By Luck W., Schick T.

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17 imply 3) and the vanishing of the L2 -cohomology of Dp∗ [M ] and Dp∗ [TR−1 ]. 5. MR is L2 -acyclic for all R > 0. Proof. 1) together with ˜ R ] vanishes. 17 show that also the L -cohomology as it is usually defined is trivial. 6. 1) at ∗ = p independently of R. Proof. 1. 12). 10). 11) observe |ω|M |2H 1 + |ω|E R R−1 |2H 1 ≤ 2 |ω|2L2 + |dω|2L2 + |δω|2L2 + i∗ (∗ω) 2 H 1/2 (∂ MR ) + i∗ (∗ω) = 2|ω|2H 1 + 2 H 1/2 (∂ ER−1 ) 2 i∗ (∗ω) H 1/2 (∂ M ) R + i∗ (∗ω) 2 . H 1/2 (∂ ER−1 ) Therefore, we only have to deal with restriction to the boundary.

But also (y, d∗ x) ∗ ∞ because δHabs L2 = 0 because d x ∈ δCabs and 0 ∞ ∞ dC0 ⊥ δCabs . 19] implies now 1 . e. dx, δx ∈ L2 and i∗ (∗x) = 0. that x ∈ Hloc abs Now dx = 0 ∈ L2 ; δx = (d + δ)x = d∗ x ∈ L2 . dx=0 ¨ W. LUCK AND T. SCHICK 554 (x, dδy) = (d∗ x, δy) = (δx, δy) = (x, dδy) ± GAFA ˜ ∂N ˜) δy ∧ ∗x ∀y ∈ C0∞ (N =⇒ i∗ (∗x) = 0 . Clearly δd|D(d∗ d) = ∆ = ∆⊥ . e. that ∞ = D d∗ (d| D(∆) ∩ δCabs δC ∞ ) . abs 2 = {x ∈ H 2 ; i∗ (∗x) = 0 = i∗ (∗dx)} ⊂ D(d∗ d). Now D(∆) = Habs 1 ∩ δC ∞ and dx ∈ H 1 .

Then i∗ (∗ω) = 2 H 1/2 (∂ MR ) 2 ∗ i (∗χR ω) H 1/2 (∂ M ) R ≤ C2 TR−1 ∪TR |χR ω|2H 1 (M) ≤ C 2 TR−1 ∪TR Cχ2R |ω|2H 1 . 12 which map TR−1 ∪ TR to T0 ∪ T1 interchange χR and χ1 . Because the Sobolev norms are defined locally in terms of the geometry, the existence of these isometries shows that we can choose CT ∪T CχR independently of R. 11). 12) is similar. 13), for every ω1 ⊕ ω2 ∈ L2 (MR ) ⊕ L2 (ER−1 ) with ω1 |TR−1 = ω2 |TR−1 we must find an element ω ∈ L2 (M ) with j(ω) = ω1 ⊕ ω2 and |ω|2L2 ≤ |ω1 |2L2 + |ω2 |2L2 .