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

where the map ι(h) is defined by
CONNES-CHERN CHARACTER AND ETA COCHAINS 13
ι(h)(a 0 ⊗ a 1 ⊗ . . . ⊗ a l )
l∑
= (−1) i (a 0 ⊗ . . . ⊗ a i ⊗ h ⊗ a i+1 ⊗ . . . ⊗ a l ).
i=0
(1.16)
A relative K-theory class in K 0 (A, B) can be represented by a triple (p, q, h) with
projections p, q ∈ Mat N (A) and h : [0, 1] → Mat N (B) a smooth path of projections
with h(0) = σ(p), h(1) = σ(q) (cf. [HiRo00, Def. 4.3.3], see also [LMP09, Sec. 1.6]).
The Chern character of (p, q, h) is represented by the relative cyclic cycle
( ) )
ch • p, q, h =
(ch • (q) − ch • (p) , − T/ch •
(h) , (1.17)
where
T/ch •
(h) =
∫ 1
0
/ch •
(
h(s), (2h(s) − 1) ḣ(s) ) ds. (1.18)
That the r.h.s. of Eq. (1.17) is a relative cyclic cycle follows from the transgression
formula Eq. (1.14). From a secondary transgression formula [LMP09, (1.43)] one deduces
that (1.17) indeed corresponds to the standard Chern character on K 0 (J ) under
excision.
1.3.2. Odd case. The odd case parallels the even case in many aspects. Given an
element g ∈ GL ∞ (A) := lim GL N(A), the odd Chern character is the following normalized
periodic cyclic cycle:
N→∞
∞∑
(
ch • (g) = (−1) k k! tr 2k+1 (g −1 ⊗ g) ⊗(k+1)) . (1.19)
k=0
If (g s ) 0≤s≤1 is a smooth path in GL ∞ (A), the transgression formula (cf. [Get93b,
Prop. 3.3]) reads
d
ds ch •(g s ) = (b + B) /ch •
(g s , ġ s ), (1.20)
where the secondary Chern character /ch •
is defined by
/ch •
(g, h) = tr 0 (g −1 h)+ (1.21)
∞∑
k∑ (
+ (−1) k+1 k! tr 2k+2 (g −1 ⊗ g) ⊗(j+1) ⊗ g −1 h ⊗ (g −1 ⊗ g) ⊗(k−j)) .
k=0
j=0
A relative K-theory class in K 1 (A, B) can be represented by a triple (U, V, h), where
U, V ∈ Mat N (A) are unitaries and h : [0, 1] → Mat N (B) is a path of unitaries joining
σ(U) and σ(V ). Putting
T/ch •
(h) =
∫ 1
0
/ch •
(
hs , ḣs)
ds, (1.22)
the Chern character of (U, V, h) is represented by the relative cyclic cycle
( ) )
ch • U, V, h =
(ch • (V ) − ch • (U) , − T/ch •
(h) . (1.23)