RELATIVE INDEX PAIRING AND ODD INDEX THEOREM FOR EVEN DIMENSIONAL MANIFOLDS

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1 RELATIVE INDEX PAIRING AND ODD INDEX THEORE FOR EVEN DIENSIONAL ANIFOLDS ZHIZHANG XIE Abstract. We prove an analogue for even dimensional manifolds of the Atiyah- Patodi-Singer twisted index theorem for trivialized flat bundles. We show that the eta invariant appearing in this result coincides with the eta invariant by Dai and Zhang up to an integer. We also obtain the odd dimensional counterpart for manifolds with boundary of the relative index pairing by Lesch, oscovici and Pflaum. Introduction In this article, we will prove an analogue for even dimensional manifolds of the Atiyah-Patodi-Singer twisted index theorem for trivialized flat bundles over odd dimensional closed manifolds [3, Proposition 6.2], and some related results. For notational simplicity, we will restrict the discussion mainly to spin manifolds. However all results can be straightforwardly extended to general manifolds. Unless we specify otherwise, we always fix the Riemannian metric for each manifold in this article and use the associated Levi-Civita connection to define its characteristic classes. To motivate the subject matter of this paper, we begin by recalling the APS twisted index theorem for odd dimensional closed manifolds in the following form, cf. [12, Corollary 7.9]. For p s s 1 k C N, s [, 1], a smooth path of projections over N, one has 1 1 d 2 ds ηp sdp s ds = N ÂN Tch p s. Here p s Dp s is the Dirac operator twisted by p s, ηp s Dp s its η-invariant, ÂN the Â-genus form of N and Tch p s is the Chern-Simons transgression form of p s s 1, cf. Section 3. To prove our analogue for even dimensional closed manifolds, we shall replace a path of projections by a path of unitaries. The more interesting issue is what should replace the η-invariant appearing on the left hand side of the above formula. To answer this, let us first consider the case where the manifold in question bounds, that is, it is the boundary of some spin manifold. In this case, the η-invariant by Dai and Zhang [9, Definition 2.2] is the right candidate, cf. Section 6 below. Indeed, suppose the even dimensional manifold Y is the boundary of a spin manifold X and 2 athematics Subject Classification. 58Jxx; 46L8. Key words and phrases. APS twisted index theorem, manifolds with boundary, relative index pairing. The author was partially supported by the US National Science Foundation awards no. DS

2 2 ZHIZHANG XIE U s s 1 is the restriction to Y of a smooth path of unitaries over X. Denote the η-invariant of Dai and Zhang by ηy, U s for each s [, 1], then 1 1 d 2 ds ηy, U sds = ÂY Tch U s..1 When Y bounds, it follows from the cobordism invariance of the index of Dirac operators that IndD + =, where D + is the restriction of the Dirac operator over Y to the even half of the spinor bundle according to its natural Z 2 -grading. The condition IndD + = is crucial for the definition of the η-invariant by Dai and Zhang, however is often not satisfied by even dimensional closed spin manifolds in general. To cover the general case, we shall use another approach where we lift the data to S 1 Y. The main ingredient of the method of proof is using an explicit formula of the cup product K 1 S 1 K 1 Y K S 1 Y, inspired by the Powers-Rieffel idempotent construction, cf. [15]. In fact, the formula given for the case when Y = S 1 by Loring in [14] also works for all manifolds in general, cf. Section 2 below. Our analogue for even dimensional closed spin manifolds of the APS twisted index theorem Theorem 4.1 below is as follows. Theorem I. Let Y be an even dimensional closed spin manifold and U s s 1 U k C Y a smooth path of unitaries over Y. For s [, 1], e s 2k C S 1 Y is the projection defined as the cup product of U s with the generator e 2πiθ of K 1 S 1. Let D S1 Y be the Dirac operator over S 1 Y. Then 1 1 d 2 ds ηe sd S 1 Y e s ds = ÂY Tch U s..2 The formula for e s is given in Section 2. A priori, the η-invariants in the formulas.1 and.2 appear to be different, we however will show that they are equal to each other modulo Z Theorem 5.7 below in the case where Y bounds. Theorem II. Suppose Y is the boundary of an odd dimensional spin manifold. If U U k C Y is a unitary over Y and e U is the cup product of U with [e 2πiθ ] K 1 S 1, then one has ηy, U = ηe U D S1 Y e U mod Z. The method of proof is based on a slight generalization of a theorem by Brüning and Lesch [6, Theorem 3.9], see Proposition 5.6 below. In this sense, ηe U D S1 Y e U can be thought of as the extension to general even dimensional manifolds of the definition of the η-invariant by Dai and Zhang. The same technique used above also allows us to prove the following analogue Theorem 6.3 below for odd dimensional manifolds with boundary of the relative index pairing formula by Lesch, oscovici and Pflaum [12, Theorem 7.6]. Suppose is an odd dimensional spin manifold with boundary. By a relative K-cycle [U, V, u s ] K 1,, we mean U, V U n C are two unitaries over with u s U n C, s [, 1], a smooth path of unitaries over such that u = U and u 1 = V. We denote by T U, resp. T V, the Toeplitz operator on with respect to U, resp. V see Section 5 for details. Theorem III. Let [U, V, u s ] be a relative K-cycle in K 1,. If U and V are constant along the normal direction near the boundary, then Ind [D] [U, V, u s ] = IndT V IndT U + SF u 1 s D [,1] u s ; P us Y Y

3 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 3 where SF u 1 s D [,1] u s ; P us is the spectral flow of the path of elliptic operators u 1 s D [,1] u s ; P us, s [, 1], with Atiyah-Patodi-Singer type boundary conditions determined by P us as in 5.4. This uses Dai and Zhang s Toeplitz index theorem for odd dimensional manifolds with boundary [9]. We shall give the details in Section 6. It should be mentioned that, although the objects we work with are from classical geometry, the spirit of the proofs is very much inspired by methods from noncommutative geometry, cf. [8]. A brief outline of the article is as follows. In Section 1, we recall some results about index pairings for manifolds with boundary. Section 2 is devoted to the explicit formula of the cup product in K-theory mentioned earlier. This allows us to carry out explicit calculations for Chern characters in Section 3. With these preparations, we prove an analogue for even dimensional manifolds of the APS twisted index theorem in Section 4. In Section 5, we show the equality of the two a priori different eta invariants. In the last Section, we prove the odd-dimensional counterpart of the relative index pairing formula by Lesch, oscovici and Pflaum [12, Theorem 7.6]. Acknowledgements. I am greatly indebted to Henri oscovici for his continuous support and advice. This paper grew out of numerous conversations with him. I want to thank Nigel Higson for helpful suggestions. I am grateful to Alexander Gorokhovsky for a careful reading of the first version of this paper as well as for many helpful comments. I started working on this problem during my visit at the Hausdorff Center for athematics in Bonn, Germany. I want to express my thanks to the center for its hospitality and to atthias Lesch for the invitation, as well as for generously sharing with me his insights into the subject. 1. Relative Index Pairing Let be a compact smooth manifold with boundary. Following [4, Sec. 2], consider an elliptic first order differential operator D : C c \, E C c \, E where Cc \, E is the space of compactly supported smooth sections of the Hermitian vector bundle E. Such an operator has a number of extensions to become a closed unbounded operator on H = L 2 \, E, e.g. D min and D max the minimum extension and the maximum extension respectively. Consider D e a closed extension of D such that that is, DD min DD e DD max. Let and D min D e D max, 1.1 B = D e D e F e = BB /2 T = T with T = D e D ed e + 1 1/2 and T = D ed e D e + 1 1/2. Denote by C \ the space of continuous functions vanishing at infinity. Then the -representation of C \ on H H given by scalar multiplication, together with F e, defines

4 4 ZHIZHANG XIE an element in KKC \, C, see [4] for the precise construction. Such a K-homology class turns out to be independent of the choice of a closed extension of D [4, Proposition 2.1 ], and will be denoted [D]. Similarly for each formally symmetric elliptic operator, one constructs a cycle in KKC \, Cl 1 [4, Section 2], where Cl 1 is the Clifford algebra with one generator. For each [D] KKC \, Cl, one has the index pairing map Ind [D] : K \ Z. An element in K \ is represented by a triple E, F, α with E, F vector bundles over \ and α : E F a bundle homomorphism whose restriction near infinity is an isomorphism, cf. [1]. oreover, we can choose connections over the bundles E and F such that the forms Ch E and Ch F coincide near infinity. Under this assumption, one can write down an explicit formula for the index pairing map: Ind [D] [E, F, α] = ω D [Ch E Ch F ]. Here ω D HdR even \ is the dual of the Chern character of the K-homology class [D], as explained in the introduction of Chap. I in [7]. In the case where is a spin manifold and D the Dirac operator over, one has ω D = Â. Similarly, in the odd case, an element in K 1 \ consists of two unitaries U and V over \ and a homotopy h between U and V near infinity. oreover, we can assume that U and V are identical near infinity and the homotopy h becomes the identity map near infinity, cf. e.g.[1, Prop ], in which case the index pairing map has the following cohomological expression 1 : Ind [D] [V, U, h] = ω D [Ch V Ch U]. Note that the boundary data are conspicuously absent in the above formulas. Indeed, by definition, K \ is essentially the reduced K-group of the one point compactification of \. The information from the boundary is therefore completely eliminated from the picture. In order to recover that, we shall turn to the relative K theory of the pair,, denoted K,, cf. [12]. A relative K-cycle [p, q, h s ] K, is a triple where p, q n C are two projections over and h s n C, s [, 1], is a path of projections over such that h = p and h 1 = q. Similarly, a relative K-cycle [U, V, u s ] K 1, is a triple where U, V U n C are two unitaries over with u s U n C, s [, 1], a smooth path of unitaries over such that u = U and u 1 = V. First notice that K, = K \. Hence the above index pairing induces a map Ind [D] : K, Z. The issue now is to find an explicit formula which incorporates geometric information of the boundary. For even dimensional manifolds with boundary, this is done by Lesch, oscovici and Pflaum[12, Theorem 7.6]. We shall give an analogous formula for odd dimensional manifolds with boundary in the Section 6. 1 We adopt the negative sign here in order to be consistent with our sign convention throughout the article.

5 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 5 2. Cup Product in K-theory Let A and B be local Fréchet algebras. The cup product between K 1 A and K 1 B is defined by : K 1 B K 1 A = K SB K SA K SB SA = K B A where SA resp. SB is the suspension of A resp. B, the isomorphism is the Bott isomorphism and K SB K SA K SB SA is given by [p] [q] = [p q]. 2.1 In the case where B = C S 1, we shall give an explicit formula of this cup product. Since e 2πiθ is a generator of K 1 C S 1 = Z, it suffices to give this formula for [e 2πiθ ] [U] with U U k A. Lemma 2.1 cf. [14]. With the above notation, [e 2πiθ ] [U] = [e U ] f g + hu where e U = hu + g 1 f 2k C S 1 A is a projection with f, g and h nonnegative functions on S 1 satisfying the following conditions 1 f 1, 2 f = f1 = 1 and f1/2 =, 3 g = χ [,1/2] f f 2 1/2 and h = χ [1/2,1] f f 2 1/2. Proof. It is not difficult to see that : K 1 C S 1 K 1 A K C S 1 A is the same as the standard isomorphism [16, Section 7.2] Θ A : K 1 A K SA K C S 1 A after identifying K 1 C S 1 with Z. The inverse of this map is constructed as follows, cf. [1, Proposition 4.8.2][16, Section 7.2]. The group K SA is generated by formal differences of normalized loops of projections over A. Such a loop is a projection-valued maps p : [, 1] n A with p = p1 n C. For each loop, there is a path of unitaries u : [, 1] U n A such that pt = utp1ut 1 and u = 1 n. Without loss of generality, we can assume p = p1 =. v This implies that u1 is of the form. Then one checks that [p] [v] is a w well-defined inverse to Θ A. To see that our formula agrees with the usual definition, it suffices to show that Θ 1 1 A e U = U. First notice that e U = e U 1 = and e U θ is a projection over A for each θ S 1 = R/Z, hence e U is a normalized loop of projections. Now consider the following path of unitaries over A, f1 θ + f Uθ = 2 θu 1 f 1/2 θ 1 f 1/2 θ f 1 θ f 2 θu

6 6 ZHIZHANG XIE where f 1 = χ [,1/2] f 1/2 and f 2 = χ [1/2,1] f 1/2. In particular, U = U U1 = U. By a direct calculation, one verifies 1 e U θ = Uθ Uθ from which the lemma follows. 1 and 1 We will also make use of the following lemma in Section 3, cf. [14, Lemma 2.2]. Lemma 2.2. For f, g and h nonnegative functions on S 1 = R/Z satisfying the following conditions 1 f 1, 2 f = f1 = 1 and f1/2 =, 3 g = χ [,1/2] f f 2 1/2 and h = χ [1/2,1] f f 2 1/2, we have 1 Proof. Notice that 1 f h 2k dθ = [ 2 4fh h 2k 1 + 4f h 2k] dθ = 1 1/2 and integration by parts gives 1 fθ f 2 θ k dfθ = 2 4fh h 2k 1 dθ = 2 k 1 1 k 1!k 1! 2k 1! x x 2 k dx = k!k! 2k + 1!, f h 2k dθ. 3. Chern Characters and Transgression Formulas Throughout this section, although we deal with commutative algebras, we shall use the similar formalism for the Chern character in K-theory as in cyclic homology [7, Chap. II], [13, Chap. VIII]. Let be a compact smooth manifold with or without boundary. The even resp. odd Chern characters of projections resp. unitaries in n C can be expressed as follows. For p n C such that p 2 = p and p = p, Ch p := trp + 1 k 1 1 2πi k k! tr pdp 2k HdR even. 3.1 k=1 For U U n C, 1 k! Ch U := 2πi k+1 2k + 1! tr U 1 du 2k+1 HdR odd. 3.2 k= For each U U n C, let e U be the projection as in Lemma 2.1. If no confusion is likely to arise, we also write e instead of e U.

7 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 7 Lemma 3.1. Ch e U = k=1 1 k 4f 2πi k h 2k + 2 4fh h 2k 1 dθ tru 1 du 2k 1 k! Proof. Notice that f de = g + h U h U + g f dθ + which implies trede 2k f g + hu = tr hu + g 1 f j=2k f g + hu + tr hu + g 1 f j=1 f g + h U h du h du, 2k h du h du dθ j 1 h du h du 3.3 2k j h du h du. 3.4 Since most of the matrices appearing in the above summation only have off diagonal entries, a straightforward calculation gives the following equalities. 3.3 = h 2k tr U 1 du 2k 3.4 = 1 k k 2 4fh h 2k 1 + 4f h 2k dθ tru 1 du 2k 1. On the other hand, tru 1 du 2k = tr U 1 duu 1 du 2k 1 from which it follows that 3.3 vanishes. This finishes the proof. As a consequence of lemma 2.2 and lemma 3.1, one has the following corollary. From now on, integration along the fiber S 1 will be denoted by π. Corollary 3.2. π Ch e U = Ch U. Consider a smooth path of unitaries U s U n C with s [, 1], or equivalently U U n C [, 1]. The secondary Chern character Ch U s is given by the formula 1 k! Ch U s := 2πi k+1 2k! tr Us 1 U s Us 1 du s 2k. k= Then Ch U can be decomposed as Ch U = Ch U s + ds Ch U s where ChU s see 3.2 above and Ch U s do not contain ds. Applying de Rham differential to both sides gives us the following transgression formula s Ch U s = d Ch U s. Similarly, if e s m C is a smooth path of projections, or equivalently a projection e m C [, 1], then Ch e = Ch e s + ds Ch e s.

8 8 ZHIZHANG XIE with Ch e s := 1 k πi k+1 k! tr 2e s 1ė s de s 2k+1. k= Applying Corollary 3.2 to Ch U and Ch e U, one has π Ch e U = Ch U, which implies ds π Ch e s = ds Ch U s. Denote the Chern-Simons transgression forms by Tch e s s 1 := Tch U s s 1 := 1 1 ds Ch e s, ds Ch U s. We summarize the results of this section in the following proposition. Proposition 3.3. Consider U U n C and U s U n C for s [, 1]. Let e, resp. e s, be the cup product of U, resp. U s, with e 2πiθ a generator of K 1 S 1 as in Lemma 2.1. Then π Ch e = Ch U and π Tch e s s 1 = Tch U s s Odd Index Theorem on Even Dimensional anifolds In this section, we shall prove our analogue for even dimensional closed manifolds of the APS twisted index theorem. Let us first recall the APS twisted index theorem and fix some notation. Let N be a closed odd dimensional spin manifold and D/ its Dirac operator. If p is a projection in n C N, then p induces a Hermitian vector bundle, denoted E p, over N. With the Grassmannian connection on E p, let pd/ I n p be the twisted Dirac operator with coefficients in E p. For notational simplicity, we also write pd/p instead of pd/ I n p. Then by [12, Corollary 7.9], for p s k C N a smooth path of projections over N, one has ξp 1 D/p 1 ξp D/p = ÂN Tch p s + SFp s D/p s s where N ξp i D/p i = ηp id/p i + dim kerp i D/p i 2 the reduced eta invariant of p i D/p i. Here SFp s D/p s s 1 is the spectral flow of p s D/p s s 1. Notice that the vector bundle on which p s D/p s acts may vary as s moves along [, 1]. To make sense of the definition of such a spectral flow, we introduce a path of unitaries u s U n C N over N with u s p u s = p s so that p u sd/u s p acts on the same vector bundle E p. SFp s D/p s s 1 is then defined to be SFp u sd/u s p s 1 the spectral flow of the family p u sd/u s p s 1. Now by [11, Lemma 3.4], formula 4.1 is equivalent to 1 1 d 2 ds ηp sd/p s ds = ÂN Tch p s. 4.2 N

9 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 9 Theorem 4.1. Let Y be a closed even dimensional spin manifold and U s s 1 U k C Y a smooth path of unitaries over Y. For s [, 1], let e s 2k C Y be the projection defined as the cup product of U s with the generator e 2πiθ of K 1 S 1. Let D S1 Y be the Dirac operator over S 1 Y. Then 1 1 d 2 ds ηe sd S 1 Y e s ds = ÂY Tch U s. 4.3 Proof. Applying formula 4.1 to S 1 Y, one has ξe 1 D S 1 Y e 1 ξe D S 1 Y e = ÂS 1 Y Tch e s + SFe s D S 1 Y e s. S 1 Y Notice that ÂS1 = π 1ÂS1 π 2Â and ÂS1 = 1, where π 1 : S 1 S 1, resp. π 2 : S 1, is the projection from S 1 to S 1, resp.. By Proposition 3.3, the integral on the right side is equal to Y ÂY Tch U s. Now the formula 1 1 d 2 ds ηe sd S 1 Y e s ds = ÂY Tch U s Y follows from the equality [11, Lemma 3.4] ξe 1 D S 1 Y e 1 ξe D S 1 Y e = SFe s D S 1 Y e s + Y 1 1 d 2 ds ηe sd S 1 Y e s ds. Remark 4.2. od Z, the reduced η-invariant ξe s D S 1 Y e s is equal to the reduced η-invariant ξy, U s defined by Dai and Zhang, cf. [9, Definition 2.2], at least when Y bounds. See Theorem 5.7 below. 5. Equivalence of Eta Invariants Throughout this section, we assume is an odd dimensional spin manifold with boundary. Denote by S the spinor bundle over. Let D be the Dirac operator over, then near the boundary d D = cd/dx dx + D where D is the Dirac operator over and cd/dx is the Clifford multiplication by the normal vector d/dx. Then D I n is the Dirac operator acting on S C n, when we use the trivial connection on the bundle C n over. If no confusion is likely to arise, we shall write D instead of D I n. Now a subspace L of ker D is Lagrangian if cd/dxl = L ker D. In our case, since bounds, the existence of such a Lagrangian subspace follows from the cobordism invariance of the index of Dirac operators. Let L 2 >S C n be the positive eigenspace of D, i.e. the L 2 -closure of the direct sum of eigenspaces with positive eigenvalues of D. Then the projection P := P L = P + P L imposes an APS type boundary condition for D, where P, resp. P L, is the orthogonal projection L 2 S C n L 2 >S C n, resp. L 2 S C n L. Let us denote the corresponding self-adjoint elliptic operator by D P.

10 1 ZHIZHANG XIE Let L 2 S C n ; P be the nonnegative eigenspace of D P and P P the orthogonal projection P P : L 2 S C n L 2 S C n ; P. ore generally, for each unitary U U n C over, the projection UP U 1 imposes an APS type boundary condition for D and we shall denote the corresponding elliptic self-adjoint operator by D UP U 1. Similarly, let P UP U be the 1 orthogonal projection P UP U 1 : L2 S C n L 2 S C n ; UP U 1 where L 2 S C n ; UP U 1 is the nonnegative eigenspace of D UP U 1. With the above notation, we define the Toeplitz operator on with respect to U as follows, cf.[9, Definition 2.1]. Definition 5.1. T U := P UP U 1 U P P. Dai and Zhang s index theorem for Toeplitz operators on odd dimensional manifolds with boundary [9, Theorem 2.3] states that IndT U = Â Ch U ξ, U + τ µ UP U 1, P, P 5.1 where P is the Calderón projection associated to the Dirac operator D on cf. [5] and τ µ UP U 1, P, P is the aslov triple index [11, Definition 6.8]. The reduced η-invariant ξ, U will be defined after the remarks. Remark 5.2. Notice that the integral in 5.1 differs from Dai and Zhang s by a constant coefficient 2πi dim +1/2. This is due to the fact that our definition of characteristic classes follows topologists convention, i.e. factors such as 1 2πi k/2 are already included. Remark 5.3. The aslov triple index τ µ UP U 1, P, P is an integer. For unitaries U, V U n C, if there a path of unitaries u s U n C with s [, 1] such that u = U and u 1 = V, one has cf.[11, Lemma 6.1]. τ µ UP U 1, P, P = τ µ V P V 1, P, P, To define ξ, U, let us first consider D [,1] the Dirac operator over [, 1]. If no confusion is likely to arise, we shall write U for both U and the trivial lift of U from to [, 1]. Let [,1] := D [,1] + 1 ψu 1 [D [,1], U] 5.2 over [, 1], where ψ is a cut-off function on [, 1] with ψ 1 near {} and ψ near {1}. With APS type boundary conditions determined by P on {} and Id U 1 P U on {1}, [,1] becomes a self-adjoint elliptic operator, denoted [,1] ; P U. See proposition 5.6 for an explanation of the choice of notation. Similarly, [,1] t := D [,1] + 1 tψu 1 [ D [,1], U ]. 5.3 Denote by [,1] t; P U the elliptic operator [,1]t with boundary condition P U. Note that [,1]1 = Dψ,U [,1].

11 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 11 Definition 5.4. [9, Definition 2.2] ξ, U := ξ [,1] ; P U SF [,1] t; P U t 1 where dim ξ kerdψ,u [,1] [,1] = ; P U + η [,1] ; P U. 2 Remark 5.5. ξ, U is independent of the cut-off function ψ [9, Proposition 5.1]. In order to show the equality ξ, U = ξe U D S1 e U mod Z, we need to relate the operator e U D S1 e U to [,1], where D S 1 = cd/dθ d dθ + D is the Dirac operator over S 1 and e U is the cup product of U with e 2πiθ K 1 S 1. Recall that f g + hu DS1 e U D S 1 e U = hu f g + hu + g 1 f D S1 hu + g 1 f 1 = U U DS 1 1 U U D S1 where f 1/2 U = 1 + f 1/2 2 U 1 f 1/2 1 f 1/2 f 1/2 1 f 1/2 2 U with f 1 = χ [,1/2] f and f 2 = χ [1/2,1] f. Then viewed as an operator over [, 1], U e U D S1 e U U = D [,1] + f 2 U 1 [ D [,1], U ] with the boundary condition β, x = Uβ1, x, for x and β Γ[, 1] ; S C n. Let H := L 2 {} ; S C n L 2 {1} ; S C n, then the above boundary condition can be written as 1 1 U 2 U 1 β =, for β H. 1 From now on, let us assume ψ = 1 f 2. In particular, one has U e U D S 1 e U U = [,1]. Now consider Pt U cos = 2 tp + sin 2 ti P cos t sin tu cos t sin tu 1 cos 2 tid U 1 P U + sin 2 tu 1 P 5.4 U for t π/4cf.[11, Equation 5.13], [6, Section 3]. This is a path of projections in BH such that P U P = Id U 1 P U and Pπ/4 U = 1 1 U 2 U 1. 1 For each t [, π/4], the Dirac operator [,1], with the boundary condition P U t, is a self-adjoint elliptic operator, denoted by [,1] ; P U t.

12 12 ZHIZHANG XIE With the above notation, we have the following slight generalization of a theorem by Brüning and Lesch [6, Theorem 3.9]. Proposition 5.6. d dt ηdψ,u [,1] ; P U t =. Proof. Following [6, Section 3], we define U U τ := U 1 = U, cd/dθ γ :=, cd/dθ D à := U 1 D. U where à is determined by Dψ,U [,1] near the boundary, by noticing that d [,1] = cd/dθ dθ + D near {} and d [,1] = cd/dθ dθ + U 1 D U near {1}. Since cd/dθ U = U cd/dθ EndS C n, it follows that τã + Ãτ = = τ γ + γτ, τ 2 = 1, τ = τ. oreover, one verifies by calculation cf. [6, Eqs to 3.13] γp t U = I Pt U γ; [Pt U, Ã2 ] = ; P U t ÃP U t = cos2t à P U t. Then by [6, Theorem 3.9], it suffices to find a unitary µ : H H such that Let This finishes the proof. µ 2 = I, µτ + τµ = µ γ + γµ = µã + õ =. µ := U U 1. Now the equality ξ, U = ξe U D S1 e U mod Z follows as a corollary. To be slightly more precise, we have the following result. Theorem 5.7. ξ, U = ξe U D S 1 e U SF [,1] ; P U t SF [,1] t; P U t 1. In particular, ξ, U = ξe U D S 1 e U mod Z.

13 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 13 Proof. By [11, Lemma 3.4], ξe U D S1 e U ξ [,1] ; P U = SF [,1] ; P U t t π/4 + π/4 d 1 dt 2 ηdψ,u [,1] ; P U t dt. The formula now follows from the definition of η, U and the proposition above. 6. Relative Index Pairing for Odd Dimensional anifolds with Boundary In this section, we shall use the Toeplitz index theorem for odd dimensional manifolds with boundary by Dai and Zhang to prove our analogue of the index pairing formula by Lesch, oscovici and Pflaum [12, Theorem 7.6]. First let us recall the even case. Let X be an even dimensional spin manifold with boundary X. We assume its Riemannian metric has product structure near the boundary. The associated Dirac operator takes of the following form D X = D D + = d dx + D X d dx + D X near the boundary, where D X is the Dirac operator over X, cf. Appendix A. Definition 6.1. Let P = χ [, D X and D + P be the elliptic operator D + with the APS boundary condition P, cf.[2]. Then Ind AP S D + := IndD + P. Recall that a relative K-cycle in K X, X is a triple [p, q, h s ] such that p, q n C X are two projections over X and h s n C X, s [, 1], is a path of projections over X such that h = p X and h 1 = q X. If p and q are constant along the normal direction near the boundary, then the relative index pairing by Lesch, oscovici and Pflaum [12, Theorem 7.6] states that Ind [DX ][p, q, h s ] = Ind AP S qd + q Ind AP S pd + p + SFh s D X h s s 1. Now let be an odd dimensional spin manifold with boundary. We assume its Riemannian metric has product structure near the boundary. The Dirac operator D over naturally induces an element in KKC \, c 1 = K 1, cf. [4, Section 2], from which one has the relative index pairing map Ind [D] : K 1, Z. 6.1 As an intermediate step, let us first show a pairing formula by using the lifted data on S 1. The method of proof is similar to the one used in proving Theorem 4.1. Denote the Dirac operator over S 1 by D and its restriction to the half-spinor bundles by D +. We shall explain in detail the structure of D near the boundary in appendix A. Lemma 6.2. For a relative K-cycle [U, V, u s ] K 1,, that is, U, V U n C are two unitaries over with u s U n C, s [, 1], a smooth path of unitaries over such that u = U and u 1 = V. If U and V are constant along the normal direction near the boundary, then Ind [D] [U, V, u s ] = Ind AP S e V D+ e V Ind AP S e U D+ e U + SFe us D S1 e us s 1

14 14 ZHIZHANG XIE Proof. A relative K-cycle [U, V, u s ] K 1, naturally induces a relative K- cycle [e U, e V, e us ] K,. By [12, Theorem 7.6], [U, V, u s ] 6.2 Ind AP S e V D+ e V Ind AP S e U D+ e U + SFe us D S1 e us s 1 is a well-defined map from K 1, to Z. We need to show that it does agree with the relative index pairing induced by that of K 1 \. As before cf. Section 1 above, we can assume U [,ɛ = V [,ɛ and u s = U = V, for all s [, 1]. It suffices to prove the lemma for representatives of relative K- cycles of this special type. Notice that such a representative also defines an element in K 1 \ by its restriction to \ and recall from Section 1 that the index map 6.1 has the following explicit formula: Ind [D] [V, U, u s ] =  [Ch V Ch U]. Now by the APS index theorem for manifolds with boundary, Ind AP S e D+ U e U = = S 1 ÂS 1 Ch e U ξe U D S 1 e U 6.3  Ch U ξe U D S 1 e U. 6.4 where the second equality follows from Proposition 3.3. There is a similar equation where we replace U by V. It follows that the image of a representative of the special type as above, under the map 6.2, is equal to  [Ch V Ch U]. This agrees with the relative index map 6.1. Using this lemma and another two lemmas below, we shall now prove our main result in this section. Theorem 6.3. For a relative K-cycle [U, V, u s ] K 1,, that is, U, V U n C are two unitaries over with u s U n C, s [, 1], a smooth path of unitaries over such that u = U and u 1 = V. If U and V are constant along the normal direction near the boundary, then where SF u 1 s D [,1] u s ; P us u 1 s Ind [D] [U, V, u s ] = IndT V IndT U + SF u 1 s D [,1] u s ; P us D [,1] u s ; P us is the spectral flow of the path of elliptic operators us, s [, 1], with APS type boundary conditions P as in 5.4.

15 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 15 Proof. By formula 5.1, we have IndT V IndT U = Â Ch V ξ, V + τ µ V P V 1, P, P + Â Ch U + ξ, U τ µ UP U 1, P, P = Â [Ch V Ch U] + ξ, U ξ, V since τ µ UP U 1, P, P = τ µ V P V 1, P, P by [11, Lemma 6.1]. Notice that ξ, U ξ, V + SF u 1 s D [,1] u s ; P us s 1 = ξe U D S1 e U SF [,1] ; P t U SF [,1] t; P U ξe V D S1 e V + SFD ψ,v [,1] ; P t V + SF D ψ,v [,1] t; P V which is equal to + SF u 1 s D [,1] u s ; P us s 1 ξe U D S 1 e U ξe V D S 1 e V + SFe us D S 1 e us s 1 by the lemmas below. Hence IndT V IndT U + SF u 1 s D [,1] u s ; P us = Â [Ch V Ch U] s 1 ξe V D S 1 e V + ξe U D S 1 e U + SFe us D S 1 e us s 1 = Ind AP S e V D+ e V Ind AP S e U D+ e U + SFe us D S1 e us s 1 which is equal to Ind [D] [U, V, u s ] by Lemma 6.2. Lemma 6.4. SF D ψ,us [,1] ; P us s 1 = SF [,1] ; P U t SFD ψ,v [,1] ; P V t + SFe us D S1 e us s 1 Proof. Consider the t, s-parametrized family of operators D ψ,us [,1] ; P t us where P us t Hence P us = t π/4 ; s 1 is defined as in Eq Note that P Id u 1 s P and u s e us D S 1 e us P us π/4 = 1 2 = D ψ,us [,1] ; P us π/4. 1 us u 1 s 1.

16 16 ZHIZHANG XIE Consider the following diagram D ψ,v D ψ,v [,1] ; P V [,1] ; P t V D ψ,us [,1] ; P us D ψ,v [,1] ; P V π/4 D ψ,us [,1] ; P us π/4 [,1] ; P U [,1] ; P U t [,1] ; P π/4 U where the arrows stand for smooth paths connecting the corresponding vertices. Now the lemma follows from the homotopy invariance of the spectral flow. Now let D ψ,us [,1] t := D [,1] + 1 tψu 1 [ ] s D[,1], u s, then the same argument above proves the following lemma. Lemma 6.5. SF u 1 s D [,1] u s ; P us = SF [,1] t; P U s 1 SF D ψ,us [,1] D ψ,v [,1] t; P V Proof. Consider the s, t-parametrized family of operators us t, P cf. the following diagram D ψ,v [,1] ; P V D ψ,us [,1], P us + SF D ψ,us [,1] ; P us s 1 t,s 1 D ψ,v [,1] t; P V D ψ,v [,1] 1; P V D ψ,us [,1] ; P us [,1] ; P U D ψ,v [,1] t; P U [,1] 1; P U Notice that D ψ,us [,1] 1 = Dψ,us [,1] and D ψ,us [,1] = u 1 s D [,1] u s. The lemma follows by the homotopy invariance of the spectral flow. Appendix A. Spinor Bundles and Dirac on anifolds with boundary The material in this appendix is well known. The purpose is to clarify the relations among various Dirac operators arising in this article for the convenience of the reader. Suppose is an odd dimensional spin manifold with boundary. Its Riemannian metric assumes a product structure near the boundary. Let S resp. S be the spinor bundle over S 1 resp.. Then ClT the Clifford

17 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 17 algebra over is identified with the even part of ClT S 1 the Clifford algebra over S 1 by c e i ce i cd/dθ. where c, resp. c, is the Clifford multiplication on S, resp. S. This way S, the restriction of S to {}, is identified with S = S,+ S, the spinor bundle over. Notice that Ŝ, the spinor bundle over [, 1 S1, is naturally isomorphic to C 2 S. Here C 2 = C + C and stands for graded tensor product. Denote the Dirac operator over [, 1 S 1 by D. Then D = d dx + i d dθ d dx + i d dθ I S + I C 2 D. We identify ClT S 1 the Clifford algebra over S 1 with the even part of ClT S 1 the Clifford algebra over S 1 by ce i ĉe i ĉd/dx for e i T S 1, where ĉ is Clifford multiplication on Ŝ. From this, one has Ŝ + = C + S,+ C S, = S,+ S, S Ŝ = C S,+ C + S, = cd/dx Ŝ +, Lemma A.1. With the idenfications of spinor bundles as above, D d = dx + D S 1 d dx + D S 1 d dx + i d dθ id S, = id S,+ d dx i d dθ d dx + i d dθ id S,+ id d S,+ dx i d dθ where D S 1 resp. D is the Dirac operator over S 1 resp.. In particular, d D S 1 = cd/dθ dθ + D with i cd/dθ = and D D = S, i D. S,+ Proof. With the identification Ŝ = ĉd/dxŝ+, one has ĉd/dx D S + = ĉd/dx ĉd/dx d dx + ĉd/dθ d dθ + ĉe i ei i = d dx ĉd/dx ĉd/dθ d dθ ĉd/dx ĉe i ei i = d dx + cd/dθ d dθ + ce i ei i = d dx + cd/dθ d dθ + c e i ei i

18 18 ZHIZHANG XIE Similarly, D S ĉd/dx = ĉd/dx d dx + ĉd/dθ d dθ + ĉe i ei ĉd/dx i = d dx + cd/dθ d dθ + c e i ei i Notice that cd/dθ d dθ + D is the Dirac operator over S 1, hence D d = dx + D S 1 d dx + D. S 1 To finish the proof, one notices that i cd/dθ = ĉd/dθ ĉd/dx = I i S i = I i S 1 I 1 S References [1]. F. Atiyah, K-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, [2]. F. Atiyah, V. K. Patodi, and I.. Singer, Spectral asymmetry and Riemannian geometry. I, ath. Proc. Cambridge Philos. Soc., , pp [3], Spectral asymmetry and Riemannian geometry. III, ath. Proc. Cambridge Philos. Soc., , pp [4] P. Baum, R. G. Douglas, and. E. Taylor, Cycles and relative cycles in analytic K- homology, J. Differential Geom., , pp [5] B. Booß-Bavnbek and K. P. Wojciechowski, Elliptic boundary problems for Dirac operators, athematics: Theory & Applications, Birkhäuser Boston Inc., Boston, A, [6] J. Brüning and. Lesch, On the η-invariant of certain nonlocal boundary value problems, Duke ath. J., , pp [7] A. Connes, Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. ath., 1985, pp [8], Noncommutative geometry, Academic Press Inc., San Diego, CA, [9] X. Dai and W. Zhang, An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary, J. Funct. Anal., , pp [1] N. Higson and J. Roe, Analytic K-homology, Oxford athematical onographs, Oxford University Press, Oxford, 2. Oxford Science Publications. [11] P. Kirk and. Lesch, The η-invariant, aslov index, and spectral flow for Dirac-type operators on manifolds with boundary, Forum ath., 16 24, pp [12]. Lesch, H. oscovici, and. J. Pflaum, Connes-Chern character for manifolds with boundary and eta cochains. [13] J.-L. Loday, Cyclic homology, vol. 31 of Grundlehren der athematischen Wissenschaften [Fundamental Principles of athematical Sciences], Springer-Verlag, Berlin, Appendix E by aría O. Ronco. [14] T. A. Loring, K-theory and asymptotically commuting matrices, Canad. J. ath., , pp [15]. A. Rieffel, C -algebras associated with irrational rotations, Pacific J. ath., , pp [16] N. E. Wegge-Olsen, K-theory and C -algebras, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, A friendly approach.

19 RELATIVE INDEX PAIRING AND ODD INDEX THEORE 19 Department of athematics, The Ohio State University, Columbus, OH, , USA address: