Connes-Chern Character for Manifolds with Boundary and ETA ...

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64 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM Recall that a relative K-theory class in K 0 (M, ∂M) is represented by a pair of bundles (E, F ) over M whose restrictions E ∂ , F ∂ to ∂M are related by a homotopy h. We will explicitly write the formulæ in the even dimensional case and only point out where the odd dimensional case is different. The Chern character of [E, F, h] ∈ K 0 (M, ∂M) is then represented by the relative cyclic homology class ( ) ( ) ch • [E, F, h] = ch • (p F ) − ch • (p E ) , − T/ch • (h) , (7.11) cf. Eq. (1.17). By Theorem 6.1 we have for any t > 0 〈〉〈( [D], [E, F, h] = b ch n t (D), ch n+1 t (D ∂ ) ) , ch • ([E, F, h]) 〉 = 〈 ∑ b Ch n−2j (tD) + B b T/ch n+1 t (D), ch • (p F ) − ch • (p E ) 〉 j≥0 − 〈 ∑ j≥0 Ch n−2j+1 (tD ∂ ) + B T/ch n+2 t (D ∂ ), T/ch • (h) 〉 . Letting t ↘ 0 yields, again by Theorem 6.1, the local form of the pairing: 〈 [D], [E, F, h] 〉 ∫ = b M ω D ∧ ( ch • (p F ) − ch • (p E ) ) ∫ − ∂M ω D∂ (∂M) ∧ T/ch • (h). (7.12) (7.13) If D ∂ is invertible then, at the opposite end, letting t ↗ ∞ gives in view of Theorem 6.2 〈〉〈 ∑ [D], [E, F, h] = κ 2k (D) + B b T/ch n+1 ∞ (D), ch •(p F ) − ch • (p E ) 〉 0≤k≤l − 〈 B T/ch n+2 ∞ (D ∂), T/ch • (h) 〉 , (7.14) where 2l = n. By equating the above two limit expressions (7.13) and (7.14) one obtains the following identity: Corollary 7.7. Let n = 2l ≥ m and assume that D ∂ is invertible. Then 〈 κ 0 (D), p F − p E 〉 + + ∑ (−1) k (2k)! 〈 κ 2k (D), (p F − 1 k! 2 ) ⊗ p⊗2k F 1≤k≤l ∫ = ω D ∧ ( ch • (p F ) − ch • (p E ) ) ∫ − ω D∂ ∧ T/ch • (h) b M − (−1) n/2 n! (n/2)! + 〈 B T/ch n+2 ∞ (D ∂), T/ch n+1 (h) 〉 . ∂M 〈 B b T/ch n+1 ∞ (D), (p F − 1 2 ) ⊗ p⊗n F − (p E − 1 2 ) ⊗ 〉 p⊗2k E − (p E − 1 2 ) ⊗ 〉 p⊗n E (7.15)
CONNES-CHERN CHARACTER AND ETA COCHAINS 65 The left hand side plays the role of a ‘higher’ relative index, while the right hand side contains local geometric terms and ‘higher’ eta cochains. The pairing formula acquires a simpler form if one chooses special representatives for the class [E, F, h]. For example, one can always assume that E ∂ = F ∂ , in which case one obtains 〈[D], [E, F, h0 ] 〉 = 〈 ∑ κ 2k (D) + B b T/ch n+1 ∞ (D), ch •(p F ) 〉 0≤k≤l − 〈 ∑ 0≤k≤l κ 2k (D) + B b T/ch n+1 ∞ (D), ch •(p E ) 〉 . (7.16) Specializing even more, one can assume F = C N . Then the pairing formula becomes 〈 [D], [E, C N , h 0 ] 〉 = − 〈 ∑ κ 2k (D) + B b T/ch n+1 ∞ (D), ch •(p E ) 〉 0≤k≤l + N(dim Ker D + − dim Ker D − ). On the other hand, applying Theorem 7.6 〈 [D], [E, C N , h 0 ] 〉 = − Ind APS (p E D + p E ) + N Ind APS D + , (7.17) one obtains an index formula for the b-Dirac operator which is the direct analogue of Eq. (3.4) in [CoMo93]: Corollary 7.8. Let E be a vector bundle on M whose restriction to ∂M is trivial and assume D ∂ to be invertible. Then for any n = 2l ≥ m Ind APS (p E D + p E ) = 〈 ∑ κ 2k (D) + B b T/ch n+1 ∞ (D), ch •(p E ) 〉 . (7.18) 0≤k≤l The expression Ind APS (p E D + p E ) is to be understood as follows: if p ∂ E D ∂p ∂ E is invertible, then it is the Fredholm index of p E D + p E . If p E D + p E is not Fredholm, then chose a metric ˜g smooth up to the boundary and construct on the Clifford module of D the Dirac operator ˜D to the Riemannian metric ˜g. Then, by Theorem. 6.1 the Connes–Chern characters of ˜D and D coincide and thus 〈 [D], [E, C N , h 0 ] 〉 = − Ind APS p ˜D+ E p E + N Ind ˜D+ APS . As a by-product of the above considerations, we can now establish the following generalization of the Atiyah-Patodi-Singer odd-index theorem for trivialized flat bundles (comp. [APS76, Prop. 6.2, Eq. (6.3)]. Corollary 7.9. Let N be a closed odd dimensional spin manifold, and let E ′ , F ′ be two vector bundles which are equivalent in K-theory via a homotopy h. With D g ′ denoting the Dirac operator associated to a Riemannian metric g ′ on N, one has ∫ ξ(D F ′ g ) − ′ ξ(DE′ g ) = Â(∇ 2 ′ g ′) ∧ T/ch • (h) + SF(h, D g ′) , (7.19) N or equivalently, ∫ 1 1 d ( η(ph(t) D g ′ p h(t) ) ) ∫ dt = Â(∇ 2 g 2 dt ′) ∧ T/ch •(h) , (7.20) 0 N
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CONNES-CHERN CHARACTER FOR MANIFOLD
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CONNES-CHERN CHARACTER AND ETA COCH
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∫ b M CONNES-CHERN CHARACTER AND
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where CONNES-CHERN CHARACTER AND ET
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