Construcción de las categorías y subcategorías de las entrevistas

As with the compact case, it will be important to understand the linearisation operatorDA for the instanton equation

overY×R. Firstly, we will need to understand the case whereAis a connection constant alongR, equal to the pullback of someA∈ AP. Hence we shall spend some time demonstrating how to calculate it. Throughout this section, we shall useΛk(Y)to denoteΓ(R,Ωk(Y,adP)).

We first recall some identifications. As above, we haveΩ0_g(X)= Λ∼ 0(Y)andΩ1(X,adP)∼= Λ0(Y)⊕Λ1(Y). When Y is an oriented Riemannian manifold, it has a Hodge star∗3and a volume formωY. The orientation ofY induces a

canonical orientation onX =Y ×Rgiven by taking the volume form ofXto beωX =dt∧ωY. This yields a Hodge

star∗onXwith the property that

∗(ϕ∧dt) =− ∗3ϕ (4.3)

∗ψ=− ∗3ψ∧dt (4.4)

forϕ∈Λ1(Y)andψ ∈Λ2(Y), as can easily be seen. Then the self-dual 2-forms onX(with values ing) are exactly those having the formα∧dt− ∗3αfor someα ∈Ω1_g(Y). Hence we have an identification ofΩ2g,+(X)withΛ1(Y). Therefore the linearisation operatorDA : Ω1(X,adP) →Ω0(X,adP)⊕Ω2,+(X,adP)can be instead regarded as

4.2. EXAMPLE: THE CHERN-SIMONS FUNCTIONAL 37 THEOREM 4.3. Under the above identifications, the linearisation operatorDA: Λ0(Y)⊕Λ1(Y)→Λ0(Y)⊕Λ1(Y)is given by

DA(α, β) = ( −d dt +LA ) (α, β) =−d dt(α, β) + (−d ∗ Aβ,−dAα+∗3dAβ)

Proof. Firstly, it is clear that we will have

dAα(t) =

dα

dt ∧dt+dAα

and hencedA(α) = (α′(t), dAα) ∈Λ0(Y)⊕Λ1(Y). Now we can write an inner product inΛ0(Y)⊕Λ1(Y)in the

form

⟨(η, ξ), dAω⟩=⟨(η, ξ),(ω′, dAω)⟩=⟨η, ω′⟩+⟨ξ, dAω⟩

Observe that we can write the inner product onΛk(Y)as

⟨α, β⟩= ∫ _∞

−∞(α, β)

for(α, β)the inner product onΩk(Y,adP). Hence using the (3-dimensional) adjoint ofdAand integration by parts

will yield

⟨(η, ξ), dAω⟩=⟨−η′+d∗Aξ, ω⟩

and therefore the (4-dimensional) adjointd∗_A: Λ0(Y)⊕Λ1(Y)→Λ0(Y)is given byd∗_A(α, β) =−α′+d∗_Aβ. Next we can computed+_A : Λ0(Y)⊕Λ1(Y)→Λ1(Y)by taking

d+_A(α, β) = (1 +∗)dA(αdt+β) = (1 +∗)(β′∧dt+dAβ+dAα∧dt)

essentially using the definition of the exterior covariant derivative. Expanding out the Hodge star(1 +∗)and using the formulas in equation4.3will give

(

(β′+dAα)∧dt+dAβ

)

−(− ∗3β′− ∗3dAβ∧dt− ∗3dAα

) which we can rewrite as

(( d dt− ∗3dA ) β+dAα ) ∧dt− ∗3 (( d dt − ∗3dA ) β+dAα )

Hence under the isomorphism given above,d+_A : Λ0(Y)⊕Λ1(Y)Λ1(Y)is given byd+_A(α, β) = dAα+β′− ∗3dAβ.

Finally, we see that the linearisation operatorDAis

DA(α, β) =−(d∗A⊕d

+

A)(α, β) = (−α′−d∗Aβ,−β′−dAα+∗3dAβ)

exactly as claimed. ■

Remark4.1. The above formula seems to be incorrect in [Don02], equation (2.24). The final results are entirely equiv- alent if we reverse the direction oft∈R.

It will be important to observe thatLAis a self-adjoint first-order elliptic operator. Furthermore, the following propo-

sition is not difficult to show using the techniques from Chapter 3, section 2:

PROPOSITION 4.1. [Don02, p.24] The kernel of the operatorLA: Ωg0(Y)⊕Ω1g(Y)→Ω0g(Y)⊕Ω1g(Y)is the direct sum of H0

AandHA1, the twisted cohomology groups associated toA.

We shall say that the connectionAisnon-degenerateifH_A1 is zero, and callAreducible ifH_A0 = 0(we shall see in Chapter 6 that this is equivalent to the definition of reducibility given in Chapter 3). When both are the case, the operatorLAwill be invertible and we shall callAacyclic.

Remark4.2. Usually with Morse homology we would simply have the kernel ofLAequal toHA1, which is the kernel of

the Hessian. Here the extraH_A0 term arises from the group action and measures the size of the stabiliser ofA. Now we can state the main theorem we aim to prove in this section:

THEOREM 4.4. Under the definitions of Sobolev spaces on the tubeY ×Rgiven in Appendix B,

1. WhenAis an acyclic connection, thenDA= _dd_t+LAis a bounded, invertible operatorW1,2(X)→L2(X);

2. WhenAis a connection overXwith flat acyclic limits at±∞, then the operatorDA is bounded and Fredholm onW1,2(X)→

Proof. The proof of (1) involvesseparation of variables. Firstly, using the theory of Sobolev spaces on tubular man- ifolds considered in Appendix B, we see immediately thatDAdefines a bounded linear operatorW1,2(X)→ L2(X).

Now, sinceLAis an invertible self-adjoint elliptic operator on thecompactmanifoldY, the spaceL2(Y)has an orthogo-

nal decomposition into smooth eigenfunctionsϕλofLA, with eigenvaluesλthat are real, non-zero, of finite multiplicity

and unbounded (see for instance [Nic07, Theorem 10.4.19]). To see thatDAis surjective, supposeρis a given smooth

compactly supported section onY ×R. We can decomposeρas an infinite sum over eigenfunctions ρ(t) =∑

λ

ρλ(t)ϕλ

forρλ :R→Rcompactly supported smooth functions. With respect to this eigenfunction decomposition, the equation

DAf =ρbecomes

dfλ

dt +λfλ =ρλ

if we imagine taking a similar eigenfunction decomposition off into{fλ(t)}λ. This is simply an ordinary differential

equation, with solution given explicitly by

fλ(t) =e−λt ∫ t −∞e λs_ρ λ(s)ds whenλ <0, and by fλ(t) =−e−λt ∫ _∞ t e−λsρλ(s)ds

Since all these solutions decay exponentially ast→ ±∞, it is easy to argue that the series defined by f(t) =∑

λ

fλ(t)ϕλ

converges uniformly over Y ×Rto a weak solution ofDAf = ρ. Elliptic regularity then implies thatf is actually

smooth. By the density result proved in Appendix B, it follows thatDAis surjective. It is also immediate from separation

of variables thatf = 0is the onlyL2solution toDf = 0. HenceDAis injective also.

The proof of (2) involves two steps. Firstly, we consider the case where A actually reaches the limiting flat acyclic connections at finite values of t, that is, there exists someT > 0 such that for all|t| > T, A(t) is constant at some acyclic flat connection. It is clear thatDAis still a bounded operator; to show thatDAis Fredholm, we shall show that

it is invertible modulo compact operators, it has aparametrix. The proof of this step involves a ‘patching’ argument to glue together different inverses forDAover different domains, as is done, for instance, to construct a parametrix for

an elliptic pseudodifferential operator on a compact manifold (see [LM89] for instance). See [Don02, p.50] for the full argument.

To finish the proof of (2), we now letAbe an arbitrary connection with flat acyclic limits. Then some small perturbation

A′ofAwill actually satisfy the condition of the previous step, that is, be exactly flat over the ends. Since invertibility is an open condition for operators, the operatorD_A(t)will hence still be invertible for all sufficiently large|t|. The proof

for the previous step then applies exactly as before. ■

A very similar argument to the third step above, using the invariance of the Fredholm index under perturbations, shows that the Fredholm index ofDAis independent of the choice ofAconnecting the flat acyclic limits at either end ofY ×R

[Don02, p.51, Proposition 4.5]. Hence we can associate, to any pair of flat acyclic connections on Y, the invariant ind(DA)∈Zgiven by the Fredholm index. This will provide the relative grading in Floer’s instanton theory. However,

we still need to consider the action of the gauge transformations on these flat connections. Consider an instantonAon Y ×RconnectingA∈ AP toAg ∈ AP whereg∈GP is a gauge transformation. Topologically, we may regard this

instead as a connectionA′ on the principal SU(2)bundle overY ×S1 obtained by gluing the two ends ofPtogether via the gauge transformationg. The Atiyah-Singer index theorem applied to the new operatorDA′overY ×S1yields,

as in Chapter 3:

indDA′ = 8c2(PY×S1)−3(1−b₁(Y ×S1) +b+₂(Y ×S1)) = 8deg(g)

where we have used Theorem 4.2 to identify PY×S1 with deg(g), and the observation that b₁(Y ×S1) = 1 and

b2(Y ×S1) = 0wheneverY is a homology3-sphere. The fact that indDA = indDA′ is justified by appealing to the

4.2. EXAMPLE: THE CHERN-SIMONS FUNCTIONAL 39 THEOREM 4.5. (Additivity of the Index) [Don02, Proposition 3.8] SupposeX is a4-manifold with two ends, of the formY ×R

andY¯ ×R, along with a principalSU(2)bundleP→Xand a connectionAonPwith flatacycliclimits over the two ends differing by a gauge transformation. Then the4-manifoldX′obtained by gluing together the two infinite ends with reverse orientation has a principal

SU(2)bundleP′ →X′with a connectionA′, and the Fredholm index ofDAis equal to that ofDA′.

Now suppose there is a connectionBoverY ×Rconnecting a flat acyclic connectionB at one end toAat the other. Taking X = Y ×R⨿Y ×Rand following the procedure described in the theorem above to glue the connection

B⨿AoverXalong the flat connectionA, we obtain a new connectionB′overY ×RconnectingBat one end toAg at the other. From the additivity of the index, we find that indDB+indDA =indB′ and hence the index ofDB′ and

DBdiffer only by a multiple of8.