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

58 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM
Given a de Rham current C of degree j then C defines naturally a cochain ˜C ∈
C j ( b C ∞ (M)) by putting ˜C(a 0 , . . . , a j ) := 1 j! 〈C, a 0da 1 ∧ · · · ∧ da j 〉. One has b ˜C = 0 and
B ˜C = ˜∂C, where ∂ is the codifferential. Because of this identification we will from now
on omit the ∼ from the notation if no confusion is possible.
Given a closed b-differential form ω on M of even degree. By C ω we denote the de
Rham current ∫b ω ∧ −. There is a natural pullback M ι∗ ω at ∞ (cf. Definition and
Propostion 1.5), which is a closed even degree form on ∂M. We now find
∫
∫
∫
〈∂C ω , α〉 = ω ∧ dα = d(ω ∧ α) = ι ∗ (ω ∧ α) =
b M
b M
∂M
∫
(6.21)
= ι ∗ (ω) ∧ ι ∗ (α) = 〈C ι ∗ ω, ι ∗ (α)〉.
∂M
In view of Subsection 1.2 this means that the pair (C ω , C ι ∗ ω) is a relative de Rham cycle
or via the above mentioned identification between de Rham currents and cochains that
(˜C ω , ˜C ι ∗ ω) is a relative cocycle in the relative total complex Eq. (6.14).
If ω = ∑ j≥0 ω 2j with closed b-differential forms of degree 2j then the pair (ω, ι ∗ ω)
still gives rise to a relative cocycle of degree dim M in the relative total complex.
These considerations certainly apply to the even degree forms b ω D , ω D∂ which satisfy
ι ∗ ( b ω) = ω D∂ . The limit results (6.18), (6.19), (6.5) can then be summarized as
lim
t→0+
( b ˜ch
k
t (D), ch k−1
t (D ∂ ) ) =
( ∫b M
∫
b ω D ∧ −,
∂M
)
ω D∂ ∧ − . (6.22)
The limit on left is understood pointwise for each component of pure degree.
Finally we need to relate ( k
b ˜ch t (D), ch k−1
t (D ∂ ) ) to the Chern character of [D] ∈
K m (M, ∂M). First recall from Eq. (1.11) that HP •( C ∞ (M), C ∞ (∂M) ) is naturally
isomorphic to HP •( J ∞ (∂M, M) ) . Under this isomorphism, the class of the
pair ( k
b ˜ch t (D), ch k−1
t (D ∂ ) ) is mapped to b k
˜ch t (D) |J ∞ (∂M,M), just because elements of
J ∞ (∂M, M) vanish on ∂M. We note in passing that by (A.5) a smooth function f
on M ◦ lies in J ∞ (∂M, M) iff in cylindrical coordinates one has for all l, R and every
differential operator P on ∂M
∂ l xDf(x, p) = O(e Rx ),
x ↦→ −∞.
In view of (6.17), the class of ( k
b ˜ch t (D), ch k−1
t (D ∂ ) ) in HP •( J ∞ (∂M, M) ) equals
that of ˜Cb ω D
. As explained in Section 1.8, HP •( J ∞ (∂M, M) ) is naturally isomorphic
to H• dR (M, ∂M; C). Under this isomorphism, ˜Cb ω D
corresponds to the relative
de Rham cycle ( ) Cb ω D
, C ωD∂ . Finally, note that under the Poincaré duality isomorphism
H• dR (M, ∂M; C) ∼ = H dR • (M \ ∂M; C), the relative de Rham cycle ( )
Cb ω D
, C ωD∂
is mapped onto the closed form b ω D . This line of argument shows that the class of
( k b ˜ch t (D), ch k−1
t (D ∂ ) ) depends only on the absolute de Rham cohomology class of the
closed form b ω D = c·Â(b ∇g) 2 on the open manifold M \∂M. The transgression formula
in Chern–Weil theory shows that the (absolute) de Rham cohomology class of  ( )
∇ 2 g
is independent of the metric g on M ◦ . Thus the de Rham class of b ω D equals that of