the gromov-lawson-rosenberg conjecture 375 M. In particular, the kernel of D(M) is a Z=2-graded module over C`n. The element represented by ker D(M) i

Transcription

1 j. differential geometry 45 (1997) THE GROMOV-LAWSON-ROSENBERG CONJECTURE FOR GROUPS WITH PERIODIC COHOMOLOGY BORIS BOTVINNIK, PETER GILKEY & STEPHAN STOLZ 1. Introduction By a well-known result of Lichnerowicz [18], there are manifolds which do not admit positive scalar curvature metrics. Lichnerowicz observes that the existence of such a metric on a closed spin manifold M of dimension n congruent to 0 mod 4 implies that the (chiral) Dirac operator D + (M) (cf. x4) is invertible (cf. [17, Ch. II, Thm. 8.8]). In particular, index (D + (M)) = dim ker D + (M) ; dim coker D + (M) vanishes. Note that index (D + (M)), unlike the dimension of the kernel and the dimension of the cokernel of D + (M), is independent of the metric used in the construction of D + (M). In fact, according to the Atiyah-Singer Index Theorem, it is equal to a topological invariant ^A(M), the ^A-genus of M (cf. [17, Ch. III, Thm ]). We recall that ^A(M) isacharacteristic number dened by evaluating a certain polynomial in the Pontrjagin classes of the tangent bundle on the fundamental class of M. Lichnerowicz' result was generalized by Hitchin [12], who constructs aversion of the Dirac operator D(M) which is selfadjoint and commutes with an action of the Cliord algebra C`n, wheren is the dimension of Received August 31, 1995, and, in revised form, December 4, The second author was partially supported by NSF grant DMS , IHES (France) and MPIM (Germany), and the third author by NSF grant DMS

2 the gromov-lawson-rosenberg conjecture 375 M. In particular, the kernel of D(M) is a Z=2-graded module over C`n. The element represented by ker D(M) in the Grothendieck group Z(C`n) of nitely generated Z=2-graded modules over C`n in general will depend on the metric. However, it can be shown that the class [ker D(M)] 2 Z(C`n)=i Z(C`n+1 ) is independent of the metric, where i is the map induced by the inclusion i: C`n! C`n+1 (cf. [17, Ch. III, Prop. 10.6]). This class is known as the Cliord index of D(M). We note that the cokernel of i is isomorphic to KO(S n ), the real K-theory of the n-sphere by a result of Atiyah-Bott-Shapiro (cf. [17, Ch. I, Thm. 9.27]). If M has nite fundamental group, the manifold M in the discussion above can be replaced by its universal covering f M. The -action on fm by deck transformations induces a -action on the kernel of the Dirac operator D(f M). This action commutes with the C`n-module structure, and hence ker D(f M) is a module over C`n R, where R is the real group ring of. As in the case = f1g discussed above, the element (1:1) (M) :=[ker D(f M)] 2Z(C`n R)=i Z(C`n+1 R) is independent of the metric. The quotient of these Grothendieck groups is isomorphic to KO n (R), the real K-theory of R (more or less by the denition of these K-theory groups). Let be an arbitrary fundamental group. Rosenberg constructs an invariant (M) in KO n (C r ), where C r is a suitable completion of R, known as the (reduced) group C -algebra C r. This invariant can be thought of as the `equivariant Cliord index' of the Dirac operator on the universal covering of M and agrees with (1.1) for nite groups. We refer to [21], [22], [23] for details. We hope the discussion above makes it clear that the existence of a positive scalar curvature metric on M implies the vanishing of (M). Modifying a conjecture of Gromov and Lawson, Rosenberg conjectures that the converse holds as well: Conjecture 1.1 (Gromov-Lawson-Rosenberg Conjecture). A closed connected smooth spin manifold M of dimension n 5 with fundamental group admits a positive scalar curvature metric if and only if (M) vanishes. This conjecture has been proved for simply connected manifolds [28]. It has been proved for manifolds whose fundamental group is odd order cyclic [23], [15], is of order 2 [24], or which is one of a (short) list of in-

3 376 boris botvinnik, peter gilkey & stephan stolz nite groups, including free groups, free abelian groups, and fundamental groups of orientable surfaces [25]. The main result of this paper is the following theorem. Theorem 1.2. The Gromov-Lawson-Rosenberg Conjecture is true for nite groups with periodic cohomology. We remark that by a theorem of Kwasik-Schultz [15] the Gromov- Lawson-Rosenberg Conjecture is true for a nite group if and only if it is true for all Sylow subgroups of. It is well known that nite groups with periodic cohomology are precisely those nite groups whose p-sylow subgroups are cyclic for odd p and cyclic or (generalized) quaternion groups for p =2. So the new result is that the conjecture is true for cyclic groups of order 2 k for k 2 and for generalized quaternion groups. We recall that a smooth space form is a manifold whose universal covering is dieomorphic to the sphere S n. The fundamental group of a space form has periodic cohomology, and space forms are spin manifolds for n 3 mod 4. Using the fact that KO n (Cr ) vanishes for nite groups and n 3 mod 8, Theorem 1.2 implies in particular that every smooth spherical space form of dimension n 3 mod 4 admits a positive scalar curvature metric, a result due to Kwasik and Schultz [16]. In Section 2, we outline the proof of Theorem 1.2 and explain the role of the remaining sections. 2. Outline of the proof Let M be a spin manifold of dimension n (all manifolds considered in this paper are smooth and compact their boundary is empty unless mentioned otherwise). In the introduction, we described the invariant (M), the `equivariant Cliord index' of the Dirac operator on the universal covering of M, which lives in KO n (Cr 1(M)). Generalizing slightly, if f : M! B is a map to the classifying space of a discrete group, we dene (M f) 2 KO n (Cr ) to be the -equivariant Clifford index of the Dirac operator on f M, where f M! M is the pull back via f of the covering E! B. It turns out that (M f) depends only on the bordism class (M f) represents in the spin bordism group spin n (B). Hence the -invariant induces a homomorphism : spin n (B)! KO n (Cr ). This homomorphism can be factored as

4 the gromov-lawson-rosenberg conjecture 377 follows [30] (2.1) = A p D : spin n (B)! ko n (B)! KO n (B)! KO n (Cr ): Here ko n (B) (resp. KO n (B)) is the connective (resp. periodic) real K-homology of B, D and p are natural transformations between these generalized homology theories, and A is known as the assembly map. The proof of Theorem 1.2 will be based on the following result. Let spin + n (B) be the subgroup consisting of bordism classes represented by pairs (N f) for which N admits a positive scalar curvature metric, and let ko + n (B) = D( spin + n (B)). See Jung [13] and Stolz [29], cf. [30, Thm 4.5] for the following Theorem 2.1. Let M be a spin manifold of dimension n 5 with fundamental group, andlet u: M! B be the classifying map of the universal covering. Then M has a positive scalar curvature metric if and only if D[M u] is in ko + n (B). Corollary 2.2. The Gromov-Lawson-Rosenberg conjecture is true for a group if and only if we have ko + n (B)=Y n (B) for n 5, where Y n (B) denotes the kernel of the map A p: ko n (B)! KO n (C r ). We note that the vanishing of the invariant for manifolds with positive scalar curvature metrics implies the inclusion ko + n (B) Y n (B). To prove the converse inclusion when is a cyclic group C l or a quaternion group Q l of order l = 2 k, we consider lens spaces and lens space bundles over S 2 (in the case = C l ), respectively lens spaces and quaternionic lens spaces (for = Q l ). We furnish these manifolds with spin structures and maps to B, and consider the spin -submodule of M (B) espin (B) generated by their bordism classes. These manifolds admit metrics of positive scalar curvature, and hence we have the following chain of inclusions: (2.2) D(M n (B)) f ko + n (B) e Yn (B) where e Yn (B) denotes the kernel of A p restricted to the reduced kohomology group f kon (B), and f ko + n (B) is the intersection of f kon (B) with ko + n (B). Theorem 2.3. Let be a cyclic or (generalized) quaternion group whose order is a power of 2. Then D(M (B)) = e Y (B).

5 378 boris botvinnik, peter gilkey & stephan stolz To prove this result, we basically compare the order of the group fko n (B) and the order of the subgroup D(M n (B)). For this calculation, we need invariants that allow us to identify the elements represented by lens spaces, resp. lens space bundles, resp. quaternionic lens spaces in f ko (B). It turns out that the eta invariant provides the desired invariant. We recall that the eta invariant (D) of a selfadjoint elliptic operator D is a real number which measures the asymmetry of the spectrum of D with respect to the origin (navely, it is the number of non-negative eigenvalues of D minus the number of negative eigenvalues). We recall that a nite dimensional complex representation of a discrete group determines a vector bundle V over the classifying space B. Given a spin manifold M of odd dimension n, and a map f : M! B, wedenoteby D(M f ) the Dirac operator on M twisted by the at vector bundle f V, and by (M f)() 2 R the eta invariant of D(M f ). Since the eta invariant is additive with respect to the direct sum of representations, we may extend (M f)() to virtual representations. In Section 4 we show that the eta invariant induces a homomorphism (2.3) (): KO n (B)! R=Z: If n 3 mod 8 and is of real type (i.e., if is the complexication of a real representation), or if n 7 mod 8 and is of quaternionic type (i.e., if is obtained from a representation of on a quaternionic vector space by restricting the scalars), we still get a well dened map if we replace the range of (2.3) by R=2Z. Choosing a basis for the free Z-module of -representations of virtual dimension 0 appropriately (keeping track of the type of these representations), and combining the eta invariants corresponding to these representations in a single map, we get a homomorphism (2:4) : KO n (B)! R=Z R=Z R=2Z R=2Z: The proof Theorem 2.3 is based on comparing the order of the group fko n (B) with the order of p(e Y(B)): Theorem 2.4. Let be the cyclic group C l or the (generalized) quaternion group Q l. Let n be a positive integer, and assume n 3 mod 4 for = Q l. Then the order of f kon (B) is (n ) where wedene

6 the gromov-lawson-rosenberg conjecture 379 n 8j +1 8j +2 8j +3 8j +5 8j +7 (n C l ) 2(l=2) 2j+1 2 2(2l) 2j+1 (l=2) 2j+2 (2l) 2j+2 (n Q l )?? 2 4j+4 l 2j+1? 2 4j+4 l 2j+2 and we set (n C l )=1and (n Q l )=? otherwise. We prove this result in Section 3 using the Atiyah-Hirzebruch spectral sequence. For the quaternion group, we only determine the order of f kon (BQ l ) for n 3 mod 4, since the calculation in the other cases would involve determining some higher dierentials. This can be done, but is not done in this paper, since the above statement turns out to suce for our purposes. Theorem 2.5. If is the cyclic group C l and n odd, n 6 1 mod 8, or if is the (generalized) quaternion group Q l and n 3 mod 4, then the order of p(d(m n (B))) is greater or equal to (n ). For n 1 mod 8, the order of p(d(m n (C l ))) is greater or equal to 1 (n ), 2 and the order of p(f kon (BC l )) is greater or equal to (n ). This Theorem is proved by explicit eta-calculations in Section 5 (for = C l ) and in Section 6 ( for = Q l ), respectively. We note that taken together, Theorems 2.4 and 2.5 imply the following result, which might be of independent interest. Corollary 2.6. The composition p : f kon (B)! g KOn (B)! R=Z R=Z R=2Z R=2Z is injective if is the cyclic group and n is odd, or if is the (generalized) quaternion group and n 3 mod 4. Also, Theorems 2.4 and 2.5 imply Theorem 2.3 for = C l, n mod 8, and for = Q l, n 3 mod 4. In fact, they yield the stronger statement D(M n (B)) = f kon (B) in these cases. In particular, the map A p : f kon (B)! g KOn (B)! KO n (C r ) is trivial in these cases. To settle the remaining cases, we need some general facts about the assembly map A: KO (B)! KO (R) for nite 2-groups which are proved in [26] by dualizing the well-known result of Atiyah-Segal concerning the real K-cohomology of B. The group g KOn (B) is a direct sum of copies of Z=2 and Z=2 1. The number of summands in each dimension n is determined by the irreducible complex representations

7 380 boris botvinnik, peter gilkey & stephan stolz of as follows. Each irreducible representation of complex type (i.e., a representation not isomorphic to its complex conjugate) contributes a copy of Z=2 1 for n odd. A non-trivial irreducible representation of real type contributes a Z=2 1 in dimensions n 3 mod 4 and a Z=2 indimensions n 1 2 mod 8. Each irreducible representation of quaternionic type contributes a Z=2 1 in dimensions n 3 mod 4 and a Z=2 in dimensions n 5 6mod8. We will need the following fact see [26]. Proposition 2.7. The assembly map A: g KO (B)! KO (C r ) for a nite 2-group is injective on the Z=2-summands. Lemma 2.8. For n 1 2 mod 8, n 1, there are elements in fko n (BC l ), which map to indivisible elements in g KOn (BC l ). This shows that for n 1 2 mod 8, n 1 the kernel of A p from fko n (BC l )toko n (Cr C l ) is a subgroup of index at least 2. This implies Theorem 2.3 in these cases. To prove the result for = Q l, n 6 3 mod 4, we recall that all the irreducible complex representations of the quaternion group Q l, l =2 k, are all of real or quaternionic type. Hence the assembly map for Q l is injective in degrees n 6 3 mod 4, which implies that Yn e (BQ l ) = ker p: kon f (BQ l )! KOn g (BQ l ) for n 6 3 mod 4. Theorem 2.3 for = Q l in degrees n 6 3 mod 4 follows from the next lemma. Lemma 2.9. The kernel of p: f kon (BQ l )! g KOn (BQ l ) is trivial for n 2 mod 4 for n 0 mod 4, ker p is equal to the image of the map : f kon;1 (BQ l )! f kon (BQ l ) given by multiplication by the generator 2 ko 1 = Z=2 forn 1 mod 4, ker p is equal to the image of the map f kon;2 (BQ l )! f kon (BQ l ) given by multiplication by 2 2 ko 2 = Z=2. 3. ko-homology calculations The goal of this section is to provide the necessary information about the connective KO-homology of the classifying spaces of cyclic and quaternion groups. We do not attempt to calculate these groups, but rather content ourselves with extracting all the information necessary for the proof of Theorem 2.3. The method we use is the Atiyah-Hirzebruch

8 the gromov-lawson-rosenberg conjecture 381 spectral sequence, and for the convenience of the reader we recall its basic properties. Atiyah-Hirzebruch spectral sequence. Let X be a CWcomplex. The Atiyah-Hirzebruch spectral sequence converging to ko f (X) has the following form: (3.1) E 2 p q(x) = e Hp (X ko q )=) f kop+q (X): We recall, for q 0, that ko q = Z if q 0 mod 4, that ko q = Z=2 if q 1 2 mod 8, and that ko q = 0 otherwise. It is well-known (cf. [1, Proof of Lemma 5.6] for the dual statement concerning KO-cohomology) that for q 1 mod 8 the dierential (3.2) d 2 : E 2 p q(x) = e Hp (X Z=2)! E 2 p;2 q+1(x) = e Hp;2 (X Z=2) is the map dual to the cohomology operation Sq 2 : H p;2 (X Z=2)! H p (X Z=2): Similarly, for q 0 mod 8, the dierential (3.3) d 2 : E 2 p q(x) = e Hp (X Z)! E 2 p;2 q+1(x) = e Hp;2 (X Z=2) is mod 2 reduction composed with the homomorphism dual to Sq 2. Multiplicative Structure. We remark that E r (X) = M p q E r p q(x) is a graded module over the graded ring ko = M n ko n = Z[! ]=( 3 2!! 2 ; 4) where,!, are elementsofdegree1,4,8respectively. On E 2 (X) = e H (X ko ) this module structure comes from the action of ko on the coecients. On the E 1 -term, the module structure is induced from the ko -action on f ko (X). These module structures are compatible with the dierentials (i.e., d r (xy) = xd r (y) for x 2 ko and y 2 E r ), and the module structure on E r induces the module structure on E r+1.

9 382 boris botvinnik, peter gilkey & stephan stolz q p Figure 1. e Hp (BC l ko q )=) f kop+q (BC l ) Homology of BC l. Let l be a power of 2. Then e Hp (BC l Z)=Z=l for p odd, e Hp (BC l Z) =0for p even, and e Hp (BC l Z=2) = Z=2 for all p 1. We claim that the homomorphism (3.4) eh p (BC l Z=2)! e Hp;2 (BC l Z=2) dual to Sq 2 is non-trivial if and only if p 0 1mod4,p 4. To see this we recall that H (BC l Z=2) is the polynomial algebra Z=2[x] for l = 2, and the tensor product (x) Z=2[y] of an exterior algebra (x) and a polynomial algebra for l = 2 k, k > 1. Here x is a 1-dimensional generator, and y is a 2-dimensional generator. Using Sq 1 (y) = 0 and the Cartan formula, we conclude that Sq 2 : H p;2 (BC l Z=2)! H p (BC l Z=2) is non-trivial if and only if p 0 1mod4,p 4. This homology information leads to the following picture of the Atiyah-Hirzebruch spectral sequence converging to f ko (BC l ) see Figure 1. Here a black dot at the location (p q) means that E 2 p q is isomorphic to Z=l, while a white dot means E 2 p q = Z=2. The arrows in the picture represent non-trivial dierentials originating and terminating in the range shown. The d 2 -dierentials are determined as explained above, for the d 3 -dierentials we use the following argument.

10 the gromov-lawson-rosenberg conjecture 383 d 3 -dierentials. We observe that for dimensional reasons the only possibly non-trivial d 3 -dierentials are (3.5) d 3 : E 3 p q(bc 2 k)! E 3 p;3 q+2(bc 2 k) for p 0 mod 4, p 4, q 2 mod 8, and we claim that these are in fact non-trivial. For k =1,p =4,q = 2, the dierential (3.5) has to be non-zero, since otherwise the non-trivial element in E would survive to the E 1 -term contradicting ko6 f (BC 2 )=0(cf. [19, Lemma 7.3]). To settle the dierentials for k =1,p =4m+4, and q =2,we replace BC 2 = RP 1 by the quotient RP 1 =RP 4m;1, which does not aect the dierential (3.5). We note that this quotient is homeomorphic to the Thom space of the Whitney sum of 4m copies of the Hopf line bundle over RP 1. This is a spin bundle which implies a homotopy equivalence between ko ^ RP 1 =RP 4m;1 and ko ^ 4m RP 1 +. It follows that the dierential (3.5) can be identied with the dierential d 3 : E 3 (BC 4 2 2)! E 3 (BC 1 4 2), and hence is non-trivial. Since the dierentials in the Atiyah-Hirzebruch spectral sequence are compatible with multiplication by elements in ko, this implies that the dierential (3.5) is non-trivial for k =1,p 0mod4,q2mod4. Finally, to prove that the dierential (3.5) is non-trivial for any k, we consider the map f : BC 2! BC 2 k induced by the inclusion C 2! C 2 k. In even dimensions, the induced map f : H (BC 2 Z=2)! H (BC 2 k Z=2) is an isomorphism in odd dimensions f : H (BC 2 Z)! H (BC 2 k Z) is non-trivial. This yields by naturality of the spectral sequence that the dierential (3.5) is non-trivial for all k. Proof of Theorem 2.4 for cyclic groups. Inspection of the picture shows Ep q 4 is trivial for p + q even, except for the term E1 q 4 = Z=2 with q 1 mod 8. Hence all dierentials d r are trivial for r > 3 for dimensional reasons except dierentials with range E1 q r, q 1 mod 8. However, multiplication by acts trivially on Ep q, r r > 3, p > 1, but non-trivially on E1 q r. Hence the multiplicative structure as dened in x3 forces these dierentials to be zero. Hence the order of the group kon f (BC l ) can be read o from the picture above by multiplying the orders of the groups Ep q 1 = Ep q 3 on the line p + q = n. Inspection leads to us to see jf kon (BC l )j = (n C l ) and proves Theorem 2.4 for the cyclic group. q.e.d. Proof of Lemma 2.8. Let x 2 ko1 f (BC l ) be the generator. We claim that the elements j i x 2 ko8j+i+1 f (B), i = 0 1, j = 0 1 ::: map

11 384 boris botvinnik, peter gilkey & stephan stolz to indivisible elements in g KOn (BC l ). Clearly these elements are nonzero, since in the E 1 -term of the spectral sequence, they represent the generators of E1 8j+i 1. The same is true in the Atiyah-Hirzebruch spectral sequence converging to KO g (BC l ) (which is obtained by extending the picture of the spectral sequence above by periodicity to negative q- values), and hence these elements map non-trivially to KO g (BC l ). Furthermore, p( j x) 2 KO8j+2 g (B) is indivisible, since KO8j+2 g (B) = Z=2, and p( j x) 2 KO8j+1 g (B) is indivisible since otherwise its product with would be zero. q.e.d. A stable splitting of BQ l. Let SL 2 (Fq )bethe group of 2 by 2 matrices with determinant 1 over the eld Fq of order q. It turns out that for q odd the 2-Sylow subgroup of SL 2 (Fq ) is Q 2 k, where k the exponent of2 in the prime factor decomposition of q 2 ; 1. A standard transfer argument shows that localized at the prime 2, BSL 2 (Fq ) is a stable summand of BQ 2 k. Mitchell and Priddy [20] identify the other summands they show that there is a stable, 2-local splitting (3.6) BQ 2 k = BSL 2 (Fq ) _ ;1 BS 3 =BN _ ;1 BS 3 =BN where N S 3 is the normalizer of S 1 S 3. To calculate the order of f kon (BQ l ), wemake use of the splitting (3.6) and deal with each summand separately. The connective K-theory of ;1 BS 3 =BN is known. We refer to Bayen and Bruner [5] for the proof of the following result. Theorem ) ko n ( ;1 BS 3 =BN) =Z=2 if n =8j +1 or n =8j +2. 2) ko n ( ;1 BS 3 =BN) =Z=2 2j+2 if n =8j +3 or n =8j +7. 3) ko n ( ;1 BS 3 =BN) =Z=2 otherwise. Proof of Theorem 2.4 for quaternion groups. Without further mention, all groups in the proof are localized at 2. We have eh n (BSL 2 (Fq ) Z)=Z=l if n 3 mod 4 and is 0 otherwise. Hence the E 2 -term of the Atiyah- Hirzebruch spectral sequence converging to f ko (BSL 2 (Fq )) looks as follows, where as above a black dotatthe location (p q) means that E 2 p q is isomorphic to Z=l, while a white dot means E 2 p q = Z=2.

12 the gromov-lawson-rosenberg conjecture 385 q p Figure 2. The E 2 -term of the spectral sequence eh p (BSL 2 (Fq ) ko q )=) f kop+q (BSL 2 (Fq )) We note that for dimensional reasons, there are no non-trivial d r dierentials originating in or mapping to a group E r p q for p+q 3mod 4 (there are non-trivial dierentials in other total degrees!). Hence by multiplying the orders of the groups E 1 p q = E 3 p q on the line p+q =4i+3 we see that ko4i+3 f (BQ l ) has order l i+1. Together with Theorem 3.1 this implies Theorem 2.4 for the quaternion group. q.e.d. Proof of Lemma 2.9. Let x be a non-trivial element in the kernel of p: f kon (BQ l )! g KOn (BQ l ) for n 6 3mod4. Assume that x has skeletal ltration p, which means x 2 F p, but x =2 F p;1, where F p is the image l )! kon f (BQ l ) induced by the inclusion of the p-skeleton of BQ l. Then x projects to a non-zero element x 2 Ep n;p 1 = F p =F p;1. We note that this implies p 3 mod 4, since if x were an element in the only other non-zero group Ep n;p, 1 p 0 mod 4, then x would map to a non-zero element in the Atiyah-Hirzebruch spectral sequence converging to periodic K-theory KOn g (BQ l )(fordimensional reasons, elements in Ep n;p, r p 0 mod 4 of that spectral sequence can never be hit by a dierential). For n 2 mod 4, p 3 mod 4 the vanishing of Ep n;p 1 implies x 2 F p;1 contradicting our assumption. For n 0 mod 4 (resp. n 1 mod 4) the multiplicative structure of the spectral sequence implies that modulo elements in F p;1 the element x is in the image of (resp. 2 ). Inductively it follows that the kernel of of the map f kon (BQ (p)

13 386 boris botvinnik, peter gilkey & stephan stolz p is equal to the image of (resp. 2 ) for n 0 mod 4 (resp. n 1 mod 4). q.e.d. 4. Analytic results concerning the eta invariant The goal of this section is twofold: on one hand we want to show that the eta invariant leads to a well dened homomorphism (4.1) (): KO n (B)! R=Z resp. R=2Z for n odd (cf. 2.3). Here is a virtual representation of a discrete group of virtual dimension zero, and the range of () depends on n mod 8, and on the `type' of the representation the eta invariant takes values in R=2Z if n 3mod8 and is of real type or if n 7mod8 and is of quaternion type, and it takes values in R=Z otherwise. On the other hand, we want toshowhow the eta invariant of a pair (M f) consisting of a spin manifold M and a map f : M! B can be computed in terms of the xed point data of a -manifold W whose boundary is f M,thecovering of M determined by f. We begin by reviewing Dirac operators and the Atiyah-Singer Index Theorem for manifolds with boundary. Dirac operator. Let M be an n-dimensional spin manifold and let S be the (complex) spinor bundle over M. The Dirac operator is a rst order dierential operator D(M): ;(S)! ;(S) acting on the space ;(S) of sections of the spinor bundle (a general reference for the construction of D is the book of Lawson and Michelsohn [17], where this operator is called the \Atiyah-Singer operator"). More generally, given a vector bundle E over M with a unitary connection, one can `twist' the Dirac operator by E to get a self-adjoint elliptic operator D(M) E :;(SE)!;(S E): If the dimension of M is even, the spinor bundle decomposes S = S + S ;. With this decomposition, the twisted Dirac operator D(M) E can be written in the form D = D + + D ;,whered (M) E :;(S E)! ;(S E): Index Theorem. Assume M is the boundary of an even dimensional spin manifold W over which thebundle E extends. Atiyah, Patodi and Singer showed that D + (W ) E is a Fredholm operator if one imposes a certain (global) boundary condition. In particular, the index is well dened: index (D + (W ) E) := dim ker D + (W ) E ; dim coker D + (W ) E:

14 the gromov-lawson-rosenberg conjecture 387 We use the Index Theorem [2]: index (D + (W ) E) = R W ^A(W )ch(e) ; (D(M) E): Here the integrand is a certain polynomial in the Pontrjagin forms of W and the Chern forms of E, and (D(M) E) is the eta-invariant which is a measure for the asymmetry of the spectrum of the operator D(M) E with respect to the origin. Denition of (M f)(). Let be a nite dimensional complex representation of a discrete group. We assume that is unitary henceforth to ensure the resulting operators are self-adjoint this assumption is automatic for nite groups of course. Let M be a Riemannian spin manifold of odd dimension n, and let f : M! B be a map. The map f and the representation determine a at vector bundle V := f M, where f M! M is the -covering of M classied by f (i.e., the pull back of the universal covering E! B via f). We denoteby D(M f )the Dirac operator on M twisted by V, and by (M f)() its eta invariant. Often, the map f is understood and we will just write (M)(). It is clear from the denition that (M f)() is additive in, and hence its denition can be extended to virtual representations. We would like to show that the eta-invariant gives rise to a homomorphisms from KO (B) tor=z. To do this, we need the following. Geometric interpretation of KO n (X). pd discussed in x2 induces an isomorphism [14] ; spin (X)=T (X) [B ;1 ] = KO (X) The homomorphism here T n (X) is the subgroup of spin n (X) represented by pairs (M f q), q : M! N is an HP 2 -bundle over a(n ; 8)-dimensional spin manifold N, and f : N! X is a map. By HP 2 -bundle we mean a ber bundle whose ber is the quaternionic projective plane HP 2 and whose structure group is the isometry group of HP 2 with its standard metric. The Bott manifold B = B 8 is any simply connected spin manifold of dimension 8with ^A(B) =1. By denition, spin (X)=T (X) [B ;1 ] is the direct limit of the sequence given by Cartesian product with the Bott manifold B: spin (X)=T (X)! spin (X)=T (X)! spin (X)=T (X)! ::: :

15 388 boris botvinnik, peter gilkey & stephan stolz Theorem 4.1. Let be a virtual representation of of virtual dimension 0. Then the homomorphisms (): spin n (B)! R=Z and (): KO n (B)! R=Z which send a class represented by a spin manifold M of dimension n (resp. n +8k), and by a map f : M! B to (M f)(), are well dened. Moreover, if is of real type and n 3 mod 8, or if is of quaternionic type and n 7 mod 8, then we can replace the range of () by R=2Z. For the proof of this result, we need to show: 1) (M f)() depends only on the spin bordism class of (M f), 2) (M f)() =0for[M f] 2 T (B), and 3) (M B f)() =(M f)(). This follows from the next three lemmas. Lemma 4.2. (M f)() 2 Z. Moreover, if is of real type and n 3 mod 8, or if is of quaternionic type and n 7 mod 8, then(m f)() 2 2Z. If (M f) is the zero element in spin n (B), then Proof. Let (W F) be a zero bordism for (M f). Then the at bundle V extends to a at bundle V over W. We note that the Chern character of this bundle vanishes identically, and hence the Index Theorem discussed above implies index (D + (W ) V )=;(M f)(): This proves the rst part of the assertion of the lemma. To prove the second part, we show that under those assumptions, the space of sections ;(S W V ) has a quaternionic structure and that the Dirac operator D + (W ) V is H -linear, which implies in particular that the (complex) dimension of its kernel and cokernel is even. Let J 1 :! be complex conjugation (resp. multiplication by j 2 H ) if is a representation of real type (resp. quaternionic type). We note that J 1 is C -antilinear, it commutes with the action of the real group ring R, and J 2 1 = 1 (resp. J 2 1 = ;1) if is a real (resp. quaternionic) type. We callj 1 a real (resp. quaternionic) structure on. We claim that the C C`n-module has a real (resp. quaternionic) structure J 2 :! for n 7 mod 8 and a real structure for n 3mod8(i.e., J 2 is a C -antilinear homomorphism which commutes with the action of the real Cliord algebra C`n such that J 2 = 1). This follows from the fact that 2 C`n is the sum of two matrix algebras over R (resp. H ) for n 7 mod 8 (resp. n 3mod8). It follows that J 1 J 2 : C! C is a quaternionic structure if either is of real type and n 3mod8,or is of quaternionic type

16 the gromov-lawson-rosenberg conjecture 389 and n 7 mod 8. This quaternionic structure induces a quaternionic structure on the space of sections ;(S W V ), which makes the Dirac operator H -linear. q.e.d. Lemma 4.3. Let p: E! B be a Riemannian submersion with totally geodesic bers such that the induced metric on the bers has positive scalar curvature. Let g be themetric on E, andfor t>0 let g t be the canonical variation of g (cf. [6, Ch. 9 G]), i.e., the new metric obtained from g by rescaling in ber direction by multiplication by t. Let f : B! B be a map and let (E t f p)() be the eta invariant with respect to the metric g t. Then lim (E t f p)() =0: t!0 Proof. Let = 1 ; 0 where 0 and 1 are actual representations of the group of the same dimension. Let B and 0 B 1 be the vector bundles over B corresponding to the representations 0 and 1. Let E 0 and E 1 be the pull back of these bundles to E. We give these bundles the natural at unitary connections. The restriction of E and 0 E to any ber F 1 x is trivial since these bundles arise from a structure on the base. Since the scalar curvature of F x is positive, the kernel of the Dirac operator on F x with coecients in E is trivial. Let D E be the Dirac operator on the total space E with coecients in E for =0 1. We apply the Adiabatic limit theorem of Bismut and Cheeger [7, (0.5)]. (Note that Bismut and Cheeger use a somewhat dierent convention for rescaling the metric R than does Besse). We use this result to conclude that lim t!0 (D E )= ^A(iR B B )~, where ^A(iR B ) is the dierential form representing the ^A genus of B obtained from the Chern-Weil homomorphism, and ~ is an explicit dierential form on B whose value at x 2 B only depends on global information on F x and on the horizontal and vertical distributions of the bration. However, since the restriction of E and 0 E to F 1 x are trivial bundles of the same dimension, ~ 0 = ~ 1. Since the eta invariant is independent of the metric chosen, we complete the proof by checking lim (E t f p)() =limf(d E 0 ) ; (D E 1 )g = t!0 t!0 Z B ^A(iR B )(~ 0 ; ~ 1 )=0: We note that the standard metric on HP 2 has positive scalar curvature, and that we can equip the total space of an HP 2 -bundle p: M! N with a Riemannian metric which makes p a Riemannian submersion with

17 390 boris botvinnik, peter gilkey & stephan stolz totally geodesic bers such that the induced metric on the bers is the standard metric. Hence the previous lemma implies that the -invariant vanishes on T (B). We refer to [4, p.84] for the proof of the following result. Lemma 4.4. Let N be an even dimensional spin manifold and let M be anodd dimensional spin manifold with a given map f : M n! B. Then for any virtual representation of virtual dimension 0, we have that (M N)() = b A(N)(M)(): Remark 4.5. If the structure on M is trivial, then the eta invariant vanishes. Thus the eta invariant is carried by the reduced theories in Theorem 4.1. There is a result of H. Donnelly [9] which permits us to compute the eta invariant in terms of Dedekind sums. Let be a nite group and let be an action of on an odd dimensional spin manifold M without boundary by xed point free isometries. Let be a representation of. Let M = M= with the induced structures. Let E( M) be the eigenspaces of the Dirac operator on M and E( M) be the eigenspaces of the Dirac operator on M. The eigenspaces E( M) inherit a natural action and we may identify E( M) with E( M) : Consequently dim(e( M)) = jj ;1 X g2 Tr((g)) on E( M) so ( M)=jj P ;1 g2 ((g)) where we dene 2((g)) = X 6=0(Tr((g)) on E( M))sign()jj ;s j s=0 + (Tr((g)) on E(0 M)): A similar argument shows that if we twist by representation of then (4.2) ( M)() =jj ;1 X g2 Tr((g))((g)): Let R() be the group representation ring of a nite group and let R 0 () be the augmentation ideal of all virtual representations of virtual dimension 0. If we restrict to 2 R 0 (), then we can assume that g 6= 1 in equation (4.2). We shall assume henceforth that the metric on M

18 the gromov-lawson-rosenberg conjecture 391 has positive scalar curvature so that there are no harmonic spinors and E(0 M)=f0g this will be crucial for our analysis. We now assume that M is the boundary of a compact spin manifold N and that the action of the group extends to an action of by spin isometries on N. We do not, however, assume that (g) is xed point free for g 6= 1. We assume that the metric on N has positive scalar curvature and that M is totally geodesic. We wish to compute ((g)) in terms of the xed points of the action on N. Suppose for g 6= 1 that (g) has isolated xed points in the interior of N. At such a xed point x, det(i ; d(g))(x) 6= 0. We denote the rotation angles of d(g)(x) by 2 (0 2) and dene the defect relative to the spin complex by def((g) spin)(x) =(2i) ;1 Y cosec( =2) since cosec is not periodic mod, def((g) spin)(x) depends on the lift to spin. If (g) in fact arises from a complex matrix with complex eigenvalues f g then (4.3) def((g) spin)(x) = fq g 1=2 Q (1 ; ) the choice of a square root in equation (4.3) in this instance is equivalent to a lift from the unitary to the spin group see Hitchin [12] for details. Donnelly [9] has generalized the index formula of Atiyah, Patodi, and Singer [2], [3], [4] to the equivariant setting. Since the metric on the = M has positive scalar curvature, the Lichnerowicz- Weitzenbock formula [18], shows there are no harmonic spinors on M. Since the metric on N also has positive scalarcurvature, a similar similar argument shows there are no harmonic spinors on N see [8] for details. Consequently, the Lefschetz xed point formula for manifolds with boundary becomes in this setting for g 6= 1 (see [3, Section 2]): (4.4) ((g)) = X i def((g) spin)(x i ): Let :! U(m) be a xed point free representation. Let M be the quotient manifold S 2m;1 =(). The standard metric on S 2m;1 descends to dene a metric of constant sectional curvature+1onm, and every compact odd dimensional manifold admitting such a metric arises in this way. These are the odd dimensional spherical space forms the sphere

19 392 boris botvinnik, peter gilkey & stephan stolz and real projective spaces are the only even dimensional spherical space forms and will play no role in our discussion. The stable tangent bundle T (M) 1 is naturally isomorphic to the underlying real vector bundle of the complex vector bundle dened by over M, and thus T (M) admits a stable almost complex structure. The lift described by Hitchin of U(m) tothe group spin c (2m) provides M with a natural spin c structure. This structure can be reduced to a spin structure if and only if one can take the square root of the associated determinant line bundle or equivalently if and only if we can take the square root of the linear representation det(). This can also be expressed in topological terms. The second Stiefel-Whitney class of M is the mod 2 reduction of the rst Chern class of the determinant line bundle. This vanishes if and only if c 1 (det()) is divisible by 2 or equivalently if it is possible to take the square root of the determinant line bundle. Since every line bundle over a spherical space form arises from a linear representation, this is possible if and only if we can take the square root of the representation det(). If jj is odd, this is always possible if jj is even, this is possible if and only if m is even, i.e., the dimension of M is congruent to 3 mod 4. In this case, there are inequivalent spin structures on M, and the choice of a spin structure is equivalent tothechoice of the square root of det(). This square root plays an important role in the formula for the eta invariant in the following theorem. Theorem 4.6. Let 2 R 0 () and let :! U(m) be axedxed point free representation. Assume that there exists a representation of, which we denote by det() 1=2, whose tensor square istherepresentation det(). Let M = S 2m;1 =() with the inherited structures. Then (M)() =jj ;1 X g2 g6=1 Tr((g)) det((g)) 1=2 : det(i ; (g)) Proof. We regard S 2m;1 as the boundary of the upper hemisphere S 2m + of the sphere S 2m and extend the orthogonal action to S 2m + with an isolated xed point at the north pole. Theorem 4.6 then follows from the equations (4.2), (4.3), and (4.4). q.e.d. If the xed submanifold is higher dimensional, the defect formula is a bit more complicated. We assume that the xed point set of (g) inn is a Riemann surface X and that the normal bundle X decomposes as a direct sum of complex line bundles X = H 1 :::H k : We further assume

20 the gromov-lawson-rosenberg conjecture 393 that the action of d(g) on X respects this decomposition so d(g) X is the diagonal matrix diag( 1 ::: k ) the eigenvalues are constant and independent of the point of X which is chosen. The spin complex is multiplicative with respect to products. If H 1 is non-trivial, and all the other line bundles are trivial, then the defect in the Atiyah-Singer index theorem is given by: def((g) spin) X = fq i ig 1=2 Qi (1 ; i) F( 1)c 1 (H 1 )[X] see equation (4.3), where F is some universal function from S 1 ;f1g to C which is independent ofallthechoices made. This is easily seen from standard heat equation proofs of the Lefschetz xed point formulas or also directly from the equivariant index theorem. On the other hand, since X is 2-dimensional, the Chern classes do not interact and thus the formula in general can be written in the form (4.5) fq i def((g) spin) X = Q ig X 1=2 i (1 ; i) i F( i )c 1 (H i )[X]: We apply this observation as follows. Let ~a =(a 1 ::: a k ) be a collection of integers coprime to l. Let M be the total space of the sphere bundle S(H 1 H k ) where H i are complex line bundles over S 2 M is spin if and only if c 1 (H 1 )+::: + c 1 (H k )iseven assume this condition henceforth. Let (~a)() = diag( a 1 ::: 1 a k k ) dene a xed point free action of the cyclic group C l on M, and let M(l ~a H 1 H k ):=S=(~a)(C l ): We shall be interested in the case l even. The same discussion as that given for spherical space forms shows that these lens space bundles are spin if and only if k is even. Again, the spin structure is determined by the square root of the dening representation. Theorem 4.7. Let 2 R 0 (). The following formula holds: (M(l ~a H 1 H 2i ))() = 1 l X l =1 6=1 X j (a 1++a 2i )=2 Tr() (1 ; a 1 ) (1 ; a 2i) 1 2 c 1(H j )[C P 1 ] 1+aj 1 ; a j :

21 394 boris botvinnik, peter gilkey & stephan stolz Proof. We take D to be the disk bundle of H 1 H 2i with a suitable metric. We stated the Lefschetz xed point formula in equation (4.4) for isolated xed points, however it holds in a more general context. We use equations (4.2), (4.4), and (4.5) to see that (M(l ~a H 1 H 2i ))() = 1 l X l =1 6=1 (a 1++a 2i )=2 Tr() (1 ; a 1 ) (1 ; a 2i) X j c 1 (H j )[C P 1 ]F( j ): We apply this to the special case k =2,a 1 = a 2 = 1 to see (M(l 1 1 H H)) = 1 l X l =1 6=1 Tr() (1 ; ) 2 ~F() 2 where H is the Hopf bundle. On the other hand, we computed (see equation (2.12) of [8]) that (M(n 1 1 H H)) = 1 l X l =1 6=1 (1 + ) Tr() (1 ; ) : 3 We compare these two equations to see ~F() =(1+)=2(1;). q.e.d. 5. The range of the eta invariant for cyclic groups In this section, we prove Theorem 2.5 for cyclic groups, see Proposition 5.1 below. It then follows that the eta invariant completely detects the connective K-theory of cyclic groups in odd dimensions. We adopt the following notational conventions. Let C l be the cyclic group of order l =2 k, which we will identify with the subgroup of S 1 consisting of l-th roots of unity, i.e., C l = f 2 S 1 j l =1g: Given an integer a, let a be the complex 1-dimensional representation of S 1, where 2 S 1 acts by multiplication by a. Given a tuple of integers ~a =(a 1 ::: a t ), we note that the representation a1 ::: at restricts to a free C l -action on the unit sphere S 2t;1 of C t if and only if all the a i 's are odd. Let t =2i be even. The quotient manifold X 4i;1 (l ~a) :=S 4i;1 =( a1 ::: a2i )(C l )

22 the gromov-lawson-rosenberg conjecture 395 is a lens space of dimension 4i ; 1. The discussion of Section 4 shows it inherits a natural spin structure, and we use Theorem 4.6 to compute its eta invariant. The corresponding lens spaces of dimension 4i + 1 do not admit spin structures. To get spin manifolds of dimension n 1 mod 4, we consider lens space bundles these manifolds also play an important role in [8]. Let H be the (complex) Hopf line bundle over S 2, and let H 2 (2i;1)C be the Whitney sum of H H and 2i ; 1 copies of the trivial complex line bundle C. Given a tuple of integers ~a =(a 1 ::: a 2i ), we let2s 1 act on this sum by multiplication by a in the -th summand. This action restricts to a free action of the cyclic group C l on the sphere bundle S(H 2 (2i ; 1)C ) if and only if all the a 's are odd. The quotient ; X 4i+1 (l ~a) :=S H 2 (2i ; 1)C =Cl is a manifold of dimension 4i + 1, which bers over S 2 with ber X 4i;1 (l ~a). It is spin. The corresponding lens space bundles in dimensions 4i ; 1 are not spin. There is a natural identication 1 (X 4i1 (l ~a)) = C l that gives these manifolds natural C l structures. Since l is even, these manifolds admit two inequivalent spin structures. We x the spin structure by dening the square root of the canonical (or determinant) line bundle to be det( a1 ::: a2i ) 1=2 := (a1 +:::+a 2i )=2: We may then use Theorems 4.6 and 4.7 to compute the eta invariant. In taking the square root of the canonical bundle, we needed the a to be integer valued and not just dened mod l. We give these manifolds the natural metrics of positive scalar curvature. Let X n be 0 Xn with the trivial C l structure, and let X ~ n := X n ; X n 0 dene a bordism class [ X ~ n ]ine spin n (BC l ). Let M (BC l ) spin (BC l ) be the spin -submodule generated by the elements [ X ~ n (l ~a)] see (2.2). Dene the eta invariant ~(M) :=((M)( 1 ; 0 ) (M)( 2 ; 0 ) ::: (M)( l;1 ; 0 )): Since extends to maps in bordism and in K-theory, let ~([M]) = ~(M) take values in R l;1 =Z l;1. Let q = l=2 and let q (M) =(M)( q ; 0 ). If n 3 mod 8, the virtual representation q ; 0 is real, so by Theorem 4.1, q extends to a map in bordism taking values in R=2Z. It is convenient to separate out the eta invariant with values in R=2Z = ~ + q if

23 396 boris botvinnik, peter gilkey & stephan stolz n =8j +3, and = ~ otherwise (see x2). Let L n (BC l )ands n (BC l )be the range of the eta invariant onm n (BC l )ande spin n (BC l ) respectively. If n 3 mod 8, let K n (BC l ) be the range of the extended eta invariant ~ q. That means L n (BC l ) := span f~([m]) : [M] 2M n (BC l )g(r=z) l;1 S n (BC l ):=span n ~([M]) : [M] 2 e spin n (BC l )o (R=Z) l;1 K 8j+3 (BC l ) := span f~([m]) q ([M]) : [M] 2M 8j+3 (BC l )g (R=Z) l;1 (R=2Z): Theorem 2.5 and Corollary 2.6 for cyclic 2-groups are immediate consequences of the following result and Theorem 4.1. Proposition 5.1. Let l =2 k. a) If i 1, then jl 4i+1 (BC l )j2 (i+1)(k;1). b) If j 0, then js 8j+1 (BC l )j2 1+(2j+1)(k;1). c) If i 2, thenjl 4i;1 (BC l )j2 i(k+1). d) If j 0, then jk 8j+3 (BC l )j2 1+(2j+1)(k+1). We shall need some technical results to prove Proposition 5.1. Let B 8 be the Bott manifold B 8 is a simply connected spin manifold with ^A(B 8 )=1whichplays a central role in our discussion. If f is a complex function on C l, let X X f() := f() and ~ f() := f() l=1 l =1 6=1 be the sum of f over C l and the reduced sum of f over C l ;f1g respectively. If f(1) = 0, we may replace ~ by. If 2 R(C l ), the orthogonality relations imply l ;1 Tr(()) 2 Z. Let f n (1) = 0, and for 6= 1let f 4i;1 (~a)() := (a 1+:::+a 2i )=2 (1 ; a 1 ) ;1 (1 ; a 2i ) ;1 f 4i+1 (~a)() =f 4i;1 (~a)(1 + a 1 )(1 ; a 1 ) ;1 : Lemma 5.2. Let n = 4i 1 = 2j ; 1 3, ~ b j = (a 1 ::: a j ), 2 R 0 (C l ), and = ;3 ( 0 ; 3 ) 2. a) If s 0, then( ~ X n (l ~ b 2i )(B 8 ) s )() =l ;1 ~ f n ( ~ b 2i )()Tr(()): b) We have the identity l ;1 ~ (1 ; ) ;1 =(l ; 1)=(2l). c) If 2 R 0 (C l ) j+1, then ( ~ X n (l ~ b 2i ))() 2 Z.

24 the gromov-lawson-rosenberg conjecture 397 d) If n 7, then ( ~ X n (l ~ b 2i;2 3 3))() =( ~ X n;4 (l ~ b 2i;2 ))(): e) If n 3, then ( ~ X n (l ~ b 2i;1 1) ; 3 ~ X n (l ~ b 2i;1 3))() = ( ~ X n (l ~ b 2i;1 3))(( 1 ; ;1 )): Proof. The eta invariant vanishes if the C l structure is trivial, so we may replace ~ X n by X n in computing the eta invariant. Assertion (a) now follows from Lemma 4.4, Theorem 4.6, and Theorem 4.7. We note that the lack of symmetry between the rst and the remaining indices for f 4i+1 is caused by taking H 2 as the rst line bundle, and the trivial line bundle as the remaining line bundles in the denition of X 4i+1. Since l ;1 ~ (1 ; ) ;1 =l ;1 ~ f(1 ; ) ;1 +(1; ) ;1 g=2 =l ;1 ~ (2 ; ; )(1 ; ) ;1 (1 ; ) ;1 =2 =(l ; 1)=(2l) assertion (b) follows. If a is odd, then R 0 (C l )=( a ; 0 )R(C l ): Suppose 2 R 0 (C l ) 2i+1. Then there exists 2 R 0 (C l ) such that =( 0 ; a1 ):::( 0 ; a2i ): This shows that f 4i;1 ( ~ b 2i ) 2 R 0 (C l ). Similarly if 2 R 0 (C l ) 2i+2 then f 4i+1 ( ~ b 2i ) 2 R 0 (C l ). Let 4i 1=2j ; 1. If 2 R 0 (C I ) j+1, then Since Tr()(1) = 0, f 4i1 ( ~ b 2i ) = 2 R 0 (C l ): ( ~ X 4i1 (l ~ b 2i ))() =l ;1 ~ Tr(()) = l ;1 Tr(()) 2 Z and assertion (c) follows. Assertion (d) follows from the identity Tr()f n ( ~ b 2i;2 3 3) = f n;4 ( ~ b 2i;2 ) and assertion (a). As (1 ; ) ;1 =( )(1; 3 ) ;1, f n ( ~ b 2i;1 1) = ( ;1 )f n ( ~ b 2i;1 3) and assertion (e) follows. q.e.d.

25 398 boris botvinnik, peter gilkey & stephan stolz We dene Y 3 = ~ X 3 (l 1 1) ; 3 ~ X 3 (l 1 3) Y 8j+3 = Y 3 (B 8 ) j for j>0 Z 3 = ~ X 3 (l 1 1) Z 5 = ~ X 5 (l 1 1) ; 3 ~ X 5 (l 1 3) Z 7 = ~ X 7 (l ) ; 3 ~ X 7 (l ) Z 9 = ~ X 9 (l ) ; 3 ~ X 9 (l ) ; 3( ~ X 9 (l ) ; 3 ~ X 9 (l )) Z n = Z n;8 B 8 for n>9 = ;3 ( 0 ; 3 ) 2 (M) =((M)(( 1 ; 0 )) ::: (M)(( l;1 ; 0 ))): Lemma 5.3. a)if j 0, then q (Y 8j+3 )=1 and ~(Y 8j+3 )=0. b) If n 3, then (Z n )=0. c) If i 1, then~(z 4i;1 ) has order at least 2 k+1 in R l;1 =Z l;1. d) If i 1, then ~(Z 4i+1 ) has order at least 2 k;1 in R l;1 =Z l;1. e) The vector ( ~ X 5 (l 3 3)) has order at least 2 k;1 in R l;1 =Z l;1. Proof. We use Lemma 5.2 in the proof throughout. We prove assertions (a)-(d) in dimensions 3, 5, 7, and 9 the remaining cases then follow from Lemma 4.4. To prove assertion (a), we compute that q (Y 3 )=( ~ X 3 (l 1 3))(( q ; 0 )( 1 + ;1 ; 2 0 )) = l ;1 ~ ( q ; 1) 2 ( + ; 2)(1 ; ) ;1 (1 ; 3 ) ;1 = l ;1 ~ ( 3q ; 1)(1 ; )(1 ; 3 ) ;1 = ;l ;1 ~ ( ::: + 3q;3 )(1 ; ): We replace l ;1 ~ by l ;1 since (1 ; ) =0if =1. Since l ;1 j = j l for 0 <j 3q ; 3+2< 2l, q (Y 3 )=1 as claimed. If 2 R 0 (C l ), then ( 1 + ;1 ;2 0 ) 2 R 0 (C l ) 3 so (Y 3 )() =0. This proves assertion (a). Since 2 R 0 (C l ) 2 and ( 1 + ;1 ; 2 0 ) 2 R 0 (C l ) 2, assertion (b) follows from Lemma 5.2 (c,e). We use Lemma 5.2 (a,b) to prove assertion (c). We compute that ( ~ X 3 (l 1 1))( s ( 0 ; 1 )) = l ;1 ~ s+1 (1 ; ) ;1

26 the gromov-lawson-rosenberg conjecture 399 has order 2l in R=Z for s = ;1. Similarly, we compute that (Z 7 )( s ( 0 ; 3 )) =( ~ X 7 (l ))( s ( 0 ; 3 )( 1 ; ;1 )) =( ~ X 7 (l ))( s;1 ( 0 ; 3 )( 0 ; 1 ) 2 ) =l ;1 ~ s+2 (1 ; ) ;1 has order 2l in R=Z for s = ;2. To check ( ~ X 5 (l 3 3)) has order at least 2 k;1,we compute that ( ~ X 5 (l 3 3))(( 0 ; 3 ) = l ;1 ~ (1 + 3 )=;2=l + l ;1 (1 + 3 ) ;2=l mod Z: To check that ~(Z 5 ) has order at least 2 k;1,welet = s ( 0 ; 3 )for suitably chosen s and compute (Z 5 )() =( ~ X 5 (l 1 3))(( 1 ; ;1 )) =l ;1 ~ (1 + ) ;2=l mod Z: To check that ~(Z 9 ) has order at least 2 k;1,welet = s ( 0 ; 3 )for suitably chosen s. Choose a so 3a 1modl. Then (1 ; )=(1 ; 3 )= ::: + 3a;3 : Consequently we may complete the proof by computing (Z 9 )() =( ~ X 9 (l ))(( 1 ; ;1 ) 2 ) =l ;1 ~ (1 + )(1 ; )=(1 ; 3 ) =l ;1 ~ (1 + )( ::: + 3a;3 ) ;2a=l mod Z: Proof of Proposition 5.1. By Lemma 4.4 and Lemma 5.2 (d), induces a surjective map from L n (BC l ) to L n;4 (BC l ) if n 7. Let n =5. By Lemma 5.3 (b,d,e), the range of 5 and the kernel of 5 both have order at least 2 k;1 and thus jl 5 (BC l )j2 2k;2. We now proceed by induction. Since 4i+1 is surjective, the order of the range of 4i+1 is at least jl 4i;3 (BC l )j. By Lemma 5.3 (b,d), the kernel of 4i+1 has order at least 2 k;1 Proposition 5.1 (a) now follows. If n =8j +1, we prove Proposition 5.1 (b) by picking up an extra factor of 2 in ker ( n ). Let M = S 1 (B 8 ) j with the trivial (i.e., bounding) spin structure and

27 400 boris botvinnik, peter gilkey & stephan stolz canonical C l structure. It is then an easy calculation to show [M] 2 ker ( n ) and that ~(M) isanelementoforder2 k. Let n = 3. By Lemma 5.3 (e), ~( X ~ 3 (l 3 3)) has order at least 2 k+1. Since 4i;1 is surjective, the order of the range of 4i;1 is at least jl 4i;5 (BC l )j. By Lemma 5.3 (b,c), the kernel of 4i;1 has order at least 2 k+1 Proposition 5.1 (c) now follows by induction. To prove Proposition 5.1 (d), we must pick up an extra factor of 2 using the rened eta invariant again, it suces to perform this calculation for n = 3 using the periodicity mod 8. This follows since ~(Y 3 ) = 0 and q (Y 3 ) 6= 0by Lemma 5.3 (a). q.e.d. 6. The range of the eta invariant for generalized quaternion groups In this section, we complete the proof of Theorem 2.5 by dealing with the generalized quaternion groups see Proposition 6.1 below. We also show that the eta invariant completely detects the connective K- theory of these groups in dimensions congruent to 3mod4. We adopt the following notational conventions. Let i, j, and k be the standard elements of the quaternions H. For q 3, identify the generalized quaternion group Q l of order l = 2 q with the subgroup of S 3 H generated by the elements = e i=2q;2 and j these are the generalized quaternion groups we shall be studying. This denes a natural representation : Q l! SU(2). We note that i = 2q;3. Any subgroup of order 4 in Q l is conjugate to one of the following groups: H 1 := hii H 2 := hji H 3 := hji: There are 4 inequivalent real representations 0, 1, 2 and 3 of Q l which are given on the generators by the formula: 0 (j) :=1 0 () :=1 1 (j) :=1 1 () :=;1 2 (j) :=;1 2 () :=1 3 (j) :=;1 3 () :=;1: We notice that the representations t, t =1 2 3 are nontrivial on the subgroup H 1 if and only if q = 3, so this case is slightly exceptional. To have a uniform notation that covers this case as well, we dene the virtual representations 2 := 0 ; 1 if q =3, 2 := 3 ; 0 if q>3,