Monopole Floer homology and SOLV geometry

We study the monopole Floer homology of a SOLV rational homology sphere Y from the point of view of spectral theory. Applying ideas of Fourier analysis on solvable groups, we show that for suitable SOLV metrics on Y, small regular perturbations of the Seiberg-Witten equations do not admit irreducible solutions; in particular, this provides a geometric proof that Y is an L-space.

Theorem 1. Let Y be a Solv-rational homology sphere, equipped with a Solv metric. If the fibers are small enough, then there are small regular perturbations for which the Seiberg-Witten equations on Y do not admit irreducible solutions.
The following is an immediate consequence of the theorem. Recall that a rational homology sphere Y is an L-space if y HM˚pY, sq " ZrU s as a ZrU s-module for each spin c structure s.
Corollary 1. Let Y be a Solv-rational homology sphere. Then Y is an L-space.
The analogous result in the setting of Heegaard Floer homology (which is known to yield isomorphic invariants, see [KLT11], [CGH12] and subsequent papers) was proved by topological means in [BGW13] with Z{2Z-coefficients, and extended to Z-coefficients in [RR17]. Let us also point out that compact Solv manifolds have either b 1 " 0 or 1; in the latter case, they are Anosov torus bundles over the circle, and their Heegaard Floer homology (with Z coefficients) was computed in [Bal08].
In our approach, we look at the monopole Floer homology of Solv-manifolds from the point of view of spectral geometry. The main ingredient in the proof of Theorem 1 is the following relation, for a rational homology sphere, between the existence of irreducible solutions to the Seiberg-Witten equations and the first eigenvalue λ1 of the Hodge Laplacian on coexact 1-forms (which improves on the main result of [Lin18]).
Theorem 2 (Theorem 3 of [LL18]). Let Y be a rational homology sphere equipped with a metric g. Denote bysppq the sum of the two least eigenvalues of the Ricci curvature at the point p. If the inequality λ1 ě´inf pPYs ppq{2 holds, then the Seiberg-Witten equations do not admit irreducible solutions.
In the case of a Solv-metric,s "´2 at every point, so in order to prove Theorem 1, we need to show that for suitable Solv-metrics on Y , λ1 ě 1. Let us describe the strategy behind the proof of this by discussing the content of each section.
In Section 1, we review some facts about the geometry and topology of Solv-manifolds. As Solv is the left-invariant metric for a solvable Lie group structure on R 3 , one can study Fourier analysis on it, and we will introduce the basic ideas behind it. In Section 2, we use the aforementioned Fourier analysis to show that, for metrics with sufficiently small fibers, λ1 " 1, so that the Seiberg-Witten equations do not admit irreducible solutions by Theorem 2. As these metrics have λ1 is exactly 1, they lie in the borderline case of Theorem 2, and transversality is a quite subtle issue. We discuss it in Section 3, where we will study explicit small perturbations of the equations and existence of harmonic spinors.
Acknowledgements. The author would like to thank Liam Watson for some helpful comments. Thisf work was partially funded by NSF grant DMS-1807242.

Compact Solvmanifolds and their Fourier analysis
We start by reviewing the basics of Solv-geometry; most of the following discussion is taken from Section 12.7 of [Mar16]. Recall that Solv is the Riemannian manifold R 3 equipped with the metric e 2z dx 2`e´2z dy 2`d z 2 .
This is the left-invariant Riemannian metric on R 3 when equipped with the solvable Lie group structure px, y, zq¨px 1 , y 1 , z 1 q " px`e´zx 1 , y`e z y 1 , z`z 1 q.
This can be though as the semidirect product corresponding to the splitting of where ppx, y, zq " z, given by seen as linear automorphisms of R 2 . The Ricci tensor is given in this coordinates by » so that both s ands are´2 at each point. We can see that the foliation in R 2 by the planes with z constant descend to any compact Solv-manifold; in fact, it descends to a foliation for which all the leaves are tori or Klein bottles.
Orientable compact solvmanifolds either have b 1 " 0 or 1. The manifolds of the latter type, which will be denoted byỸ , arise as quotients ΓzSolv for lattices Γ Ă Solv. Every such lattice is a split extension where Λ Ă R 2 is a lattice invariant under the action of " e a 0 0 e´a  . The underlying topological manifold is a torus bundle with monodromy A P SLp2, Zq; here |trA| ą 2 (i.e. A is Anosov) and e a and e´a are its eigenvalues.
Example 1. Consider A " The mapping torus is well-known to be the zero surgery on the figure eight knot. Its eigenvalues are ϕ 2 and ϕ´2 where ϕ " 1`?5 2 is the golden ratio. Recall that it satisfies ϕ 2 " ϕ`1. Consider the vectors v " pϕ, 1´ϕq w " p1, 1q.
If S is the matrix with colums v and w, we have A " S´1 " ϕ 2 0 0 ϕ´2  S; setting Λ to be the lattice generated by v and w, and a " logpϕ 2 q, we obtain the lattice Γ equipping the mapping torus of A with a Solv metric.
Remark 1. We can also think about this example from a more number theoretic viewpoint, which makes the connection with [Hir73] and [ADS83] clearer. Consider the field k " Qp ? 5q. It is totally real, and it comes with two natural embeddings φ`, φ´into R sending ? 5 tȏ ?
5. The ring of integers O k is the lattice Λ " Zrϕs which has basis ϕ and 1. The group of totally positive units is generated by ϕ 2 ; and it is easy to see that its multiplication action is given in our chosen basis by A. Finally, we can embed the lattice Λ in R 2 using pφ`, φ´q; our basis elements are mapped to the vectors v and w.
A Solv-manifold with b 1 " 0, denoted by Y , is a torus semibundle; therefore it admits a double coverỸ which is a Solv torus bundle ΓzSolv. Then Y can be described in the following way. For a choice of basis v, w of the lattice Λ " Γ X R 2 , with corresponding left-invariant extension V, W, we can consider the additional orientation-preserving isometry ofỸ sending paV`bW, zq Þ Ñ ppa`1 2 qV´bW,´zq.
In particular, the action on the fiber z " 0 (which is preserved) is obtained by pav`bwq Þ Ñ pa`1 2 qv´bw; and the quotient of the fiber is a Klein bottle. This is an order 2 isometryỸ , and the quotient is Y .
From this description, we see that on any Solv-manifold Y we obtain a one parameter family of metrics obtained by rescaling the lattice Λ; this can be seen concretely in Example 1.
Let us now introduce the basics of Fourier analysis on a compact Solvmanifold with b 1 pY q " 1. We follow the first chapter of [Bre77], to which we refer for a pleasant, more thorough, discussion.
Consider a smooth function f : ΓzSolv Ñ R. This can be thought (with a little abuse of notation) as a function f : Solv Ñ R which is left invariant under Γ. In particular, it is invariant under the action of Λ Ă Γ, i.e.
f px`m, zq " f px, zq for all m P Λ.
We can therefore expand f in Fourier series in the R 2ˆt 0u Ă Solv directions for some smooth functions a µ pzq. Here Λ 1 is the dual lattice of Λ, where we use the convention Λ 1 " tµ P R 2 |µ¨m P 2πZ for all m P Λu.
We now use the fact that f is invariant by the action of p0, aq.
This implies that a µ pzq " a µ¨A pz`aq, so a µ determines via translation a µ¨A n . In particular, the Fourier series is determined by the collection of functions for a µ pzq for µ P Λ 1 {V , V being the group of automorphisms of the dual lattice Λ 1 generated by A. While a 0 is a periodic function with period a, it can be shown that the functions a µ pzq for µ ‰ 0 are in the Schwartz-type space where f pmq denotes the mth derivative of f .
With this in mind, let us study as a warm-up example the Laplacian on functions on ΓzSolv, which can be written as ∆f "´pe´2 z f xx`e`2 z f yy`fzz q.
Let us use the decomposition in Fourier modes discussed above. We then have a L 2 -unitary decomposition ∆ " à where ∆ 0 acts on L 2 pR{aZq and ∆ µ is a diagonalizable operator on L 2 pRq. In particular, if we have µ " pµ, µ 1 q, the corresponding operator is given by substituting Therefore λ is an eigenvalue of ∆ µ if and only if While this equation is not solvable in terms of elementary functions, we can still understand the basic properties of its spectrum. Let us first recall the following well-known elementary lemma.
Lemma 1. Suppose f : R Ñ R solves the second order linear ODE where Φ is smooth and Φpzq ą 0 everywhere. Then f cannot be in L 2 pRq. Proof.
Possibly after replacing f pzq by´f pzq or f p´zq, we can assume that at x 0 both f px 0 q " c ą 0 and f 1 px 0 q ě 0. Suppose there is t 0 ą x 0 with 0 ă f pt 0 q ă f px 0 q. We can also assume f ą 0 on rx 0 , t 0 s. Then there is x 0 ă t ă t 0 with f 1 ptq ă 0. Applying again the mean value theorem, there is x 0 ă t 1 ă t with f 2 pt 1 q ă 0, which is contradiction as f 2 pt 1 q " Φpzq¨f ą 0. So f pxq ě c for x ě x 0 , and the result follows.
We then have the following.
In terms of the number theoretic description in Remark 1, the quantity µµ 1 is the norm N pµq; the only basic property we will need is that there is c ą 0 such that |µµ 1 | ě c for all µ P Λ 1 zt0u.
For completeness, let us conclude this section by discussing the zero mode µ " 0. In this case, we study the ODE f zz "´λf with f periodic with period a. It has eigenvalues λ " 4π 2 a 2 n 2 for n P Z.

The spectrum on coexact 1-forms
In this section we will perform the key computation behind our main result. Recall from the previous section that on a Solv-manifold there is a non-trivial family of metrics obtain by rescaling the lattice Λ Ă R 2 . With is in mind, we have the following.
Proposition 1. Let Y be a rational homology sphere equipped with a Solv metric such that the fibers are small enough. Then the first eigenvalue on coexact 1-forms satisfies λ1 " 1. Furthermore, the 1-eigenspace is one dimensional.
In fact, our proof will provide an explicit smallness condition for the fibers.
Let us start by considering the the case of a Solv-manifoldỸ " ΓzSolv with b 1 " 1. The 1-forms X " e z dx, Y " e´zdy Z " dz descend to a left-invariant dual orthonormal frame onỸ . We can then write any 1-form ξ as where f, g, h are functions on ΓzSolv, or equivalently left-invariant functions on Solv. We are interested in understanding for which λ the equation dξ " λξ admits non-trivial solutions. Notice that, provided λ ‰ 0, such a form necessarily satisfies d˚ξ " 0, i.e. it is coclosed. We have dξ " pe´zg x´e z f y qX^Yp´g z`g`e z h y qY^Zp f z`f´e´z h x qZ^X so that our equation is equivalent to the systeḿ while coclosedness is equivalent to e´zf x`e z g y`hz " 0.
Differentiating we get´e´2 z h xx "´e´zf xz´e´z f x`λ e´zg x e 2z h yy "´e z g yz`e z g y´λ e z f ý h zz " e´zf xz´e´z f x`e z g yz`e z g y , therefore summing we obtain ∆h " λ 2 h´2e´zf x`2 e z g y , where ∆ denotes the Laplacian on functions onỸ . Similarly for g we obtaiń e´2 z g xx "´λe´zh x´fxý e 2z g yy " f xy`e z h yź Finally, as´e´2 Notice that Z is a harmonic 1-form; as b 1 " 1, all harmonic forms are multiples of it.
Lemma 3. LeỸ be a Solv manifold with b 1 " 1 equipped with a metric for which the fibers are small enough. Then λ1 " 1, and the 1-eigenspace is spanned by X and Y .
As by assumption |µµ 1 | ą 8, λ 2 ą 1. Finally, we deal with the zero mode. Suppose 0 ă λ 2 ă 1. Then λ 2´1 ă 0, hencé have no periodic solution. It follows from Equation (2) that h is constant, so we have a multiple of the harmonic form Z. Finally, the case λ 2 " 1 corresponds to the span of X and Y.
Finally, we are ready to prove Proposition 1.
Proof of Proposition 1. Suppose Y is a Solv-rational homology sphere. Consider its double cover π :Ỹ Ñ Y whereỸ has b 1 pỸ q " 1. If ξ is a λ-eigenform on Y , the π˚ξ is a λ-eigenform onỸ . Choose a Solv-metric with fibers small enough, so that Lemma 3 applies. This implies that on Y we have λ1 ě 1, and furthermore that if ξ is a 1-eigenform on Y , then π˚ξ is a linear combination of X and Y. Finally, in the notation of Section 1, if v, w is the basis of Λ, then exactly the linear combinations of X and Y that vanish on w at z " 0 descend to Y .
We will denote by η the unique the unit length 1-eigenforms such that ηpvq ą 0 and η descends to Y . Recall (Chapter 28 of [KM07]) that there is a natural one-to-one correspondence between spin c structures unit length 1-forms up to homotopy outside balls. With this in mind, we have the following. Proof. Denote by ζ the unit length 1-form obtained from η by a couterclockwise rotation of π{2 within the fibers ofỸ . Then η, ζ and Z form a dual orthonormal frame ofỸ . By twisting ζ and Z around η on R 2ˆt 0u, so that under translation by v{2 they go into their opposite, and extending in a left-invariant fashion, we obtain a new frame η, ζ 1 and Z 1 of Solv that descends to Y . The spin c structure conjugate to s 0 corresponds to the 1-form´η; but η and η are homotopic on Y through cosptqη`sinptqζ 1 for t P r0, 2πs.

Transversality
In the previous section, we have exhibited a metric for which λ1 "´infps{2q. As this is the borderline case of Theorem 2, transversality is a quite delicate issue as small perturbation might introduce irreducible solutions. This should be compared with the discussion of flat manifolds in Chapter 37 of [KM07]. As in their setting, we will show that we can achieve transversality, while still not having irreducible solutions, by considering the perturbed functional for δ sufficiently small. The corresponding equations for the critical points are We have the following.
Lemma 5. Consider a spin c structure s ‰ s 0 . Then, for δ small enough, the perturbed Seiberg-Witten equations do not admit irreducible solutions.
Proof. Suppose we have a sequence δ i Ñ 0 with corresponding irreducible solutions pB i , Ψ i q; consider the corresponding configurations in the blow-up pB i , s i , ψ i q, where }ψ i } L 2 " 1. These admit (up to gauge transformations, and up to passing to a subsequence) a limit pB, s, ψq which solves the blown-up equations with δ " 0; in particular, as the unperturbed equations do not admit irreducible solutions by Theorem 2, s " 0, B is the flat connection, and D B ψ " 0. Recall that, setting ξ " ρ´1pΨΨ˚q 0 , it is shown in [LL18] that for solutions pB, Ψq of the unperturbed Seiberg-Witten equations the pointwise identity holds. This holds for the perturbed equations up to an error going to zero for δ i Ñ 0; hence it will apply to the limit form α " ρ´1pψψ˚q 0 . Furthermore, as it is the limit of the sequence of coexact forms 1 s 2 i 1 2 ρpF B t q, α is a coexact 1-form. Let us study the geometry of α. As ψ is a harmonic spinor, and B is flat, the Weitzenböck formula on Y implies hence the pointwise identity ∆|ψ| 2 " 2xψ, ∇B∇ B ψy´2|∇ B ψ| 2 " |ψ| 2´2 |∇ B ψ| 2 holds. Multiplying by |ψ| 2 and integrating, we obtain ż |ψ| 4´ż 2|ψ| 2 |∇ B ψ| 2 " ż |ψ| 2 ∆|ψ| 2 ě 0.
Recalling now that |α| 2 " 1 4 |ψ| 4 , we obtain, by using the Bochner formula and λ1 " 1, the chain of inequalities This implies that all inequalities are equalities, so that in particular α is a 1-eigenform, i.e. a multiple of η. Finally, by Lemma 4 this can happen if and only if the underlying spin c structure is the spin structure s 0 .
We need to understand more in detail the spin structure s 0 on Y ; before doing this, let us study the spin geometry of the double coverỸ . The manifoldỸ " ΓzSolv comes with a natural spin structure s˚coming from the left invariant orthonormal framing dual to Z, X , Y, i.e. e 1 " d dz , e 2 " e´z d dx , e 3 " e z d dy .
This defines a spin structure s˚by taking the trivial bundle S " YˆC 2 and letting these vector fields act via the Pauli matrices " i 0 0´i Let B˚the spin connection on Y induced by the Levi-Civita connection.
Lemma 6. The kernel of the Dirac operator D B˚c onsists of the constant spinors.
Proof. Let us write explicitly the Dirac operator. Our orthonormal frame satisfies the commutation relations re 1 , e 2 s "´e 2 re 1 , e 3 s " e 3 re 2 , e 3 s " 0.
Setting re i , e j s " ř k C ijk e k , we have that the Christoffel symbols are hence in our case the non-zero ones are Γ 212 "´Γ 221 " 1, Γ 313 "´Γ 331 "´1.
Hence, writing Ψ " pf, gq, we have if z´e´z g x`i e z g ý ig z`e´z f x`i e z f y  , and the equations for a harmonic spinor are f z`i e´zg x`e z g y " 0 g z`i e´zf x´e z f y " 0.
Let us now decompose the equations according to the eigenmodes µ P Λ 1 . We obtain f z´µ e´zg`iμe z g " 0 g z´µ e´zf´iμe z f " 0. so that, from the point of view of spin topology, Y looks like the more familiar three-torus. From the description in Lemma 4, it readily follows that the pullback of s 0 to Y , call it s, is the spin structure obtain from the standard one s˚by twisting by 2π around the class dual to v in H 1 pT 2 ; Z{2Zq (which is a non-trivial operation). The sublattice of Λ spanned by 2v and w is preserved by A 6 ; the corresponding mapping torus Y is a double cover of Y ; and the pullback of s is the standard spin structure s˚on Y . One can then identify the harmonic spinors on pY , sq as the harmonic spinors on pY , s˚q which change sign under translation by v; by Lemma 6, there are no such spinors. Hence, there are no harmonic spinors on the base space pY, s 0 q.
Putting pieces together, we finally conclude.
Proof of Theorem 1. By the discussion above, we have found small perturbations for which there are no irreducible solutions and the (perturbed) Dirac operator of the reducible solution has no kernel; we can then add a further small perturbation to make all of its eigenvalues simple (while preserving these properties) as in Chapter 12 of [KM07]; the proof of Theorem 1 is then completed.