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

CONNES-CHERN CHARACTER AND ETA COCHAINS 63
t
1
SF ( γ·(0), D ∂
)
SF ( γ 1 (·), D ∂
)
SF ( γ·(1), D ∂
)
0 1
SF ( )
γ 0 (·), D ∂
s
Figure 4. By homotopy invariance the signed sum of the four spectral
flows adds up to 0.
Swan’s Theorem) every relative K-theory class can even be represented by a triple
(p, q, p |∂M ) such that p |[0,ε)×∂M = q |[0,ε)×∂M and hence γ(s) = p |∂M is constant.
Then the twisted version of the APS Index Theorem gives
∫
Ind APS qD + q − Ind APS pD + p = ω D ∧ ( ch • (q) − ch • (p) ) , (7.10)
where ω D denotes the local index density of D. Note that since the tangential operators
of pD + p and of qD + q coincide the η-terms cancel.
As outlined in Section 1.8 (cf. also the proof of Theorem 6.1) the Connes–Chern
character of [D] in HP •( J ∞ (∂M, M) ) ≃ H• dR (M \ ∂M; C) is represented by ∫ ω M D.
By construction, the form ch • (q) − ch • (p) is compactly supported in M \ ∂M. Thus
the right hand side of (7.10) equals the pairing 〈[D], [p, q, p |∂M ]〉 and the first equality
in (7.8) is proved.
To prove the second equality in (7.8) we note that it represents the Poincaré duality
pairing between the de Rham cohomology class of ω D(∇,g) (note ι ∗ ω D = ω D∂ ) and the
relative de Rham cohomology class of the pair of forms (ch • (q)−ch • (p), T/ch •
(h)). Hence
it depends only on the class [p, q, h] ∈ K 0 (M, ∂M) and on [D]. In the situation above
where p and q coincide in a collar of the boundary it equals 〈[D], [p, q, γ]〉 and hence by
homotopy invariance the claim is proved in general up to Eq. (7.9).
For the proof of Eq. (7.9) note first that for a closed even b-differential form ω the
map (cf. Definition and Proposition 1.5)
∫ ∫
Ω k (M) ⊕ Ω k−1 (∂M) → C, (η, τ) ↦→ ω ∧ η − ι ∗ ω ∧ τ
descends naturally to a linear form on HdR k (M, ∂M; C). Hence the right hand side of
Eq. (7.9) is well-defined and depends only on the class of [p, q, h] ∈ K 0 (M, ∂M). As
before we may therefore specialize to (p, q, p |∂M ) such that p |[0,ε)×∂M = q |[0,ε)×∂M . Then
ch • (q) − ch • (p) has compact support in M \ ∂M and the remaining claim follows from
Theorem 6.1 (4).
□
We now proceed to express the pairing between relative K-theory classes and the
fundamental relative K-homology class in cohomological terms. We assume here that
we are in the b-setting.
M
b M
∂M