METHODOLOGY

Since the cokernels of (2.29) and (2.30) have equal dimension the image of (2.29) is precisely the orthogonal complement of H^m₊ in C_δ^k,α(Λ^m), i.e.

C_δ^k,α(Λ^m) = H^m₊ ⊕ △

C_δ^k+2,α(Λ^m)⊕ρH^m_∞⊕ρtH^m_∞ . Since H^m₊ is finite-dimensional the projections are bounded.

Now let G be a Fredholm inverse for (2.29) and forβ ∈C_δ^k,α(Λ^m) define PEβ =dd^∗Gβ,

P_E^∗β =d^∗dGβ.

PE andPE^∗ map intoC_δ^k,α[dΛ^m−1] andC_δ^k,α[d^∗Λ^m+1] respectively, andβ =PE(β) +PE^∗(β) for any β ∈ △

C_δ^k,α(Λ^m)⊕ρH^m_∞⊕ρtH^m_∞

. Therefore

△

C_δ^k+2,α(Λ^m)⊕ρH^m_∞⊕ρtH^m_∞

=C_δ^k,α[dΛ^m−1]⊕C_δ^k,α[d^∗Λ^m+1] with projections PE and PE^∗. The result follows.

Definition 2.3.29. Let H_cpt^∗ (M) be the compactly supported cohomology of M, i.e. the cohomology of the cochain complex

Ω⁰_cpt(M)−→^d Ω¹_cpt(M)−→^d Ω²_cpt(M)−→ · · ·^d of compactly supported smooth forms on M.

Recall that whenM has cylindrical ends then we can compactifyM by ‘adding a copy of X at infinity’, i.e. by including it in M = M0 ∪(X×[0,∞]). The cohomology (with coefficients R) of M relative to its boundary can be identified with H_cpt^∗ (M). The long exact sequence for relative cohomology of M can be written as

· · · −→H^m−1(X)−→^∂ H_cpt^m(M)−→^e H^m(M) ^j

∗

−→H^m(X)−→ · · · (2.32) e : H_cpt^m(M) →H^m(M) is induced by the inclusion Ω^∗_cpt(M) ֒→Ω^∗(M). The image of e is the subspace of de Rham cohomology classes with compact representatives.

Definition 2.3.30. Let H₀^m(M) = im e:H_cpt^m(M)→H^m(M) .

In the asymptotically cylindrical setting the map j^∗ : H^m(M) → H^m(X) can be de-scribed as follows: For s ∈ R⁺ let js : X ֒→ M be the inclusion x 7→ (x, s) ∈ M∞. The maps js are homotopic, so they all give the same map j^∗ :H^m(M)→H^m(X).

If α is an asymptotically translation-invariant m-form let B(α) denote its asymptotic limit, which is a translation-invariant form on the cylinder X×R. B(α) can be written as Ba(α) +dt∧B_e(α), where Ba(α),Be(α) are forms onXof degreemandm−1 respectively.

If α is a closed asymptotically translation-invariant m-form then for any m-cycle C in X

Z

C

j_s^∗([α])→ Z

C

[B_a(α)]

as s→ ∞, so

j^∗([α]) = [Ba(α)]. (2.33)

By proposition 2.3.24 any α ∈ H^m₀ is exponentially asymptotically translation-invariant, and B(α)∈ H^m_∞. By remark 2.3.22

H^m_∞∼=H_X^m⊕dt∧ H^m−1_X , so we get maps B_a :H^m₀ → H^m_X, B_e :H^m₀ → H^m−1_X .

Definition 2.3.31. Let

H^m_abs= kerBe⊆ H^m₀ , H_rel^m = kerBa ⊆ H^m₀ .

Further let H^m_E,H_E^m∗ ⊆ H^m₀ denote the spaces of bounded exact and coexact harmonic forms respectively.

It follows immediately from (2.33) that H_E^m ⊆ H^m_rel. It follows by applying the Hodge star that H^m_E^∗ ⊆ H^m_abs (using remark 2.3.28 ifM is not orientable).

Let Ω^∗_−δ(M) be the cochain complex of forms β such that e^−δtβ is uniformly bounded with all derivatives, and let H_−δ^∗ (M) be its cohomology.

Lemma 2.3.32. For δ >0 the natural map H_−δ^∗ (M)→H^∗(M) is an isomorphism.

Proof. The inclusion i: Ω^∗_−δ(M) ֒→ Ω^∗(M) is a chain map, so induces a well-defined map i:H_−δ^∗ (M)→H^∗(M).

Let ρ : R → [0,1] be a smooth function with ρ(t) = 0 for t ≤ 0 and ρ(t) = 1 for t ≥ 1. Define a map c : M → M which is the identity on the compact piece M0 and (x, t) 7→ (x,(1−ρ(t))t+ρ(t) arctant) on the cylindrical part. c is smooth, and c^∗ maps Ω^∗(M) to Ω^∗_−δ(M). It is a chain map, so induces a mapc^∗ :H^∗(M)→H_−δ^∗ (M). We deduce that i and c^∗ are inverses from the fact that cis homotopic to the identity.

The next theorem is part of [42, Theorem 6.18].

Theorem 2.3.33. Let M be an EAC manifold. The natural map H^m_abs → H^m(M) is an isomorphism.

Proof. Note that by integration by parts the elements of H^m₀ are closed, so represent cohomology classes. Pick some 0< δ < ǫ₁. By the previous lemma it suffices to check that H_abs^m →H_−δ^m(M) is an isomorphism.

We first prove injectivity. Ifα∈ H^m_abs and α =dβ for some β ∈Ω^m_−δ(M) thenα ∈ H^m₊, since H^m_E ⊆ H^m_rel and H^m_abs∩ H^m_rel=H^m₊. As α is exponentially decaying with rateδ we can integrate by parts to deduce α = 0. ThusH_abs^m →H_−δ^m(M) is an injection.

It remains to show that if γ ∈Ω^m_−δ(M) is closed then γ is cohomologous to an element of H_abs^m . C_−δ^0,α(Λ^m) =H^m₊ ⊕ △C_−δ^2,α(Λ^m) by proposition 2.3.16. Thusγ can be written as

γ =φ+△ψ

with φ ∈ H^m₊ and ψ ∈C_−δ^2,α(Λ^m). Let χ=dψ. Then

dγ = 0⇒dd^∗χ= 0 ⇒ △χ= 0.

By proposition 2.3.23,χ∈C_δ^2,α(Λ^m+1)⊕ρH^m+1_∞ ⊕ρtH^m+1_∞ sod^∗χ∈ H^m_E^∗ ⊆ H^m_abs. Therefore γ −dd^∗ψ =φ+d^∗χ∈ H^m_abs (2.34) represents the same cohomology class as γ. Hence H^m_abs →H_−δ^m(M) is surjective.

Proposition 2.3.34. H^m_abs =H^m₊ ⊕ H^m_E^∗, H^m_rel =H^m₊ ⊕ H^m_E. Furthermore H^m_E =dH^m−1₋ , H_E^m∗ =d^∗H^m+1₋ .

Proof. Theorem 2.3.33 implies that H^m_E ∩ H^m₊ = 0. Similarly H^m_E^∗∩ H₊^m = 0, so the sums are direct. If γ ∈ H^m₀ then, since γ is closed, (2.34) gives

γ =φ+d^∗χ+dd^∗ψ,

whereφ∈ H^m₊,d^∗χ∈d^∗H^m+1₋ . Analogously, sinceγ is also coclosed,dd^∗ψ ∈dH₋^m−1. Hence H^m₀ =H^m₊ ⊕d^∗H₋^m+1⊕dH^m−1₋ .

Since dH₋^m−1 ⊆ H^m_E ⊆ H^m_rel and d^∗H^m+1₋ ⊆ H^m_E^∗ ⊆ H^m_abs the result follows.

As a corollary of theorem 2.3.33 we can determine that the image of the space H^m₊ of L² harmonic forms in the de Rham cohomologyH^m(M) is precisely the subspaceH₀^m(M) of compactly supported classes. Similar results (with more general hypotheses) appears in e.g. [1, Proposition 4.9] and [38, Theorems 7.6 and 7.9].

Theorem 2.3.35. Let M be an EAC manifold. The natural map H^m₊ → H^m(M) is an isomorphism onto H₀^m(M).

Proof. H₊^mis kerBainH^m_abs, and it follows from theorem 2.3.33 it is mapped isomorphically to H₀^m(M) = kerj^∗ ⊆H^m(M).

The fact that the image of H₊^m is contained in H₀^m(M) could be seen more explicitly by applying the following lemma, which is useful for other purposes.

Ifα is a closed exponentially asymptotically translation-invariantm-form onM we can write it as α∞ +βt +dt∧γt on the cylindrical part X ×R⁺, where α∞ is translation-invariant, and βt, γt are sections of Λ^mT^∗X, Λ^m−1T^∗X respectively, and are exponentially

decaying in t. Let

η(α) =ρ Z ∞

t

γsds, (2.35)

where ρ is a smooth cut-off function for the cylindrical end of M, equal to 1 for t > t0. η(α) is a well-defined exponentially decaying (m−1)-form on M.

Lemma 2.3.36. Let M be a manifold with cylindrical ends, and α a closed exponentially asymptotically translation-invariant m-form onM. Thenα+dη(α)is translation-invariant on {y ∈M :t > t0}.

Proof. Closure of α means that _∂t^∂βt+d_Xγt = 0 where d_X is the exterior derivative on X.

Thus βt =−R∞

t dXγsds. Since X is compact the dominated convergence theorem ensures that for t > t0

dη =dXη+dt∧ _∂t^∂η=−β_t−dt∧γt.

Definition 2.3.37. LetA^m =Ba(H^m₀ )⊆ H^m_X, E^m =Be(H^m+1₀ )⊆ H^m_X, and letA^m, E^m be the subspaces of H^m(X) that they represent (A^m is of course justj^∗(H^m(M))⊆H^m(X)).

When Mⁿ is oriented the Hodge star on M identifies H^m_abs and H^m−n_rel . Ifβ ∈ H^m₀ then Be(∗β) =∗Ba(β), so the Hodge star onX identifiesA^m with E^n−m−1 (and henceA^m with E^n−m−1).

By proposition 2.3.23 anyψ ∈ H₋^m can be written as χ+α+tβ+dt∧γ+tdt∧δ with χ exponentially decaying, α, β ∈ H^m_X and γ, δ ∈ H^m−1_X . Thus we can define a ‘boundary data’ map

BD :H^m₋ →(H^m_X)²⊕(H^m−1_X )², ψ 7→ β δ α γ

! .

Let BDa, BDe be the composition of BD with the projection to (H_X^m)² and (H_X^m−1)² respectively. Let ˜A^m = BDa(H^m₋) and ˜E^m−1 = BDe(H₋^m). The following proposition is a refinement of [42, Lemma 6.15].

Proposition 2.3.38. Let M be an EAC manifold.

(i) If ψ1, ψ2 ∈ H^m₋ with BD(ψi) = βi δi

αi γi

!

then in the L² inner product on X

<α1, β2>_L² =<α2, β1>_L², (2.36a)

<γ1, δ2>L² =<γ2, δ1>L² . (2.36b)

(ii) BD(H₋^m) = ˜A^m ⊕E˜^m, and A˜^m ⊂ (H^m_X)² and E˜^m−1 ⊂ (H^m−1_X )² are Lagrangian subspaces.

Proof. First assume that M is oriented. Then ∂ : H^m(X) → H_cpt^m+1(M) is the Poincar´e dual of j^∗ :H^n−m+1(M)→H^n−m(X), so

<αi, βj>L² = ([αi],[∗β_j])X = ([αi], j^∗[∗dψ_j])X = (∂[αi],[∗dψ_j])M. Note that ∂[αi] = [d(ψi+η(dψi)−ρtβi)]∈H_cpt^m+1(M). Hence

<α1, β2>L² −<α2, β1>L² = Z

M

(dψ1 −d(ρtβ1))∧ ∗dψ₂−(dψ2−d(ρtβ2))∧ ∗dψ₁

= Z

M

d(ρt(β2∧ ∗dψ1−β1∧ ∗dψ2)). (2.37) Since

β₂∧B(∗dψ₁)−β₁∧B(∗dψ₂) = β₂∧ ∗β₁−β₁ ∧ ∗β₂ = 0

the integrand in the RHS of (2.37) is the exterior derivative of an exponentially decaying form. The vanishing of the integral proves (2.36a), and (2.36b) follows by applying ∗.

This proves (i) in the oriented case. WhenM is not orientable we use remark 2.3.28.

(i) implies that ˜A^m ⊂(H^m_X)² and ˜E^m−1 ⊂(H^m−1_X )² are null spaces. In particular dim ˜A^m ≤b^m(X), dim ˜E^m−1 ≤b^m−1(X).

Since BD(H^m₋)⊆A˜^m⊕E˜^m−1 and has dimensionb^m−1(X) +b^m(X) equality must hold, so A˜^m and ˜E^m−1 are Lagrangian.

Proposition 2.3.39. If Mⁿ is an EAC manifold with cross-section X then H^m_X =A^m⊕ E^m

is an orthogonal direct sum.

Proof. Ifψ ∈ H^m₋ with BDa(ψ) = β1

α1

!

∈A˜^m thenβ1 =Be(dψ)∈ E^m. Thus the second projection ˜A^m → H^m_X has image E^m, and kernel A^m, so

dimA^m+ dimE^m = dim ˜A^m = dimH^m_X

Furthermore if α2 ∈ A^m then 0 α2

!

∈ A˜^m, and (2.36b) implies that <β1, α2> = 0, so A^m and E^m are orthogonal.

It follows from proposition 2.3.39 that we can define an isomorphism H^m_rel →H_cpt^m(M), α7→

( [α+dη(α)] for α∈ H^m₊

∂([Be(α)]) for α∈ H^m_E .

Corollary 2.3.40. Let Mⁿ be an asymptotically cylindrical manifold which has a single end (i.e. the cross-section X is connected). Then e:H_cpt¹ (M)→H¹(M) is injective.

In particular H¹_E = 0, and H¹₀ →H¹(M) is an isomorphism.

Proof. Consider the start of the long exact sequence for relative cohomology H_cpt⁰ (M)→H⁰(M)→H⁰(X)→^∂ H_cpt¹ (M)→^e H¹(M).

The dimensions of the first three terms are 0,1, and 1, so ∂ = 0, and thus e is injective.

H_rel¹ ↔H_cpt¹ (M) identifies H¹_E with kere, so the result follows.

Finally we make two simple observations for the case whenM is Ricci-flat, as it is when the holonomy is G2 orSpin(7).

Proposition 2.3.41. If M is a Ricci-flat EAC manifold then H¹₀ is the space of parallel 1-forms on M. In particular H¹₊= 0, and j^∗ :H¹(M)→H¹(X) is injective.

Proof. This is proved by the same standard ‘Bochner argument’ as corollary 2.1.9, adapted to the EAC setting. If φ is a 1-form then by proposition 2.1.8

△φ =∇^∗∇φ. (2.38)

It follows that any parallel 1-form φis harmonic, and parallel forms are of course bounded.

To show that any bounded harmonic form is parallel we use (2.38) together with an integration by parts argument, justified by lemma 2.3.15.

H¹₊ = 0 since it consists of parallel decaying forms. By theorem 2.3.35, the kernel H₀¹(M) of j^∗ :H¹(M)→H¹(X) is represented by H₊¹.

Corollary 2.3.42. If M is a Ricci-flat EAC manifold with a single end thenH_cpt¹ (M) = 0.

Proof. Follows from proposition 2.3.41 and corollary 2.3.40.