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

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56 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM By Eq. (6.11) we have ( ) b ch k t (D) − b k ˜ch t (D), ch k+1 t (D ∂ ) − ch k−1 t (D ∂ ) ( ) = − T/ch k t (D ∂) ◦ i ∗ , −(b + B) T/ch k t (D ∂) = (˜b + ˜B) ( ) 0, T/ch k t (D ∂) , (6.15) hence the two pairs differ only by a coboundary. Next let us compute ( b ch k+2 t (D), ch k+3 t (D ∂ ) ) − S ( b ch k t (D), ch k+1 t (D ∂ ) ) in the above relative cochain complex. Using Eq. (6.4) one checks immediately that b ch k+2 t (D) − b ch k t (D) = b Ch k+2 (tD) + B b T/ch k+3 t (D) − B T/ch k+1 t (D) = − (b + B) T/ch k+1 t (D) − T/ch k+2 t (D ∂ ) ◦ i ∗ . From the second line of Eq. (6.4) (or from [CoMo93, Sec. 2.1]) one thus gets ch k+3 t (D ∂ ) − ch k+1 t (D ∂ ) = −(b + B) T/ch k+2 t (D ∂ ), ( b ch k+2 t (D), ch k+3 t (D ∂ ) ) − S ( b ch k t (D), ch k+1 t (D ∂ ) ) = (˜b + ˜B) ( − b T/ch k+1 t (D), T/ch k+2 t (D ∂ ) ) . (6.16) Hence, the relative cocycles ( b ch k+2 t (D), ch k+3 t (D ∂ ) ) and S ( b ch k t (D), ch k+1 t (D ∂ ) ) are cohomologous. Similarly, one gets b ch k t (D) − ch k τ(D) = resp. = ∑ ( b Ch k−2j (tD) − b Ch k−2j (τD) ) ∫ t + B b /ch k+1 (sD, D) ds j≥0 τ = − (b + B) ∑ ∫ t b /ch k−2j−1 (sD, D) ds − ∑ ∫ t /ch k−2j (sD ∂ , D ∂ ) ds, j≥0 τ j≥0 τ ch k+1 t (D ∂ ) − ch k+1 τ (D ∂ ) = −(b + B) ∑ j≥0 ∫ t τ /ch k−2j (sD ∂ , D ∂ ) ds, hence ( b ch k t (D), ch k+1 t (D ∂ ) ) and ( b ch k τ(D), ch k+1 τ (D ∂ ) ) are cohomologous in the total relative complex as well. Thus, we have proved (1)-(3) of the following result. ( Theorem 6.1. (1) The pairs of retracted cochains b ch k t (D), ch k+1 t (D ∂ ) ) , ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) , t > 0, t > 0, k ≥ m = dim M, k − m ∈ 2Z are cocycles in the relative total complex Tot • ⊕ BC •,• (C ∞ (M), C ∞ (∂M)). (2) They represent the same class in HC n (C ∞ (M), C ∞ (∂M)) which is independent of t > 0. (3) They represent the same class in HP • (C ∞ (M), C ∞ (∂M)) which is independent of k.
lim t→0+ CONNES-CHERN CHARACTER AND ETA COCHAINS 57 (4) Denote by b ω D , ω D∂ the local index forms of D resp. D ∂ [BGV92, Thm. 4.1], see Eq. (6.17) below. Then one has a pointwise limit ( b k ˜ch t (D), ch k−1 t (D ∂ ) ) ∫ ) = b ω D ∧ −, ω D∂ ∧ − . ( ∫b M Moreover, ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) represents the Connes–Chern character of [D] ∈ KK m (C 0 (M); C) = K m (M, ∂M). The pointwise limit will be explained in the proof below. Up to normalization constants b ω D is the Â form Â(b ∇ 2 g) and ω D∂ is the Â form Â(∇2 g ∂ ) on the boundary. Note also that ι ∗ ω D = ω D∂ . Proof. It remains to prove (4). So consider a 0 , a 1 , · · · ∈ b C ∞ (M ◦ ). Using Getzler’s asymptotic calculus (cf. [Get83], [CoMo90, §3], and [BlFo90, Thm. 4.1]) one shows that the local heat invariants of a 0 e −σ 0tD 2 [D, a 1 ]e −σ 1tD 2 . . . [D, a j ]e −σ jtD 2 (cf. Proposition 5.2) satisfy ∫ ( t j str q,Wp a 0 e −σ 0tD 2 [D, a 1 ]e −σ 1tD 2 . . . [D, a j ]e −σ jtD 2) (p, p) d vol gb (p) ∆ j = 1 (6.17) ( b ) ω D ∧ a 0 da 1 · · · ∧ da j j! + |p O(t1/2 ), t → 0 + . Here str q,Wp denotes the fiber supertrace in W p , q indicates the Clifford degree of D, cf. Subsection 1.7, the factor 1 j! is the volume of the simplex ∆ j . This statement holds locally on any Riemannian manifold for any choice of a self-adjoint extension of a Dirac operator. So it holds for D and accordingly for D ∂ . From Theorem 5.3 and its well-known analogue for closed manifolds, cf. [CoMo93, Sec. 4], we thus infer ∂M lim b Ch j (tD)(a 0 , . . . , a j ) t→0+ resp. = 1 ∫ b ω D ∧ a 0 da 1 · · · ∧ da j , a j ∈ b C ∞ (M ◦ ), (6.18) j! b M lim t→0+ Chj−1 (tD ∂ )(a 0 , . . . , a j−1 ) = 1 (j − 1)! Furthermore, in view of (6.5) we have for k ≥ dim M − 1 ∫ ∂M ω D∂ ∧ a 0 da 1 · · · ∧ da j , a j−1 ∈ C ∞ (M). (6.19) lim b T/ch k+1 t→0+ t (D)(a 0 , . . . , a k+1 ) = 0, a j ∈ b C ∞ (M ◦ ), lim t→0+ T/chk t (D ∂)(a 0 , . . . , a k ) = 0, a j ∈ C ∞ (M). (6.20) To interpret these limit results we briefly recall the relation between de Rham currents and relative cyclic cohomology classes over ( b C ∞ (M), C ∞ (M)), cf. also [LMP08, Sec. 2.2].
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CONNES-CHERN CHARACTER FOR MANIFOLD
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CONNES-CHERN CHARACTER AND ETA COCH
Page 5 and 6: CONNES-CHERN CHARACTER AND ETA COCH
Page 7 and 8: ∫ b M CONNES-CHERN CHARACTER AND
Page 11 and 12: where CONNES-CHERN CHARACTER AND ET
Page 13 and 14: where the map ι(h) is defined by C
Page 21 and 22: ∫ This means that b M CONNES-CHER
Page 53 and 54: Applying (5.9) to A 1 we get e −
Page 55: CONNES-CHERN CHARACTER AND ETA COCH
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