λ )Tr( g ∗ λ ) | λ | − s converges for Re( s ) sufficiently large and has a meromorphic continuationto the entire complex s -plane. Suppose that ˆ Y → Y is a regular coveringspace with finite covering group G of order | G | . For each irreducibleunitary representation α of G, one has an associated flat vector bundle E α → Y. The invariants η α (0 , Y ) are defined using the spectrum of the operator B evα : Λ ev ( Y ) ⊗ E α → Λ ev ( Y ) ⊗ E α . These invariants were studied in [2]. Inparticular η α (0 , Y ) = | G | Σ g ∈ G η g (0 , ˆ Y ) χ α ( g ) , (4)where χ α is the character of α. We use a special version of formula (4). If wetake α to be the trivial one-dimensional representation in (4) then we have η (0 , ˆ Y ) − | G | η (0 , Y ) = − Σ g =1 η g (0 , ˆ Y ) , (5)4here the sum on the right is taken over group elements g ∈ G, g = 1 . Moreover, in our case for ˆ Y = T n − we have η (0 , ˆ Y ) = 0 , cf. [2, p. 410].In [5] the following is proved: Proposition 1 ([5, Proposition 4.6])
Let g : T n − → T n − be given by theformula (3) with A ∈ SO (4 n − , Z ) and a = 0 . If A has as an eigenvalue,then η g (0 , T n − ) = 0 . Hence Proposition 1 reduces the problem of computing η g (0 , T n − ) only tothose isometries g satisfying det( I − A ) = 0 . For such g the following is provedin [5]: Proposition 2 ([5, Proposition 4.7])
Let g : T n − → T n − be an isometryof T n − which is given by formula (3) with A ∈ SO (4 n − , Z ) . It extendsto D × T n − , by rotation through the angle πa in the first factor and theextension has only isolated fixed points. Suppose that +1 is not an eigenvalueof A. The invariants η g (0 , T n − ) are given by η g (0 , T n − ) = ν ( g )( − n cot ( πa )Π n − i =1 cot ( γ i ) (6) where ν ( g ) is the number of fixed points of the extension of g : D × T n − → D × T n − and γ i , ≤ i ≤ n − , are the rotation angles of A ∈ SO (4 n − , Z ) . The invariants ν ( g ) and η g (0 , T n − ) are independent of the translation ¯ a ∈ R n − / Z n − in formula (3) . However, our main result depends from the following observation:
Corollary 1
Propositions 1 and 2 are true for A ∈ SL (4 n − , Z ) . Proof:
It is well known that any finite order, invertible, integral matrixis conjugate to an orthogonal matrix. From the third Bieberbach theorem(see [4, Th. 4.1, Chapter I], [18, Th. 2.1 (3)], [19, Th. 3.2.2]) we knowthat an abstract isomorphism between Bieberbach groups can be realized byconjugation within
Af f ( n ) = GL ( n, R ) ⋉ R n . Equivalently, it means thatflat manifolds with isomorphic fundamental groups are affine diffeomorphic.Then it is enough to use theorem 2.4 of [2] which says, that the signature η invariant is a diffeomorphism invariant. (cid:3) Main result
In this section, with the help of formulas (5) and (6), we prove that for allseven dimensional flat manifolds with cyclic holonomy groups, which satisfycondition (3) the η invariant is an integer number. We start with the 3-dimensional case. Example 1 (See also [9].) Let M be a 3-dimensional, oriented, flat mani-fold. There are only six such manifolds. The torus and manifolds M , M , M , M , M , with holonomy groups Z , Z , Z , Z , Z × Z correspondingly, see [18,Ch. III] or [19, Th. 3.5.5]. For holonomy groups Z and Z × Z the abovematrix A , has eigenvalues ±
1. Hence the η invariant is equal to zero. Inother words η ( M ) = η ( M ) = 0 . Let us consider the case of the holonomygroup Z . Here M = R / Γ , whereΓ = gen { g = ( B ′ , (1 / , , , ( I, (0 , , , ( I, (0 , , } ⊂ SL (3 , R ) ⋉ R , where B ′ = [ B ] with B = (cid:2) − − (cid:3) . (7)The action of an element ( A, a ) ∈ SL (3 , R ) ⋉ R is standard: for x ∈ R ( A, a ) x = Ax + a. We shall use formulas (5) and (6). It means that η ( M ) = − η g (0 , R / Z ) − η g (0 , R / Z ) , where Z ⊂ Γ is the subgroup oftranslations. It is easy to see, that the number of fixed points ν ( g ) in theformula (6) is equal to 3 . In fact, these points are solutions of the matrixequation BX = X, where X ∈ R / Z . It is easy to see that a solution isrepresented by a three elements, i.e. (1 / , / , (2 / , / , (0 , . Moreover,the rotation angle of the matrix B is equal to π and cot ( π/
3) = 1 / √ . Henceand from formula (6) η g (0 , R / Z ) = − ν ( g ) cot ( π/ cot ( π/
3) = − , and η g (0 , R / Z ) = − ν ( g ) cot (2 π/ cot (2 π/
3) = − . Finally η ( M ) = ( − − − . (8)A similar calculation for the 3-dimensional flat oriented manifolds with holon-omy Z and Z gives the following version of formula (8). For Z η ( M ) = − (2 cot ( π/
6) + 6 cot ( π/ − (6 + 2) = − Z η ( M ) = − ( cot ( π/
4) + cot (3 π/ − . Theorem 1
Let T be any flat six-dimensional torus, and let a cyclic group G act freely on the Riemannian product S × T such that ( x, ¯ x ) → ( x + a, A ¯ x + ¯ a ) , where A ∈ SL (6 , Z ) descends to T and ( a, ¯ a ) , ( x, ¯ x ) ∈ S × T . (The action of G satisfies condition (3).) Let M := ( T × S ) /G be endowedwith the induced flat metric. Then η ( M ) ∈ Z . Proof:
Let us first assume that the first Betti number b ( M ) ≥ . Thenfrom condition (3) it follows that the matrix A has an eigenvalue 1 , see[8]. Hence, from Proposition 1, η ( M ) = 0 . Further we shall assume that b ( M ) = 1 . From the crystallographic restriction [7, Proposition 2.1, p. 543], thefollowing numbers are possible for an order of the holonomy group of a sevendimensional flat manifold: 2 , , , , , , , , , , , , , , , . Weshall consider these numbers case by case. In each case we shall use thecomputer program CARAT, (see [13], [10] and [11]) to determine the numberof flat manifolds which satisfy our assumptions.For a holonomy group of order two, all eigenvalues of the matrix A are equal ± . Hence, from Propositions 1, 2 and 3 η ( M ) = 0 always.If a holonomy group is equal to Z and b ( M ) = 1 , we can assume (see [17,chapter 13]) that the holonomy representation (see (2)) is a direct sum (over Q ) of the trivial representation and 3-times the two-dimensional representa-tions, which we identify with matrices B or B from the above Example 1.Hence, we can apply formula (6). With similar calculations as in the aboveExample 1 we have ν ( g ) = ν ( g ) = 3 = 27 , where g ∈ π ( M ) and p ( g ) = 1 . ( p was defined on page 1.) Finally, η ( M ) = ν ( g ) cot ( π/
3) + ν ( g ) cot (2 π/
3) = ( + ) = 2 . Let M be a flat manifold of dimension 7 with holonomy group Z . From[11], up to affine diffeomorphism, there are thirteen such manifolds. For fiveof them ,the η invariant is equal to zero since the matrix A has an eigenvalue ± , see Proposition 3. Let us consider the case of the diagonal 6 × A = h B B
00 0 B i , where B = [ −
11 0 ] . Since the set of fixed points of the action7f B on S × S is equal to { (0 , , (1 / , / } , then ν ( A ) = 8 . Hence η ( M ) = ( cot ( π/
4) + cot (3 π/ . That is the only case of a flat manifold with holonomy group Z and non-zero η invariant, which we can calculate with our methods. In Example 2 wedefine a Bieberbach group with a cyclic holonomy group of order four, whichdoes not satisfy condition (3). There are seven such manifolds, see [11].For holonomy groups Z and Z it is enough to observe that any flat manifold M of dimension 7 , with such holonomy groups, either has the first Bettinumber greater then 1 or all matrices A k , k ∈ Z , have eigenvalues ± . Infact, it follows from the crystallographic restriction (see [7]) that any faithfulintegral representation of the group Z has dimension greater than 3 . Hence η ( M ) = 0 . We should add, that a seven dimensional flat manifold with Z holonomy group and the first Betti number equal to 1 does not exist, [11].However, there are three such manifolds with holonomy group Z . From [11] there are sixteen isomorphism classes of Bieberbach groups withholonomy group Z , which are the fundamental groups of flat 7-dimensionalmanifolds with the first Betti number 1 . All of them satisfy our condition (3).Since in eleven cases the matrix A has eigenvalues ± , then the η invariantis equal to 0 , cf. Proposition 3. Let us assume that A = h B B
00 0 B i , where B i = D = [ −
11 1 ] , for i = 1 , , . For A the number of fixed points is equal to1 , for A it is equal to 3 = 27 . Hence, the final formula is the following: η ( M ) = cot ( π/
6) + cot (2 π/ cot (4 π/
6) + cot (5 π/
6) = 4 . There are still four manifolds to consider. For the first one the matrix A hason the diagonal the matrices B = D, B = B = ¯ B = [ − −
11 0 ] and the η invariant is equal to 4 . For the second case the matrix A ∈ GL (6 , Z ) is notthe diagonal type but is conjugate in GL (7 , Q ) to the above matrix. Herethe η invariant is also equal to 4 . The last two cases are the following. The matrix A is either the diagonaltype B = B = D, B = ¯ B or is an integral matrix which is conjugateto A in GL ( n, Q ) . By similar calculation as above, the η invariant is equalcorrespondingly to 2 and 2 . Here and in what follows ν ( A ) = ν ( g ) , where g = ( A, a ) , see (3). Z the matrix A = −
11 0 0 0 0 −
10 1 0 0 0 −
10 0 1 0 0 −
10 0 0 1 0 −
10 0 0 0 1 − . Then ν ( A k ) = 7 andthe eigenvalues are equal to cos (2 kπ/
7) + isin (2 kπ/ , k = 1 , , , , , . Hence, η ( M ) = 2 cot ( π/ cot (2 π/ cot (3 π/ cot ( π/ cot (2 π/ − cot (3 π/ . For holonomy group Z we have eleven isomorphism classes of Bieberbachgroups with the first Betti number one. Six of them do not satisfy condition(3). For two of them, the η invariant is equal to zero, since the matrix A haseigenvalues ± . The last three manifolds have the η invariant equal to 2 . For instance, let us present the calculation for the following matrix:¯ A = − − . We have ν (( ¯ A ) k ) = 4 for k = 1 , , , η ( M ) = cot ( π/ cot (3 π/ cot ( π/ − cot (3 π/ . For a cyclic group of order 9 the matrix A = −
11 0 0 0 0 00 1 0 0 0 00 0 1 0 0 −
10 0 0 1 0 00 0 0 0 1 0 . The characteristic polynomial of A is equal to x + x +1 . Moreover ν ( A k ) = 3for k = 1 , , , , , ν ( A k ) = 27 for k = 3 , . Hence η ( M ) = − cot ( π/ cot (2 π/ cot (4 π/
9) + cot (2 π/ cot ( π/ cot (4 π/ cot ( π/ − cot (4 π/ cot ( π/ cot (2 π/
9) = ++ cot (2 π/ cot ( π/ cot (4 π/ cot (2 π/ − cot ( π/ − cot (4 π/ − = 0 . The computations of the trigonometric sums were done with the aid of a computerand MATHEMATICA version 7. Z we have twenty nine manifolds, see [11]. This is themost non-standard case. In seven cases the η invariant is zero, because thematrix A has an eigenvalue ± . Moreover, condition (3) is not satisfied inten cases. Let us assume, that the matrix A = h C C
00 0 C i , where C i = E =[ −
11 0 ] , C i = F = (cid:2) − − (cid:3) or C i = G = [ − ] where i = 1 , , . For example, weshall consider (compare with the case n = 6) : C = F, C = C = E, and C = C = F, C = E. It is easy to see, that it is enough to consider in theformula (5) only
A, A , A and A . In fact, in all other cases eigenvaluesare equal to ± η ( M ) = cot ( π/ cot ( π/ cot ( π/
4) + cot (7 π/ cot ( π/ cot ( π/ cot (2 π/ cot (5 π/ cot ( π/
4) + cot (11 π/ cot (2 π/ cot ( π/
4) = 4 , or η ( M ) = ( cot ( π/ cot ( π/ cot ( π/ − cot (7 π/ cot ( π/ cot ( π/ cot (2 π/ cot ( π/ cot (5 π/ − cot (11 π/ cot (2 π/ cot ( π/ ( cot ( π/
12) + cot (5 π/
12) + cot (5 π/
12) + cot ( π/
12) = 4 . From [11], we know that there are six such flat manifolds. For all, the η invariant is equal to 4 . There is still another possibilty: the matrix A = [ D F ] , where D = (cid:20) − − − (cid:21) is the faithful, irreducible rational representation of the group Z , see [3, p.234], [13]. Moreover F = [ − ] k or F = [ −
11 0 ] , k = 1 , . From the classification (see [11]) there are six such flat manifolds and the η invariants are equal 0 in 4 cases and are equal 2 in 2 cases. Here, weshould mention that one manifold with the η invariant equal to zero is alsoconsidered at the end of the paper, see formula (18).For holonomy group Z , A = − − − − − . A is equal to x +1 x +1 . Moreover ν ( A k ) = (cid:26) k = 1 , , , , ,
137 for k = 2 , , , , , η ( M ) = cot ( π/ cot (3 π/ cot (5 π/ cot ( π/ − cot (3 π/ − cot (5 π/ cot ( π/ cot (3 π/ cot (2 π/ − cot ( π/ − cot (2 π/
7) + cot (3 π/ . There is only one flat manifold of this kind, see [11].For cyclic group of order 15 ,A = − − −
10 0 1 0 0 −
10 0 0 1 0 −
10 0 0 0 1 − and ν ( A k ) = 15 for k = 1 , , , , , , , . Moreover η g k (0 , T ) = 0 , for k = 3 , , , , , , where g ∈ SO (7) ⋉ R denotes an isometry defined bythe formula g (( x , x , ..., x )) = ( A ( x , x , ..., x ) , x ) + (0 , , ..., , / . Summing up, we get η ( M ) = 2 cot ( π/ cot ( π/ cot (2 π/ cot ( π/
15) + cot (2 π/ cot (4 π/ − cot (7 π/ . For holonomy group Z , A = − − − − − . We have ν ( A k ) = k = 1 , , , , , , , ,
173 for k = 2 , , , , , k = 6 , η ( M ) = cot ( π/ cot (7 π/ cot (5 π/ − cot ( π/ cot (5 π/ − cot (7 π/ cot ( π/ cot (2 π/ cot (4 π/ − cot ( π/
9) + cot (2 π/ − cot (4 π/ cot ( π/
6) + 3 cot ( π/
3) = − − + 1 + = 0 . In the cases above, for a holonomy group of order 2, 3, 7, 9, 14, 15 and 18there exists only one flat manifold with the first Betti number one. Let usconsider the last three cases of cyclic groups of order 20 ,
24 and 30 . We startwith the matrix A = − − − − −
10 0 0 0 1 0 of order 20. From previous facts we have only to consider the followingelements: A k , k ∈ { , , , , , , , } = S. For all other the η invariantis equal to zero. Finally, we have η ( M ) = Σ k ∈ S ( cot ( kπ/ cot ( kπ/ cot (2 kπ/ cot ( kπ/ cot ( π/ cot (2 π/ cot ( π/
20) + cot (3 π/
20) + cot (7 π/
20) + cot (9 π/ . From [11], there is another flat manifold with holonomy group Z , and inthis case the η invariant is also 4 . The cyclic group of order 24 is isomorphicto Z × Z or to Z × Z and there are only two such flat manifolds, see [11].For the first group the matrix A = − −
10 0 0 0 1 − and has order 24 . As in the last case we have only to consider the elements A k for k ∈ { , , , , , , , , , , , } = T. Moreover ν ( A k ) = (cid:26) k = 1 , , , , , , ,
233 for k = 2 , , , η ( M ) = Σ k ∈{ , , , , , , , } cot ( kπ/ cot ( kπ/ cot (3 kπ/ cot ( kπ/ k ∈{ , , , } cot ( kπ/ cot ( kπ/ cot (3 kπ/ cot ( kπ/
3) = 4 . For the second group the matrix A is almost the same as the matrix above.We only put in the right-down corner, the (2 ×
2) integral matrix of order 6in place of the matrix of order 3 . The η invariant is equal to 0 . In the lastcase of cyclic group of order 30 there are three manifolds with the followingmatrices A = − − − − − , A = − − − − −
10 0 0 0 1 0 and A = − − − − . We shall present calculatins for A , the other cases are similar. As in thecases above, we have only to consider elements A k , k ∈ { , , , , , , , , , , , , , , , , } = R. We have ν ( A k ) = (cid:26) k = 1 , , , , , , k = 2 , , , , , , , η ( M ) = Σ k ∈{ , , , , , , } cot ( kπ/ cot (2 kπ/ cot (4 kπ/ cot ( kπ/ k ∈{ , , , , , , , } cot ( kπ/ cot (2 kπ/ cot (4 kπ/ cot ( kπ/
6) = 0 . For the other two manifolds with holonomy group of order 30 , the η invariantis equal to 4 and 0 . (cid:3) Let us present a final table. 13olonomy number of number number of cases not calculated values of η manifolds with b = 1 all b = 1 Z
15 1 0 0 0 Z Z
87 13 37 7 0, 4 Z Z
74 16 0 0 0, 2, 4 Z Z
24 11 10 6 0, 2 Z Z
12 3 0 0 0 Z
89 29 27 10 0, 2, 4 Z Z Z Z Z Z Example 2
Let R / Γ be the seven dimensional flat oriented manifold withthe fundamental group Γ ⊂ SO (7) ⋉ R generated by { ( A, (1 / , , ..., , ( I, e i ) } , where e i , i = 1 , , ..., R and A = − . It is easy to see that the manifold R / Γ has holonomy Z and does not satisfythe condition (3). SO ( n, Z ) This part is a modified and refreshed version of some results of [5]. Weshall present the formula (6) under the assumption that the image of the14olonomy representation (see (2)) is a subgroup of SO ( n, Z ) . In this case(see [5]) a method is given for finding the number ν ( g ) of fixed points of g, where g is an isometry of the torus.Let e i , ≤ i ≤ n − R n − . Since A ∈ SO (4 n − , Z ) one has A ( e i ) = ± e j ( A,i ) , i = j ( A, i ) for each i. We denote by σ ( A ) the element of SO (4 n − , Z ) defined by σ ( A )( e i ) = e j ( A,i ) . Then σ ( A )is a permutation matrix and we may decompose σ ( A ) = σ ( A ) σ ( A ) ...σ l ( A )into disjoint cycles. Summing up we have: Proposition 3 ([5, Proposition 4.9])
Let g : T n − → T n − be an isometrygiven by the formula (3) with a = 1 . Then η g (0 , T n − ) = 0 if A has aneigenvalue equal to +1 or − . Otherwise η g (0 , T n − ) = 2 l ( − n cot ( πa )Π n − i =1 cot ( γ i where(i) The angles γ i are the rotation angles of the orthogonal matrix A. (ii) The integer l is the number of distinct cycles in the decomposition ofthe permutation matrix σ ( A ) = σ ( A ) σ ( A ) ....σ l ( A ) , corresponding to A. (cid:3) As an immediate corollary we have a special version of the formula (4).
Proposition 4 ([5, Proposition 4.12])
We keep the above notations. Thecorresponding eta invariant is given by η (0 , Y ) = | G | Σ ′ (2 l ( − n cot ( πa )Π n − i =1 cot ( γ i )) (10) where the symbol Σ ′ means summation over the group elements g whose as-sociated A has no eigenvalue equal to ± . Furthermore γ i are the rotationsangles of A ∈ SO (4 n − , Z ) , and l is the number of distinct cycles of A asabove. Let us present two families of flat manifolds which satisfy the above assump-tions. 15 xample 3
1. Let M n be an oriented flat manifold with holonomy group( Z ) n − , so called Hantzsche-Wendt manifold. From [15] it follows that n is an odd number and the image of the holonomy representation of π ( M n )(see (2)) in GL ( n, Z ) consists of diagonal matrices with ± η ( M n ) = 0 . We should add, that the same is truefor any oriented flat manifold of dimension n with holonomy group ( Z ) k , ≤ k ≤ n − Example 4
Let us recall the fundamental group Γ of a flat manifold ofdimension 4 n − Z n − . From Bieberbach theorems(see [18]) Γ defines the short exact sequence of groups.0 → Z n − → Γ → Z n − → . (11)Moreover Γ ⊂ SO (4 n − ⋉ R n − , is generated by the translation subgroupΓ ∩ { I } × R n − ≃ Z n − and an element ( A ′ , (0 , ..., , n − )) . Here A ′ = [ A
00 1 ] (12)and A = " ... −
11 0 ... ... ........... ... (13)is (4 n − × (4 n −
2) integral orthogonal matrix of order 2(4 n − . Usingthe formula (10) from Proposition 4 for a flat manifold R n − / Γ = M n − wehave η ( M n − ) = ( − n n − Σ n − k =1 l k (cot( kπ n −
2) )Π n − l =1 cot( k (2 l − π n −
2) )) , (14)where l k is the number of distinct cycles of A k , see Proposition 3. Thecharacteristic polynomial of the matrix A is equal to det( A − ( λ ) I ) = λ n − +1 . Hence, it is easy to calculate that A n − = − I, and ± A . It is to see that, for 1 ≤ m < n and k = 2 r +1 cot ( k ( m − π n − ) = cot ( kπ − k (2 m − π n − ) = tg ( k (2 m − π n − ) (15)and cot ( k (2 n − π n − ) = cot ( kπ ) = ( − r . (16)16sing elementary formulas cot ( π − α ) = tg ( α ) , cot ( π − α ) = − ctg ( α )and the above equations (15), (16) we obtain η ( M n − ) = ( − n n − Σ n − r =0 l r ( − r cot( (2 r +1) π n − ) . (17)Here l r is the number of distinct cycles of A (2 r +1) . For n = 2 , the components of the above formula are non zero only for k =1 , , , , , . It is easy to see that l = l = l = l and l = l = 3 . Summingup η ( M ) = cot ( π ) − + tg ( π ) + tg ( π ) − + (18)+ cot ( π ) = ( cot ( π ) + tg ( π )) − = − = 0 ∈ Z . References [1] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Rie-mannian geometry I, Math. Proc. Camb. Phil. Soc., 77, (1975), 43-69.[2] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Rie-mannian geometry II, Math. Proc. Camb. Phil. Soc., 78, (1975), 405-432.[3] H. Brown, R. B¨ulow, J. Neub¨user, W. Wondratschek and H. Zassenhaus,Crystallographic groups of four-dimensional space, New York, 1978.[4] L. S. Charlap, Bieberbach Groups and Flat Manifolds, Universitext,Springer-Verlag, New York, 1986.[5] H. Donnelly, Eta Invariants for G -Spaces, Indiana Univ. Math. Journal,27 (6), 1978, 889-918.[6] P. B. Gilkey, R. J. Miatello, R. A. Podest´a. The Eta Invariant andEquivariant Bordism of Flat Manifolds with Cyclic Holonomy Group ofOdd Prime Order, Ann. Global Anal. Geom. 37 (2010), no 3, 275-306.[7] H. Hiller, The Crystallographic Restriction in Higher Dimensions. Acta.Cryst. (1985), A41, 541-544. 178] H. Hiller, C. H. Sah, Holonomy of flat manifolds with b = 0 . Quart. J.Math. Oxford Ser. (2), 37 (1986), 177-187.[9] D. D. Long, A. W. Reid, On the geometric boundaries of hyperbolic4-manifolds, Geometric Topology, 4, (2000), 171-178.[10] R. Lutowski, Seven dimensional flat manifolds with cyclic holonomygroup, preprint, Gda´nsk, 2010, arXiv:1101.2633.[11] R. Lutowski, A list of 7-dimensional Bieberbach groups with cyclicholonomy, avaible online , http://rlutowsk.mat.ug.edu.pl/flat7cyclic/.[12] R. J. Miatello and R. A. Podest´a, The spectrum of twisted Dirac oper-ators on compact flat manifolds, Trans. Amer. Math. Soc., 358 (2006),10, 4569-4603[13] J. Opgenorth, W. Plesken, T. Schulz,
CARAT, Crystallographic Algo-rithms and Tables