The eta invariant and the equivariant spin. bordism of spherical space form 2 groups. Peter B Gilkey and Boris Botvinnik

Transcription

1 The eta invariant and the equivariant spin bordism of spherical space form 2 groups Peter B Gilkey and Boris Botvinnik Mathematics Department, University of Oregon Eugene Oregon USA Abstract We use the eta invariant to compute the equivariant spin bordism groups spin (Z=2 ); spin 3 (BQ); and spin 7 (BQ): MSC numbers: 8G12, 8G2, 3A0, 3C2, N22. 1 Introduction Let BG be the classifying space of a nite group G: Let M be a compact spin Riemannian manifold of dimension m without boundary. Let : M! BG dene a G structure on M: The group spin m (BG) classies spin manifolds with G structures up to bordism. Let S 3 be the unit sphere of the quaternions H: For 3; let n = 2?1 and let = e 2i=n : Let Q = Q := h; ji S 3 be the quaternion group of order 2 : Theorem 1.1 Let 3: Then (a) spin 3 (BQ ) = Z=2 Z=4 Z=4: (b) spin 7 (BQ ) = Z=2 +3 Z=2?3 Z=4 Z=4: Theorem 1.2 Let 2: Then spin (BZ=2 ) = Z=2 Z=2?2 : Remark 1.3 Methods of algebraic topology show that spin (BZ=2) = 0: Bayen and Bruner [4] have proved Theorem 1.1 independently using topological methods. 1

2 All the torsion in the coecient ring spin is 2-torsion so the prime 2 is distinguished in this subject; if n is odd, one can use the calculation of the Brown-Peterson homology groups BP (BZ=n) in [2] to compute spin (BZ=n): We have chosen the dimensions m = 3 and m = 7 in Theorem 1.1 and the dimension m = in Theorem 1.2 to illustrate the use of the eta invariant to compute invariants in algebraic topology. Although Theorems 1.1 and 1.2 concern bordism groups, which are objects of algebraic topology, the proof will be largely analytic; methods of algebraic topology are used only to obtain upper bounds on the orders of the bordism groups. These groups are interesting from the point of view of dierential geometry. If M is a manifold which admits a metric of constant positive sectional curvature, then the 2-Sylow subgroup of 1 (M) is either Z=2 or Q : These are the 2-groups with periodic cohomology. A consequence of the proof we shall give of Theorems 0.1 and 0.2 is that the bordism groups spin 3 (BQ ); spin 7 (BQ ); and spin (BZ=2 ) are completely detected by the eta invariant and are generated by manifolds which admit metrics of positive scalar curvature. The Gromov-Lawson-Rosenberg conjecture asserts that a spin manifold M of dimension at least with fundamental group G admits a metric of positive scalar curvature if and only if a generalized index of the Dirac operator vanishes; this is an invariant of equivariant spin bordism. We refer to [3] for further details where this conjecture is proved if G is a spherical space form group. In [3], it was necessary to prove that certain bordism groups were generated by manifolds which admit metrics of positive scalar curvature; to do this we needed to nd suitable lower bounds for the range of the eta invariant. In the course of this investigation, we noted in addition to determining the order of the range of the eta invariant that we could compute the additive structure of some of these groups. We present here some of these calculations which are of independent interest. In x2, we discuss the eta invariant. In x3, we prove Theorem 1.1. In x4, we prove Theorem The eta invariant Let be a G structure on a compact spin Riemannian manifold without boundary of odd dimension m: Let be a representation of G and let A be the Dirac operator with coecients in the at vector bundle dened by : Let (M)() := (A ) = 1 2 ftr L 2(A?s ) + dimker(a )gj s=0 (2:1) be the eta invariant of the Dirac operator with coecients in : Since the eta invariant is additive with respect to direct sums, we may extend (g; M)() to the 2

3 group representation ring R(G): Let R 0 (G) be the augmentation ideal of virtual representations of virtual dimension 0. We can interpret the eta invariant as a bordism invariant as follows: Lemma 2.1 Let m be odd, let jgj < 1; and let 2 R 0 (G): (a) M! (M)() denes a group homomorphism from spin m (BG) to R=Z: (b) Let be real. If m 3 (8); then M! (M)() denes a group homomorphism from spin m (BG) to R=2Z: (c) Let be quaternion. If m 7 (8); then M! (M)() denes a group homomorphism from spin m (BG) to R=2Z: Proof: If N is a compact manifold with boundary M such that the spin and G structures on M extend over N; we must prove (M)() 2 Z in general and that (M)() 2 2Z if (b) or (c) hold. We use the Atiyah-Patodi-Singer index theorem for manifolds with boundary; see [1] for details. We choose a metric on N which is product near M: Let N be the half-spin bundles over N and let W be a coecient bundle dened by a representation of the group G: Let D W : C 1 ( + N W )! C 1 (? N W ) (2:2) be the Dirac operator with coecients in W: We decompose D W = (@ n + A W ) (2:3) where is a bundle isometry from + NW to? NW and where A W is the associated Dirac operator on n is the inward unit normal. Let + be spectral projection on the non-negative eigenspaces of A W : We introduce Atiyah-Patodi-Singer boundary conditions for D W by dening Domain(D W ) = ff 2 C 1 ( + N W ) : + (fj M ) = 0g: (2:4) Since the complex vector bundle W is at, index(d W ) = dim(w ) R N ^A(N)? (A W ): (2:) If the bundles W i are dened by representations i with dim( 1 ) = dim( 2 ); then (M)( 2? 1 ) = (A W2 )? (A W1 ) (2:6) 3

4 = index(d W1 )? index(d W2 ) 2 Z: We prove (b) by showing index(d W ) is an even integer if W is a real bundle over N and if m 3 (8). Let the Cliord algebra Clif(R m+1 ) be the real unital algebra generated by R m+1 subject to the Cliord commutation rules v w + w v =?(v; w) 1: (2:7) Suppose m = 3: We note that Clif(R 4 ) is isomorphic to the algebra M 2 (H) of 2 by 2 matrices over the quaternions. This isomorphism denes a representation from SPIN(4) to M 2 (H): Let e N be the vector bundle associated with this representation. Note that e N is an 8 dimensional real vector bundle which admits a natural left Clif(T N) module structure. The Cliord action of the orientation form is self-adjoint and induces a natural real splitting e N = e + N e? N: Let W be a real vector bundle over N and let fd W : C 1 ( e + N R W )! C 1 ( e? N R W ) (2:8) be the associated elliptic complex of Dirac type. The complexication of e N gives two copies of the spin bundle. Thus index( f D W ) = 2 index(d W ): (2:9) We show index(d W ) is even by showing index( f D W ) is divisible by 4. Right and left quaternion multiplication commute. We have used left quaternion multiplication for the transition functions of e N and to dene the Clif(N) module structure on e N ; thus right multiplication gives e N a natural quaternion structure. The boundary conditions and the orientation splitting commute with this structure so ker( f D W ) and ker( f D W ) are real vector spaces which have quaternion structures. Their dimension is therefore divisible by 4 and thus index(d W ) is even. This completes the proof in the case m = 3; the general case follows from a similar argument which uses the periodicity theorem Clif(R +8 ) ' Clif(R ) R M 16 (R): (2:10) We use the observation that Clif(R 8 ) = M 16 (R) to perform a similar analysis if m = 7 and prove (c). Cartesian product makes spin (BG) into a spin module. The following Lemma relates the eta invariant to this module structure, see [1,6,7] for details. 4

5 Lemma 2.2 Let M 2 spin (BG) and N 2 spin : If 2 R 0(G) and if m is odd, then m 4k (M N)() = (M)() ^A(N): To use the eta invariant, we shall need some combinatorial formulas. For k > 1; let : G! U(k) be a xed point free representation of a nite group G: Let M = M(; G) := S 2k?1 =(G): (2:11) We give M the inherited metric of constant sectional curvature +1: The manifold M admits a spin structure if and only if det() 1=2 extends to a representation of G and then a choice of a square root denes the spin structure in question. We will be working with representations which take values in S 3 = SU(2) so det() = 1 and we will take the trivial square root to dene the spin structure. If jgj is even and if k is odd, then the manifold M(; G) does not admit a spin structure so we shall need other manifolds to construct generators of the group spin (BZ=2 ): Let a () = a dene a linear representation the cyclic group. Let L be the Hopf line bundle over CP 1 : The representation a denes an action of Z=2 on the ber C of L: Let a 1 and a 2 be odd integers. Let S(LL) be the corresponding spherical bundle. Give the lens space bundle M(2 ; a 1 ; a 2 ) := S(L L)=( a1 a2 )(Z=2 ) (2:12) the canonical spin and Z=2 structure. Let f(a 1 ; a 2 )(1) := 0: If 6= 0; let f(a 1 ; a 2 )() := (a 1+a 2 )=2 (1? a 1+a 2 ) (1? a 1 )2 (1? a 2 ) 2 : (2:13) We refer to [] for the proof of (a) and to [3] for the proof of (b) in the following Lemma. Lemma 2.3 (a) Let 2 R 0 (G): Then (M(; G))() = jgj?1 2G?f1g Tr (()) det(()) 1=2 det(i? ())?1 : (b) Let 2 R 0 (Z=2 ): Then (M(2 ; a 1 ; a 2 ))() = 2? 2Z=2?f1gTr (())f(a 1 ; a 2 )():

6 The next Lemma gives an upper bound for the orders of the bordism groups we shall be studying. The rst assertion follows from the Atiyah Hirzebruch spectral sequence; the second assertion follows from the Adams spectral sequence. These are the only results from algebraic topology that we shall need. Lemma 2.4 (a) j spin 3 (BQ )j 16jQ j and j spin 7 (BQ )j 16jQ j 2 : (b) j spin (BZ=2)j = 0: If 2; then j spin (BZ=2 )j 2 2?2 : We will use the following technique to prove Theorems 1.1 and 1.2. Let (m; G) 2 f(3; Q ); (7; Q ); (; Z=2 )g (2:14) and let i 2 R 0 (G) be suitably chosen representations. Let i 2 f1; 2g be dictated by Lemma 2.1 and let M j be suitably chosen manifolds dening [M j ] 2 spin m (BG): Let ~(M) := ((M)( 1 ); :::; (M)( k )) 2 (R= 1 Z) ::: (R= k Z) (2:1) range(~) := span j f~(m j )g: (2:16) We will show range(~) is isomorphic to the Abelian group A m of Theorem 1.1 or 1.2. Since ja m j is the upper bound of Lemma 2.4, ja m j j range(~)j j spin m (BG)j ja m j (2:17) : spin m (BG) =?!A m : (2:18) 3 The calculation of spin 3 (BQ ) and spin 7 (BQ ) Let H 1 =: hii; H 2 =: hji; and H 3 =: hji be the non-conjugate subgroups of order 4 in Q = Q : For m = 4k? 1; we embed S m in H k and dene M m Q := S m =Q and M m i := S m =H i : (3:1) Let be the inclusion map of Q into SU(2) = S 3 be the identity map. We note that det(i? ) = 2? Tr () (3:2) and that det(i? ()) = 2 if is an element of order 4 in Q: 6

7 by: There are 4 inequivalent real linear representations of Q given on the generators 0 () = 1; 0 (j) = 1; 1 () = 1; 1 (j) =?1 (3:3) 2 () =?1; 2 (j) = 1; 3 () =?1; 3 (j) =?1: (3:4) The i are non-trivial on H 1 if and only if jqj = 8 so this case is slightly exceptional. To have a uniform notation, we dene 2 := 0? 2 if jqj = 8 3? 0 if jqj > 8 and 3 := 0? 3 if jqj = 8 2? 0 if jqj > 8 (3:) Lemma 3.1 Let m = 4k? 1: Let 2 i; j 3: (a) 4 k (MQ m )( i ) 2 Z: (b) (MQ 7 )(2? ) = a2??2 for some odd integer a: (c) (MQ 7 )((2? )2 ) = (MQ 3 )(2? ) = 1? 2? : (d) (MQ 3 )(2? 2 )2 Z: (e) (M m 1? M m i )( j ) = 2?k ij : Proof: Let n = 2?1 ; let q = 2?2 ; let H 0 = hi; and let M m 0 let = Sm =H 0 : If 2 R(Q); E m 1 () := jqj?1 2H0?f1g() det(i? ())?k = 1 2 (M m 0 )(); (3:6) E m 2 () := jqj?1 2Q?H0 ()((1 + i)(1? i))?k = 2?k?2 f(j) + (j)g: (3:7) Then (M m Q )() = E m 1 () + E m 2 (): We note E m 2 ( i ) = 2?k?1 : Since for any 2 H 0 ; we see that i () = ( q? 1) = 1 2 (q? 1)( q? 1) (3:8) E m 1 ( i) = 2 1?2k jqj?1 2H0?f1g k ( q? 1) 2k (? 1)?2k (3:9) = 2?2k jh 0 j?1 2H0?f1g k (1 + + ::: + q?1 ) 2k = 2?2k q 2k jh 0 j?1 2?2k jh 0 j?1 2H0 k (1 + + ::: + q?1 ) 2k ; assertion (a) follows since q 2k jh 0 j?1 2 Z and jh 0 j?1 2H0 ` 2 Z for any `: 7

8 To prove (b), we compute E 7 1 (2? ) = jqj?1 2H0?f1g det(i? ())?1 = 1 2 (M 3 0 )( 0) (3:10) We must show (M 3 0 )( 0 ) = ~a=(4n) for ~a odd. Let E 7 2 (2? ) = 1=4: (3:11) T d 2 (~x) = ( i<j x i x j + ( k x k ) 2 )=12 (3:12) be the Todd genus. We use Rademacher reciprocity, see for example [, Theorem 2.], to prove (b) by checking that (M 0 )( 0 ) n?1 T d 2 (n; 1;?1) (n 2? 1)=(12n) mod Z: (3:13) We prove (c) and (d) by computing: (M 7 Q)((2? ) 2 ) = (M 3 Q)(2? ) = jqj?1 2Q?f1g 1 (3:14) (M 3 Q)((2? ) 2 ) = jqj?1 2Q?f1g (2? )() (3:1) = jqj?1 2Q (2? )() 2 Z: Since 2? 0 is supported on the two elements of order 4 in Z=4; (M m i )( 2? 0 ) = 4?1 2 (?2) f(1? i)(1 + i)g?k =?2?k ; (3:16) assertion (e) is now follows from the denition of the i : Since? 2 is quaternion and i is real, we dene ~ 3 (M 3 ) := (M 3 )(? 2; 2 ; 3 ) 2 R=Z (R=2Z) 2 ; (3:17) ~ 7 (M 7 ) := (M 7 )(? 2; (? 2) 2 ; 2 ; 3 ) 2 R=2Z (R=Z) 3 : (3:18) Proof of Theorem 1 for m = 3: By Lemma 3.1, ~ 3 (MQ 3 ) = ( d=jqj; b=4; b=4) ~ 3 (M 3 1? M 3 2 ) = ( 0; 1=2; 0) ~ 3 (M 3 1? M 3 3 ) = ( 0; 0; 1=2) (3:19) for d odd. The rst column is a bordism invariant mod Z; the next two columns are bordism invariants mod 2Z: We perform Gaussian elimination to see range(~ 3 ) Z=jQj Z=4 Z=4: (3:20) 8

9 Proof of Theorem 1 for m = 7 : Let N 4 be the Kummer surface. Since ^A(N 4 ) = 2; by Lemma 2.2 (M 3 Q N 4 )() = 2(M 3 Q)(): Thus ~(M Q) 7 = ( a=4jqj; d=jqj; c=16; c=16 ) ~(M Q 3 N 4 ) = ( 2d=jQj; 0; b=2; b=2 ) (3:21) ~(M 7 1? M 7 2 ) = ( 0; 0; 1=4; 0 ) ~(M 7 1? M 7) = ( 0; 0; 0; 1=4 ) 3 for a and d odd. The rst column is a bordism invariant mod 2Z; the remaining columns are bordism invariants mod Z: We multiply the third and fourth rows by 2b and subtract them from the second row to assume that b = 0: Since a is odd, we can multiply the rst column by an appropriate integer and subtract it from the third and fourth columns to assume that c = 0 as well. This puts the ~ matrix in the form 0 1 a=4jqj d=jqj 0 0 2d=jQj B C 0 0 1=4 0 A =4 The two blocks decouple; the lower block gives rise to the group Z 4 Z 4 and the upper block gives rise to the group Z=(8jQj) Z=(8?1 jqj): 4 The calculation of spin (BZ=2 ) Let n = 2 and let n = 2?1 for 2: Let L be the Hopf line bundle over CP 1 : Let M(n; a 1 ; a 2 ) and f(a 1 ; a 2 ) be as in (2.12) and (2.13). Then (M(n; a 1 ; a 2 ))() = 2Z=n?f1g Tr ()f(a 1 ; a 2 )(): (4:1) Proof of Theorem 1.2 if n = 4 or n = 8 : We compute this shows range(~) Z 4 if n = 4: We compute (M(4; 1; 1))( 1? 0 ) = 1=4; (4:2) (M(8; 1; 1))( 1? 0 ) = 7=8; (M(8; 1; 1))( 2? 0 ) = 8=8 (4:3) (M(8; 1; 3))( 1? 0 ) = 0; (M(8; 1; 3))( 2? 0 ) = 1=2: (4:4) This shows range(~) Z 8 Z 2 if n = 8: 9

10 We prove the general case of Theorem 1.2 by induction. Let ind( s ) = s + s+n : R(Z=n)! R(Z=n): (4:) Lemma 4.1 Let 2 R 0 (Z=n) and ~ 2 R 0 (Z=n): (a) (M(n; a 1 ; a 2 ))( ind(~)) = (M(n; a 1 ; a 2 ))(~): (b) (M(n; a 1 ; a 2 ) + M(n; a 1 ; a 2 + n))() = (M(n; a 1 ; a 2 ))(j Z=n ): (c) (M(n; a 1 ; a 2 ))( ind(~)j Z=n ) = 2(M(n; a 1 ; a 2 ))(~): Proof: If f and g are complex valued functions on Z=n; dene (f; g) n = n?1 2Z=n f()g(): (4:6) Since s + s+n = 0 on Z=n? Z=n and s + s+n = 2 s on Z=n; (f; ind( s )) n = (f; s ) n ; (4:7) this is Frobenius reciprocity. We prove (a) by computing (M(n; a 1 ; a 2 ))( ind(~)) = (f(a 1 ; a 2 ); ind(~)) n (4:8) = (f(a 1 ; a 2 ); ~) n = (M(n; a 1 ; a 2 ))(~): Since f(a 1 ; a 2 ) + f(a 1 ; a 2 + n) = f(a 1 ; a 2 )( 0 + n ); we prove (b) by computing (M(n; a 1 ; a 2 ) + M(n; a 1 ; a 2 + n))() (4:9) = (f(a 1 ; a 2 ); ( 0 + n )) n = (f(a 1 ; a 2 ); j Z=n ) n We prove (c) by noting that ind(~)j Z=n = 2~: Let = (M(n; a 1 ; a 2 ))(j Z=n ): (4:10) M(n) := spanfm(n; a 1 ; a 2 )g spin (BZ=n); (4:11) ~() := (()( 1? 0 ); :::; ()( n?1? 0 )) 2 R n?1 =Z n?1 ; (4:12) R(n) := f~(m) : M 2 M(n)g; (4:13) K(n) := fm 2 M(n) : ~(M) = 0g: (4:14) We complete the proof of Theorem 1.2 by proving: 10

11 Lemma 4.2 Let 4; let n = 2 ; and let n = 2?1 : (a) ~ : spin (BZ=n)! R(n)is an isomorphism. (b) jr(n)j = 2 2?2 summands. and the group R(n) has exactly two non-trivial cyclic (c) R(n) = Z=2 Z=2?2 : Proof: Let (M) := (M)(ind( 1? 0 ); :::; ind( n?1? 0 )): By Lemma 4.1 (a), denes a surjective map : R(n) = M(n)=K(n)! R(n)! 0: (4:1) Let ` be the number of non-trivial summands in the Abelian 2-group R(n); iterating the above argument shows R(n) admits a surjective map to R(8) so ` 2: Choose M i 2 R(n) for 1 i ` so that By Lemma 4.1 (b), we may choose M i 2 M(n) so that Then jspan i f~( M i )gj 2`: Furthermore span i f~(m i )g = (Z=2)`: (4:16) ( M i )() = (M i )(j Z=n ): (4:17) ( M i )(ind(~)) = (M i )(ind(~)j Z=n ) = 2(M i )(~) = 0 (4:18) so M i 2 ker(): If n > 16 we use induction and if n = 16 we use the case n = 8 which was checked separately to see jr(n)j 2 2(?2) : We use Lemma 2.4 to estimate j spin (BZ=n)j jr(n)j = jm(n)=k(n)j jker()j jr(n)j (4:19) 2` 2 2(?2) 2 (?1) j spin (BZ=n)j: Consequently all the inequalities must have been inequalities so ~ : spin (BZ=n)?!R(n) = and ` = 2: (4:20) This completes the proof of (a) and (b) for all values of n: Thus we may choose a() b() so that R(2 ) = Z=2 a() Z=2 b() : (4:21) Since we have a short exact sequence 0! Z 2 Z 2! R(2 +1 )! R(2 )! 0 (4:22) we can conclude a(+1) = a()+1 and b(+1) = b()+1 and assertion (c) follows from the case n = 8: 11

12 References 1. M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry I, II, III, Math. Proc. Cambr. Phil. Soc. 77 (197) 43? 69, 78 (197) 40? 432, 79 (1976) 71? A. Bahri, M. Bendersky, D. Davis, and P. Gilkey, The complex bordism of groups with periodic cohomology, Trans. AMS V316 (1989), B. Botvinnik, P. Gilkey, and S. Stoltz, The Gromov-Lawson-Rosenberg conjecture for groups with periodic cohomology (preprint). 4. D. Bayen and R. Bruner, The real connective K-homology of BG for groups G with Q 8 as Sylow 2-subgroup, to appear Transactions of the AMS.. P. Gilkey, The eta invariant and the K-theory of odd dimensional spherical space forms, Invent. Math. 76 (1984), P. Gilkey, Invariance Theory, the heat equation, and the Atiyah- Singer index theorem 2 nd Ed CRC press (December 94). 7. P. Gilkey, The geometry of spherical space form groups, World Scientic Press (1980). B. Botvinnik botvinn@poincare.uoregon.edu. P. Gilkey gilkey@math.uoregon.edu. Research partially supported by NSF grant DMS , by MSRI (NSF grant DMS ), and by IHES (France). 12