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

12 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
note that the ideal J ∞ (∂M, M) is H-unital, since ( J ∞ (∂M, M) ) 2
= J ∞ (∂M, M)
(cf. [BrPf08]). Hence excision holds true for the ideal J ∞ (∂M, M), and any of the
above cohomology theories of J ∞ (∂M, M) coincides with the corresponding relative
cohomology of the pair ( C ∞ (M), E ∞ (∂M, M) ) . In particular, we have the following
chain of quasi-isomorphisms
Tot • ⊕ BC •,•( J ∞ (∂M, M) ) ∼ qism Tot • ⊕BC •,•( C ∞ (M), E ∞ (∂M), M ) ∼ qism
∼ qism Tot • ⊕ BC •,• (C ∞ (M)) ⊕ Tot •+1
⊕ BC •,• (E ∞ (∂M, M)).
Next recall from [BrPf08] that the map
(1.11)
Tot k ⊕ BC per(C •,• ∞ (M)) → Tot k ⊕BC per(E •,• ∞ (∂M, M)),
( (E
ψ ↦→
∞ (∂M, M) ) ˆ⊗k+1
∋ F0 ⊗ . . . ⊗ F k ↦→ ψ ( ) F 0‖X ⊗ . . . ⊗ F k‖X
between the periodic cyclic cochain complexes is a quasi-isomorphism. As a consequence
of the Five Lemma one obtains quasi-isomorphisms
(
Tot • ⊕ BC per
•,• J ∞ (∂M, M) ) ∼ qism Tot • ⊕BC per( •,• C ∞ (M), E ∞ (∂M) ) ∼ qism
(1.12)
∼ qism Tot • ⊕ BC per(C •,• ∞ (M)) ⊕ Tot •+1
⊕ BC per(C •,• ∞ (∂M)).
In this paper we will mainly work with the relative complexes over the pair of algebras
(
C ∞ (M), C ∞ (∂M) ) , because its cycles carry geometric information about the boundary,
which is lost when considering only cycles over the ideal J ∞ (∂M, M). In this respect we
note that periodic cyclic cohomology satisfies excision by [CuQu93, CuQu94], hence
in the notation of (1.3), HP • (J ) is canonically isomorphic to HP • (A, B).
1.3. The Chern-character. For future reference, we recall the Chern character and
its transgression in cyclic homology, both in the even and in the odd case.
1.3.1. Even case. The Chern character of an idempotent e ∈ Mat ∞ (A) :=
lim
N→∞ Mat N(A) is the class in HP 0 (A) of the cycle given by the formula
ch • (e) := tr 0 (e) +
∞∑
k=1
(−1) k (2k)!
k!
( (e 1) tr 2k − ⊗ e
⊗(2k))
, (1.13)
2
where e ⊗(2k) is an abbreviation for the 2k–fold tensor product e ⊗ · · · ⊗ e, and tr 2k
denotes the generalized trace map Mat N (A) ⊗j −→ A ⊗j .
If (e s ) 0≤s≤1 is a smooth path of idempotents, then the transgression formula reads
d
ds ch •(e s ) = (b + B) /ch •
(e s , (2e s − 1)ė s ); (1.14)
here the secondary Chern character /ch •
is given by
/ch •
(e, h) := ι(h) ch • (e), (1.15)