1 Background material

We recall some essential ideas about monopole Floer homology; we refer the reader to [Reference Kronheimer and MrowkaKM07] for the definitive reference, and [Reference LinLin16] for a friendly introduction to the topic. Fix a spin $^c 3$ -manifold $(Y,\mathfrak {s})$ . Given a spin $^c$ connection B, the curvature $2$ -form $F_{B^t}$ of the connection $B^t$ induced on the determinant line bundle is always closed. Throughout the paper, we will only consider torsion spin $^c$ structures; by Chern-Weil theory, this implies that $F_{B^t}$ is an exact $2$ -form. Because of this one sees that reducible solutions $(B,0)$ of the equations (3) up to gauge are in correspondence with the torus $\mathbb {T}_Y$ of spin $^c$ connections B with $B^t$ flat, see [Reference Kronheimer and MrowkaKM07, Section 4.2].

In particular, when $b_1>0$ , the unperturbed equations are never transversely cut out. Furthermore, in order to define the monopole Floer homology groups one needs to work in the blown-up configuration space consisting of triples $(B,r,\psi )$ with $r\in \mathbb {R}^{\geq 0}$ and $\|\psi \|_{L^2}=1$ . In this setup, one needs to introduce suitable perturbations and transversality for the three-dimensional equations can be phrased in terms of the following criteria:

• • the critical points of (the perturbation of) $\mathrm {grad}\mathcal {L}$ in the blow-down are non-degenerate.

• • at each reducible critical point $(B,0)$ of the (perturbation of) $\mathrm {grad}\mathcal {L}$ , the (perturbation of the) Dirac operator $D_B$ has simple spectrum and trivial kernel.

In [Reference Kronheimer and MrowkaKM07, Ch. 11] a large space of perturbations is introduced so that these conditions can be achieved. Of course, even if one had a concrete understanding of the solutions to the unperturbed equations, this might not be the case anymore after appealing to the general results for achieving transversality. Our goal in this paper is to study a concrete instance in which one can both achieve transversality and maintain a concrete understanding of the space of solutions. For example, in the context of transversality for the flat torus [Reference Kronheimer and MrowkaKM07, Ch. 37] and mapping tori of cyclic coverings $\Sigma \rightarrow \mathbb {P}^1$ in [Reference LinLin22b], for $\delta ,\varepsilon> 0$ perturbations of the Chern-Simons functional of the form

$$ \begin{align*} \widetilde{\mathcal{L}}(B,\Psi)=\mathcal{L}(B,\Psi)-\frac{\delta}{2}\|\Psi\|^2+\varepsilon f(B) \end{align*} $$

were considered. Here for a function $f:\mathbb {T}_Y\rightarrow \mathbb {R}$ we define (abusing notation) $f(B)$ to be the evaluation at the harmonic part of $B-B_0$ (thought of as an element in $ H^1(Y,i\mathbb {R})$ ).

In those situations, the analysis of transversality begins by studying the perturbations for which $\varepsilon =0$ . In this context, the first of the Seiberg-Witten equations (3) is replaced by the eigenvalue equation

(5) $$ \begin{align} D_B\Psi=\delta\Psi. \end{align} $$

In the level of generality we are interested in, we will need to introduce additional perturbations involving only spinors that generalize the ${\delta }\|\Psi \|^2/2$ term. In particular, we will consider for a section $A\in \Gamma ( i\mathfrak {u}(S))$ the perturbed Chern-Simons-Dirac functional (4), for which the corresponding perturbed Seiberg-Witten equations are

(6) $$ \begin{align} \begin{aligned} D_B\Psi&=A\Psi\\ \frac{1}{2}\rho(\ast F_{B^t})+(\Psi\Psi^*)_0&=0. \end{aligned} \end{align} $$

We will refer to these perturbations as of spinorial type. Notice that they are not of cylinder type in the sense of [Reference Kronheimer and MrowkaKM07, Ch. 11]; nevertheless, it is readily checked that they define k-tame perturbations, hence can be used to perturb the equations in the relevant functional setup. The main advantage of using them is that, while the first equation changes in an understandable way, the second equation is left unchanged. In particular, reducible solutions $(B,0)$ to the equations are still identified with the torus $\mathbb {T}_Y$ .

Remark 1.1. Under the decomposition $A=A_0+\tau \cdot I$ with $A_0$ traceless, we can rewrite (2) as

$$ \begin{align*} D_{\tilde{B}}\Psi=\tau\cdot \Psi \end{align*} $$

where $\tilde {B}=B-\rho ^{-1}(A_0)$ is a new spin $^c$ connection. Hence we can interpret the family of equations (2) for $[B]\in \mathbb {T}_Y$ as a family of equations where $\tilde {B}$ varies in the torus of spin $^c$ connections with fixed (exact) curvature.

Of course, in general the equations (6) will admit irreducible solutions $(B,\Psi )$ (i.e. $\Psi $ not identically zero), and it is in general quite challenging to understand explicitly whether they do. In [Reference LinLin20] and [Reference Lin and LipnowskiLL22] it is proved that a suitable condition of spectral nature guarantees that there are no such solutions. While those results were focused towards the case $b_1=0$ , we will now adapt the discussion to fit our needs.

Recall that we denote by $\lambda _1^*$ the least eigenvalue of the Hodge Laplacian $\Delta =(d+d^*)^2$ on coexact $1$ -forms, and by $\tilde {s}(p)$ the sum of the two least eigenvalues of the Ricci curvature at $p\in Y$ , and we say that the metric g is spectrally large if the inequality

$$ \begin{align*} \lambda_1^*>-\frac{1}{2}\inf_{p\in Y} \tilde{s}(p) \end{align*} $$

holds (cf. Definition 0.1).

Proof. The way in which $\lambda _1^*$ comes into play in the proofs in [Reference LinLin20] and [Reference Lin and LipnowskiLL22] is through its Rayleigh quotient interpretation:

(7) $$ \begin{align} \|d\xi\|^2_{L^2}\geq \lambda_1^*\|\xi\|^2_{L^2}\text{ for any coexact} 1\text{-form } \xi. \end{align} $$

In our setup, $\ast F_{B^t}$ is coexact $1$ -form, and the second Seiberg-Witten equation is unchanged, so the proof applies without changes. For completeness, let us point out how to deal with the perturbed operator $D_B-A$ . The main effect is that the pointwise inner product $\langle \Psi ,D_B^2\Psi \rangle $ does not vanish identically. In the argument one takes its integral after multiplying by $|\Psi |^2$ ; integrating by parts, we obtain

$$ \begin{align*} \int_Y|\Psi|^2\langle \Psi,D_B^2\Psi\rangle d\mathrm{vol}&=\int_Y\langle |\Psi|^2 \Psi,D_B^2\Psi\rangle d\mathrm{vol}=\\ &=\int_Y\langle D_B|\Psi|^2 \Psi,D_B\Psi\rangle d\mathrm{vol}\\ &=\int_Y\langle D_B|\Psi|^2 \Psi,A\Psi\rangle d\mathrm{vol}. \end{align*} $$

We can then bound the integrand

$$ \begin{align*} \lvert\langle D_B|\Psi|^2 \Psi,A\Psi\rangle\lvert\leq C_A |\nabla_B\Psi|\cdot|\Psi|^3\leq \frac{C_A}{2}\cdot(|\Psi|^4+|\nabla_B\Psi|^2|\Psi|^2) \end{align*} $$

where we used $2ab\leq a^2+b^2$ and $C_A$ is a constant depending only on the $C^0$ norm of A. The rest of the argument then is carried in exactly the same way without major changes.

In the case $b_1=0$ , the perturbed Dirac operator at the (unique up to gauge) flat connection is $D_{B}-\delta $ , hence it has no kernel for generic $\delta $ ; furthermore, one can add additional small perturbations as in [Reference Kronheimer and MrowkaKM07, Ch. 12] to make its spectrum simple. Hence a space satisfying the assumption of Proposition 1.1 is an L-space.

By contrast when $b_1>0$ , the A-perturbed equations still have $\mathbb {T}_Y$ as reducible critical set, so the set of solutions to (6) is never transversely cut out in the sense of [Reference Kronheimer and MrowkaKM07]. Furthermore, even though $\mathbb {T}_Y$ is a smooth manifold, it is in general not a Morse-Bott critical submanifold: this fails exactly on the locus $\mathsf {K}\subset \mathbb {T}_Y$ where $D_{B}-A$ has kernel. Our main goal is to extract information about the Floer homology groups of $(Y,\mathfrak {s})$ in terms of the pair $(\mathbb {T}_Y,\mathsf {K})$ .

In order to do so, we begin by reviewing some well-known facts on the space $\mathrm {Op}^{sa}$ of Dirac type operators of the form $D_{B_0}+h$ where $B_0$ is a fixed reference connection and h is a $0$ -th order perturbation. For the sake of the exposition, we will not be too detailed about the functional setup, and refer the reader to [Reference Kronheimer and MrowkaKM07, Ch. 12] for details.

We can consider the subspace of $\mathcal {K}\subset \mathrm {Op}^{sa}$ consisting of operators with non-trivial kernel. This admits a stratification

$$ \begin{align*} \mathcal{K}=\coprod_{i>0}\mathcal{K}_i \end{align*} $$

where $\mathcal {K}_i$ is the subspace of operators with kernel having dimension exactly i. The latter is a smooth Banach submanifold of $\mathrm {Op}^{sa}$ of codimension $i^2$ , where the normal bundle at an operator $T\in \mathcal {K}_i$ is naturally identified with the space of self-adjoint operators on $\mathrm {ker}T$ :

(8) $$ \begin{align} N_T\mathcal{K}_i=i\mathfrak{u}(\mathrm{ker}T). \end{align} $$

Given this discussion, we can state the following well-known results (see, for example, [Reference Kronheimer and MrowkaKM07, Section 12.6]).

Lemma 1.2. For generic spinorial perturbation $A\in \Gamma ( i\mathfrak {u}(S))$ , the locus $\mathsf {K}\subset \mathbb {T}_Y$ for which $D_B-A$ has kernel admits a stratification

$$ \begin{align*} \mathsf{K}=\coprod_{i>0}\mathsf{K}_i \end{align*} $$

corresponding to the dimension of the kernel, and $\mathsf {K}_i\subset \mathbb {T}_Y$ is a smooth submanifold of codimension $i^2$ .

Proof. This follows from a standard transversality argument. In particular (again being imprecise about the functional setup) we can consider the map

$$ \begin{align*} F:\mathbb{T}_Y\times \Gamma(i\mathfrak{u}(S))&\rightarrow \mathrm{Op}^{sa}\\ (B,A)&\mapsto D_{B}-A. \end{align*} $$

Because the normal bundle of each stratum is given by (8), we readily see that F is transverse to all strata, hence by Sard-Smale for a generic $A\in \Gamma (i\mathfrak {u}(S))$ the map $F(-,A)$ is also transverse to all strata.

While the discussion above is local in nature, for our purposes we will need to discuss some global aspects of the family of operators $\{D_B-A\}$ . In our setup, the Atiyah-Singer index theorem for families [Reference Atiyah, Patodi and SingerAPS76], determines the classifying map

$$ \begin{align*} \mathrm{ind}:\mathbb{T}_Y\rightarrow U(\infty) \end{align*} $$

of the family up to homotopy in terms of the triple cup-product

$$ \begin{align*} \cup_Y^3:\Lambda^3H^1(Y;\mathbb{Z})\rightarrow \mathbb{Z}, \end{align*} $$

thought of as an element in

(9) $$ \begin{align} \cup_Y^3\in H^3(\mathbb{T}_Y;\mathbb{Z})\cong \Lambda^3 H^1(\mathbb{T}_Y;\mathbb{Z})\cong \Lambda^3 H^1(Y;\mathbb{Z})^*. \end{align} $$

We refer the reader to [Reference LinLin22a] for a quick introduction to the topic geared towards our needs. We have the following.

Remark 1.4. While the remainder of the paper we will only focus on the case in which $\mathsf {K}=\mathsf {K}_1$ is smooth, we will say a couple of words about the simplest situation in which the $\mathsf {K}$ has singularities, namely the case in which there are points for which $D_B-A$ has two-dimensional kernel. The locus where the kernel has dimension at least 3 has codimension $9$ , so this is generically true for $b_1(Y)\leq 8$ .

We will denote the locus by $\mathsf {K}_2\subset \mathbb {T}$ . By (8), the normal bundle at a point $T\in \mathsf {K}_2$ is naturally identified with $i\mathfrak {u}(\mathrm {ker}(T))$ , the real $4$ -dimensional space of hermitian operators on $\mathrm {ker}(T)$ . Identifying $\mathrm {ker}(T)=\mathbb {C}^2$ , we have

$$ \begin{align*} i\mathfrak{u}(2)=\left\{ \begin{bmatrix} a& z\\ \bar{z}& b \end{bmatrix}\lvert a,b\in\mathbb{R},z\in\mathbb{C}. \right\} \end{align*} $$

In this model, the locus of matrices with kernel is given by the quadric in $\mathbb {R}^4$

$$ \begin{align*} Q=\{ab-|z|^2=0\}, \end{align*} $$

which is a quadratic form with signature $(1,3)$ , and the only matrix with multiplicity $2$ is the zero matrix. Topologically, Q is the union over two disjoint copies of $S^2$ ; in general, when transversely cut out (cf. Lemma 1.2), the locus in $\mathsf {K}$ where the dimension of the kernel is at most $2$ is a Q bundle over $\mathsf {K}_2$ near $\mathsf {K}_2$ .

2 Morse functions on the Jacobian

Suppose now that we fixed a spinorial perturbation A small and generic so that the conclusions of Propositions 1.1 and 1.2 both hold. Let us also make the following.

This is indeed the case when $b_1\leq 3$ for dimensional reasons, and will also hold in the explicit examples with $b_1>3$ that we will consider later. By Lemma 1.3, the spectral flow between any two points in $\mathbb {T}_Y\setminus \mathsf {K}$ is well-defined, and we make the following definition.

When our family of Dirac operators (perturbed by A) has type k, we can fix a basepoint $B_0$ so that all other points have positive spectral flow from it; this gives us a inclusion of subspaces

(10) $$ \begin{align} \mathbb{T}_0\subset\mathbb{T}_1\subset\cdots\subset\mathbb{T}_k=\mathbb{T}_Y \end{align} $$

so that $\mathsf {K}=\amalg \partial \mathbb {T}_i$ and the points of $\mathrm {int}\mathbb {T}_j\setminus \mathbb {T}_{j-1}$ have spectral flow from $B_0$ exactly j.

Notice that A-transverse Morse functions always exist by Lemma 1.3 and its proof.

The fact that $\mathsf {K}=\mathsf {K}_1$ is transversely cut out in the space of operators readily implies the following.

We consider then an additional perturbation of the form

(11) $$ \begin{align} \widetilde{\mathcal{L}}(B,\Psi)=\mathcal{L}(B,\Psi)-\frac{1}{2}\langle\Psi, A \Psi\rangle_{L^2}+\varepsilon f(B) \end{align} $$

for our A-transverse Morse function $f:\mathbb {T}_Y\rightarrow \mathbb {R}$ .

Proposition 2.4. Suppose we are in the situation above. Then for $\varepsilon>0$ sufficiently small, the perturbed Seiberg-Witten equations corresponding to the critical points of (11), namely

(12) $$ \begin{align} \begin{aligned} D_{B}\Psi&=A\Psi\\ \frac{1}{2}\rho(\ast F_{B^t})+(\Psi\Psi^*)_0&=-\varepsilon\cdot \rho\left(\mathrm{grad}f(B)\right). \end{aligned} \end{align} $$

do not have irreducible solutions.

Here, $\mathrm {grad}f(B)$ is a tangent vector on $\mathbb {T}_Y$ , which we interpret as an imaginary-valued harmonic $1$ -form on Y.

Proof. Let us denote for simplicity $D_{B,A}:=D_B-A$ throughout the proof. Suppose that there is a sequence $\varepsilon _i>0$ converging to zero such that the equation have an irreducible solution $(B_i,\Psi _i)$ ; these solve the equations in the blow-up $(B_i,r_i,\psi _i)$ given by

(13) $$ \begin{align} \begin{aligned} D_{B_i,A}\psi_i&=0\\\frac{1}{2}\rho(\ast F_{B_i^t})+r_i^2(\psi_i\psi_i)^*)_0&=-\varepsilon_i\cdot \rho\left( \mathrm{grad}f(B_i)\right) \end{aligned} \end{align} $$

where $\|\psi _i\|_{L^2}=1$ . We can then extract a sequence converging to a solution $(B,r,\psi )$ with $\|\psi \|_{L^2}=1$ to the equations with $\varepsilon =0$ ; by Proposition 1.1 we necessarily have $r=0$ . In particular, $B^t$ is flat hence $B\in \mathbb {T}_Y$ , and the equation

$$ \begin{align*} D_{B,A}\psi=0 \end{align*} $$

then implies that $B\in \mathsf {K}$ .

Consider now a short piece of flowline $B(t)$ of $-\mathrm {grad}f$ in $\mathbb {T}$ such that $\gamma (0)=B$ . There is a smooth parametrization $\lambda (t)$ of the smallest eigenvalue by Lemma 2.3, and because the eigenvalue is simple we can choose a $1$ -parameter family of non-zero eigenspinors $\varphi (t)$ :

$$ \begin{align*} D_{B(t),A}\varphi(t)=\lambda(t)\varphi(t) \end{align*} $$

with $\varphi (0)=\psi $ .

Differentiating we obtain

$$ \begin{align*} D_{B(t),A}\dot{\varphi}(t)+\rho(\dot{B}(t))\varphi(t)=\dot{\lambda}(t)\varphi(t)+\lambda(t)\dot{\varphi}(t) \end{align*} $$

hence evaluating at $t=0$ we have

$$ \begin{align*} D_{B,A}\dot{\varphi}(0)+\rho(-\mathrm{grad}f(B))\psi=\dot{\lambda}(0)\psi. \end{align*} $$

Taking $L^2$ inner products with $\psi $ , and using that $D_{B,A}$ is self-adjoint, so that

$$ \begin{align*} \langle D_{B,A}\dot{\varphi}(0),\psi\rangle_{L^2}=\langle \dot{\varphi}(0),D_{B,A}\psi\rangle_{L^2}=0 \end{align*} $$

we obtain that

$$ \begin{align*} \langle \rho\left(\mathrm{grad}f(B)\right), (\psi\psi^*)_0\rangle_{L^2}=\frac{1}{2}\langle\psi ,\rho(\mathrm{grad}f(B))\psi\rangle_{L^2}=-\frac{\dot{\lambda}(0)}{2}\|\psi\|_{L^2}^2>0. \end{align*} $$

where we used Lemma 2.3 for the sign on $\dot {\lambda }(0)$ .

On the other hand, we can take the $L^2$ inner product of the second equation in (13) with $\varepsilon _i\cdot \mathrm {grad}f(B_i)$ ; because $\ast F_{B^t}$ is exact (hence orthogonal to harmonic forms), we obtain that

$$ \begin{align*} r_i^2\langle \rho\left(\mathrm{grad}f(B_i)\right), (\psi_i\psi_i^*)_0\rangle_{L^2}=-\varepsilon_i\|\mathrm{grad}f(B_i)\|^2_{L^2}\leq 0. \end{align*} $$

Because $r_i\neq 0$ , we can conclude passing to the limit that

$$ \begin{align*} \langle \rho\left(\mathrm{grad}f(B)\right), (\psi\psi^*)_0\rangle_{L^2}\leq 0, \end{align*} $$

which is a contradiction.

With this in hand, we can furthermore add a small additional perturbation as in [Reference Kronheimer and MrowkaKM07] to make first the spectra of the perturbed Dirac operators simple at the critical points of f, and then to arrange regularity for the moduli spaces of flowlines, hence achieving transversality without introducing irreducible critical points.

The Floer chain complex computing is then is generated by the stable critical points in the blow-up $\check {C}_*=C^s$ ; these form an infinite tower over each critical point of f in $\mathbb {T}_Y$ . The differential is given by

$$ \begin{align*} \check{\partial}=\bar{\partial}^s_s-\partial^u_s\bar{\partial}^s_u. \end{align*} $$

Unfortunately, in this level of generality, the latter cannot be described concretely for two reasons:

• • the term $\partial ^u_s$ involves irreducible solutions to the Seiberg-Witten equations on $\mathbb {R}\times Y$ , of which we do not have any understanding.

• • the counts of reducible trajectories $\bar {\partial }^s_s$ and $\bar {\partial }^s_u$ will in general involve families of Dirac operators parametrized by higher dimensional spaces of Morse flowlines.

We conclude this section by describing two useful tools that allow to obtain partial information about ; we will see in the next section that when combined, these provide a complete description of the homology group in the simplest type case.

First of all, one can compare the Floer chain complex $\check {C}_*$ with the Floer chain complex computing the U-localized version $\overline {\mathit {HM}}_*(Y,\mathfrak {s})$ . This is given by $\bar {C}_*=C^s\oplus C^u$ with differential

$$ \begin{align*}\bar{\partial}= \begin{bmatrix} \bar{\partial}^s_s & \bar{\partial}^u_s\\ \bar{\partial}^s_u & \bar{\partial}^u_u\\ \end{bmatrix}. \end{align*} $$

The map , which is induced by the chain map

$$ \begin{align*} i:C^s\oplus C^u\rightarrow C^s \end{align*} $$

given by $i=\mathrm {id}-\partial ^u_s$ induces an isomorphism in degrees high enough. This is helpful because even though one cannot describe explicitly the differential $\bar {\partial }$ , the resulting homology groups $\overline {\mathit {HM}}_*(Y,\mathfrak {s})$ can be readily computed in terms of the triple cup product $\cup _Y^3$ , as we briefly recall.

In [Reference Kronheimer and MrowkaKM07, Ch. 33], the authors define for a closed smooth manifold Q equipped with a family L of Dirac type operators the coupled Morse homology group $\bar {H}_*(Q,L)$ . This is an invariant of the homotopy type of L, and is obtained by choosing a Morse function $f:Q\rightarrow \mathbb {R}$ and studying the negative gradient flow equation for f suitably coupled with the Dirac equation. When $Q=\mathbb {T}_Y$ and L is the corresponding family of Dirac operators, the outcome is exactly $\overline {\mathit {HM}}_*(Y,\mathfrak {s})$ . Using the extra flexibility afforded by homotopy invariance, the authors were able to provide the following general computation of $\overline {\mathit {HM}}_*(Y,\mathfrak {s})$ over $\mathbb {Q}$ . Consider $H_*(\mathbb {T}_Y)\otimes \mathbb {Q}[U,U^{-1}]$ . Identifying $H_k(\mathbb {T}_Y)=\Lambda ^k H_1(\mathbb {T}_Y)=\Lambda ^k H^1(Y)$ , we can consider the degree $-1$ map

$$ \begin{align*} d:=\iota_{\cup_Y^3}\otimes U^{-1}, \end{align*} $$

where $\iota _{\cup _Y^3}$ is the contraction with $\cup _Y^3\in \left (\Lambda ^k H^1(Y)\right )^*$ . Then $\overline {\mathit {HM}}_*(Y,\mathfrak {s})=\mathrm {ker}d/\mathrm {im}d$ .

The second tool arises from the filtration of spaces (10). Let us introduce the following refinement of the notion of A-transverse Morse function.

Again, A-adapted Morse functions exist by Lemma 1.3 and its proof. The meaning of this condition is the following. The value of the perturbed functional (11) at a reducible critical point $(B,0)$ is given by $f(B)$ . Because of this, for a given choice of A-adapted f, we then see that the differential of a critical point lying in $\mathbb {T}_j$ still lies in $\mathbb {T}_j$ , hence we obtain a natural filtration of $\check {C}_*$ .

Our discussion in the next section (which we will spell out in detail in the case of simplest type, in which more can be said) will imply the following result (the non-obvious part being the identification of the $E^1$ -page).

Proposition 2.6. In our setup, there exists a spectral sequence whose $E^1$ page is given by

$$ \begin{align*} E^1=\bigoplus H_*(\mathbb{T}_j,\mathbb{T}_{j-1})\langle -2j\rangle\otimes \mathcal{T}_+ \end{align*} $$

and which converges to .

Here $\mathcal {T}_+$ is the $\mathbb {Q}[U]$ -module $\mathbb {Q}[U^{-1},U]/U\cdot \mathbb {Q}[U]$ , commonly referred to as ‘tower’. Let us point out that in many situations it is very important to understand the U-action on ; while the spectral sequence is compatible with the U-action, it is in general non-trivial to reconstruct the U-action on from that on the $E^{\infty }$ -page.

We conclude by pointing out the following result.

Proof. It is clear that the maps involving reducible trajectories preserve the Morse filtration, so we only need to show that the map $\partial ^u_s$ involving irreducible trajectories preserves the Morse filtration. Given two critical points x and y of the Morse function f on $\mathbb {T}_Y$ , suppose there is an irreducible trajectory from the mth unstable critical point lying over x to the nth stable critical point lying over x. Here we choose the enumeration so that $m,n\geq 0$ and the first unstable and stable critical points correspond to m and $n=0$ respectively. The value of the Chern-Simons-Dirac functional (11) at reducible critical points is given by $\varepsilon f$ , hence $f(x)>f(y)$ . Because of this, if $x\in \mathrm {int}T_j\setminus T_{j-1}$ and $y\in \mathrm {int}T_k\setminus T_{k-1}$ the second bullet in Definition 2.5 implies $j\geq k$ . Denoting the Morse index by $\mathrm {ind}_f$ , the relative grading between the reducible critical points in the blow-up we are considering is then

$$ \begin{align*} \mathrm{ind}_f(x)-1-2m-(\mathrm{ind}_f(y)+2n+2(j-k)), \end{align*} $$

where the additional term $2(j-k)$ arises from the spectral flow between x and y. This number is at least one because of the existence of a trajectory, and the result follows.

3 Spin $^c$ Manifolds of the simplest type

We focus on the case of a spin $^c$ manifold $(Y,\mathfrak {s})$ of the simplest type (where we keep the metric g and spinorial perturbation A implicit in our discussion). In this situation, we denote the filtration (10) simply by

$$ \begin{align*} \mathbb{T}_-\subset\mathbb{T}_Y. \end{align*} $$

By Lemma 1.3, the restriction of the family of Dirac operators $\{D_B-A\}$ is homotopically trivial on both $\mathbb {T}_-$ and $\mathbb {T}_Y\setminus \mathrm {int}\mathbb {T}_-$ . On the other hand, the family on $\mathbb {T}_Y$ itself is not homotopically trivial in general, the obstruction being the triple cup product of the three-manifold (9).

Remark 3.1. We can interpret this in terms of the cofibration

$$ \begin{align*} \mathbb{T}_-\stackrel{\iota}{\hookrightarrow} \mathbb{T}_Y\rightarrow \mathbb{T}_Y/\mathbb{T}_- \end{align*} $$

as follows. Families of Dirac type operators are classified up to homotopy by maps to $U(\mathbb {\infty })$ (see [Reference Atiyah and SingerAS69] and [Reference Kronheimer and MrowkaKM07, Section 33.1]). Using

$$ \begin{align*} [X,U(\infty)]=K^1(X), \end{align*} $$

we can consider then the morphism of exact sequences

where the vertical maps are given by the Chern character. In particular we know from the Atiyah-Singer index theorem for families [Reference Atiyah, Patodi and SingerAPS76] (see also [Reference Kronheimer and MrowkaKM07, Lemma 35.2.1]) that

$$ \begin{align*} \mathrm{ch}(\mathrm{ind}\{D_B-A\})=\cup_Y^3\in H^3(\mathbb{T}_Y)=\left(\Lambda^1H^1(Y)\right)^* \end{align*} $$

so that $i^*(\cup _Y^3)=0\in H^3(\mathbb {T}_-)$ , hence $\cup _Y^3$ arises from an element of $H^3(\mathbb {T}_Y,\mathbb {T}_-;\mathbb {Q})$ .

We then choose an adapted Morse function f, see Definition 2.5. The Floer chain complex $\check {C}$ is naturally relatively $\mathbb {Z}$ -graded in this situation. For our purposes it is convenient to fix the absolute grading of the first stable critical point over an index zero critical point in $\mathbb {T}_-$ to be $0$ ; this is well-defined because there is no spectral flow between points in $\mathbb {T}_-$ of f given our definition.

Let us consider the filtration on critical points induced by the value of $\widetilde {\mathcal {L}}$ . We obtain subcomplexes

$$ \begin{align*} \check{C}^-\subset \check{C}\text{ and }\bar{C}^-\subset \bar{C}. \end{align*} $$

Denoting by $\check {C}^+$ and $\bar {C}^+$ the respective quotient chain complexes, these all fit in the diagram

Calling the vertical maps i, we have the following observation.

Notice that the analogous statement does not hold for the map $i_*$ on the full complexes; indeed, in that situation the cokernel is the reduced Floer homology, which we will see to be not-trivial in concrete examples.

We now discuss the groups $H_*(\bar {C}^{\pm })$ in more detail. In order to do so, we generalize the framework of coupled Morse homology [Reference Kronheimer and MrowkaKM07, Ch. 33] from the setup of closed manifolds Q to the case of non-empty boundary $\partial Q\neq 0$ . In this situation, as in classical Morse theory with boundary, one can look at Morse functions f for which $-\mathrm {grad}f$ points either outwards or inwards at the boundary; in the former case one obtains $H_*(Q,\partial Q)$ while in the latter one obtains $H_*(Q)$ . When Q is equipped with a family L of Dirac-type operators, one can then construct the corresponding coupled Morse homology groups by $\bar {H}^{\pm }_*(Q;L)$ . These are again invariant under homotopy of L.

In our situation, we apply this discussion to $\mathbb {T}_-$ and $\mathbb {T}_Y\setminus \mathrm {int}\mathbb {T}_-$ with a Morse function on $\mathbb {T}_Y$ as above; thought of a Morse function on $\mathbb {T}_Y\setminus \mathrm {int}\mathbb {T}_-$ (resp $\mathbb {T}_-$ ), the vector field $-\mathrm {grad}f$ points outwards (resp. inwards), and the respective coupled Morse chain complexes are identified with $\bar {C}^+$ (resp. $\bar {C}^-$ ). Hence we get the exact triangle

where the horizontal arrow has degree $-1$ .

Because the restriction of the family $\{D_B-A\}$ to $\mathbb {T}_Y\setminus \mathsf {K}$ is homotopically trivial, we also have the further identifications

(14) $$ \begin{align} \begin{aligned} {\bar{H}_*^+(\mathbb{T}_Y\setminus\mathrm{int}\mathbb{T}_-,\{D_{B,A}\})}&\cong H_*(\mathbb{T}_Y,\mathbb{T}_-)\otimes \mathbb{Q}[U,U^{-1}]\\ {\bar{H}_*^-(\mathbb{T}_-,\{D_{B,A}\})}&\cong H_*(\mathbb{T}^-)\otimes \mathbb{Q}[U,U^{-1}]. \end{aligned} \end{align} $$

where we identified the homology of $\mathbb {T}_Y\setminus \mathrm {int}\mathbb {T}_-$ relative to its boundary with the homology of the pair $(\mathbb {T}_Y,\mathbb {T}_-)$ using excision.

Using this, we can obtain a description of the groups $H_*(\check {C}^{\pm })$ as follows.

Lemma 3.2. We have the identifications

$$ \begin{align*} H_*(\check{C}^-)&\cong H_*(\mathbb{T}_-)\otimes\mathcal{T}_+\\ H_*(\check{C}^+)&\cong H_*(\mathbb{T}_Y,\mathbb{T}_-)\otimes\mathcal{T}_+\langle-2\rangle \end{align*} $$

as graded $\mathbb {Q}[U]$ -modules (up to an overall shift).

In particular, we have an exact triangle

where again the horizontal arrow has degree $-1$ . Furthermore, the natural maps $i_*$ from the bar version triangle are all surjective. Because of this, in order to describe $\check {\partial }$ it will suffice to describe $\bar {\partial }$ , which we now do.

Consider the long exact sequence of the pair $(\mathbb {T}_Y,\mathbb {T}_-)$ in usual homology with $\mathbb {Q}$ -coefficients

where $\partial $ has degree $-1$ . Because we are using $\mathbb {Q}$ -coefficients, this breaks into the short exact sequences

(15) $$ \begin{align} 0 \rightarrow {i_*(H_k(\mathbb{T}_-))} \rightarrow {H_k(\mathbb{T}_Y)} \rightarrow {q_*(H_k(\mathbb{T}_Y))} \rightarrow 0 \end{align} $$

and the degree $-1$ isomorphism

(16) $$ \begin{align} \mathrm{coker}\left(q_*:H_k(\mathbb{T}_Y)\rightarrow{H}_k(\mathbb{T}_Y,\mathbb{T}_-)\right)\cong\mathrm{ker}\left(i_*:H_{k-1}(\mathbb{T}_-)\rightarrow H_{k-1}(\mathbb{T}_Y)\right) \end{align} $$

Denoting the pieces in (15) by $I^\pm _*$ respectively, and (16) by $E_*$ , we see can consider the map $\bar {\partial }$ under the identification (14) as a map

$$ \begin{align*} \bar{\partial}:(I^+_*\oplus E_*)\otimes \mathbb{Q}[U,U^{-1}]\rightarrow (I^-_*\oplus E_*\langle-1\rangle)\otimes\mathbb{Q}[U,U^{-1}]. \end{align*} $$

As we are working in the localized (bar) version of the invariants, this map is in fact independent of the specific family of operators up to homotopy and choice of Morse function. In order to obtain a concrete description, we look at the determination of the chain complex for a family pulled-back from $SU(2)$ as discussed in [Reference Kronheimer and MrowkaKM07, Ch. 34], and in particular Proposition $34.2.1$ and its proof.

Referring to the construction and notation there, because in our setup the restriction of the family on $\mathbb {T}_Y\setminus \mathrm {int}\mathbb {T}_-$ is homotopically trivial, we can begin by homotoping the whole family to a map $v:\mathbb {T}_Y\rightarrow SU(2)$ so that the restriction to it is the constant operator taking values in the north pole of $SU(2)$ . In particular, $v^{-1}(W)\subset \mathbb {T}_-$ , and we can furthermore choose the Morse function to take a fixed regular value on $\mathsf {K}$ .

With this setup, the coupled Morse chain complex is then $C_*(\mathbb {T}_Y,f)\otimes \mathbb {Q}[T,T^{-1}]$ equipped with the differential

$$ \begin{align*} \bar{\partial}x=\partial x+ (\tilde{\xi}\cap x)T \end{align*} $$

where $\tilde {\xi }$ is a Čech cochain representing $\cup ^3_Y$ supported in $\mathbb {T}_-$ . When considering the filtration induced by $\mathbb {T}_-\subset \mathbb {T}_Y$ , we see then that $\bar {\partial }x=\partial x$ on $C_*(\mathbb {T}_Y,\mathbb {T}_-)\otimes \mathbb {Q}[T,T^{-1}]$ . In particular we conclude (omitting the powers of T from the notation for simplicity) that:

• • on $I_+$ , $\bar {\partial }$ coincides with cap product with $[\tilde {\xi }]=\cup _Y^3$ ;

• • on E, $\bar {\partial }$ is given by the isomorphism (16) (which decreases the Morse filtration index by $1$ ) plus higher order terms (possibly involving $I_-$ ); in particular it is an isomorphism onto its image.

Putting everything together, we finally get the following.

Theorem 3.3. Suppose $(Y,\mathfrak {s})$ is of the simplest type. We have the direct sum decomposition of $\mathbb {Q}[U]$ -modules

where

$$ \begin{align*} i_*\left(\overline{\mathit{HM}}_*(Y,\mathfrak{s})\right)&\cong \mathrm{Cone}\left(\cup_Y^3:I^+\langle-2\rangle\rightarrow I^-\right)\otimes \mathcal{T}_+\\ {\mathit{HM}}_*(Y,\mathfrak{s})&\cong E_*\langle-2\rangle; \end{align*} $$

this holds at the level of graded modules up to an overall grading shift.

Note that we do not identify the U-action on ${\mathit {HM}}_*(Y,\mathfrak {s})$ , cf. Remark 2.2.

Proof. All the pieces are in places so we will comment on the remaining details. The grading shifts arise from the spectral flow when crossing $\mathsf {K}$ , and the reduced Floer homology arises as the kernel of $\check {\partial }$ from parts of the triangle of the form

(17) $$ \begin{align} U:\mathcal{T}^+\langle-2\rangle\rightarrow \mathcal{T}^+\langle-1\rangle, \end{align} $$

where the $-1$ shift is the one from (16). Finally, there are no extension problems to worry about (up to isomorphism of $\mathbb {Q}[U]$ -modules) because the reduced homology on the $E^{\infty }$ page lies in the top filtration level, and U-action from towers can only decrease it.

4 Concrete examples

In this section we study two concrete examples of spin $^c$ manifolds of simplest type (for suitable metric g and perturbation A), namely $S^1\times \Sigma _h$ with $h=2,3$ equipped with the unique torsion spin $^c$ structure $\mathfrak {s}_0$ , and use Theorem 3.3 to provide a computation of their Floer homology. We will suppress the genus of the surface from our notation when unnecessary.

It is well-known that the unperturbed equations on $(S^1\times \Sigma ,\mathfrak {s}_0)$ equipped with a product metric do not admit irreducible solutions for all h, see, for example, [Reference LinLin22b] for an exposition. Furthermore the torus $\mathbb {T}_{S^1\times \Sigma }$ of flat connections can be identified with ${S^1\times \mathrm {Jac}(\Sigma )}$ . The corresponding Dirac operator has kernel exactly at points in $\{0\}\times \Theta $ , where $\Theta $ is the classical theta divisor of the complex curve: after identifying $\mathrm {Jac}(\Sigma )$ with the torus of degree $h-1$ holomorphic line bundles $\Theta \subset \mathrm {Jac}(\Sigma )$ is the locus consisting of bundles admitting non-zero holomorphic sections. Notice that this depends on the metric on $\Sigma $ only through its underlying conformal structure.

Unfortunately, the argument for the absence of irreducible solutions (which involves dimensional reduction) is quite subtle and does not apply to the perturbed equations. We will therefore take an alternative route involving spectral geometry. In particular, we have the following.

Proof. The result follows from a direct computation which is well-known in classical electromagnetism in relation to the propagation of the electromagnetic field in a waveguide [Reference WaldWal22]. We denote by $\ast _\Sigma $ and $\ast $ the Hodge star on $\Sigma $ and $S^1\times \Sigma $ respectively. For a $1$ -form a on $\Sigma $ , we have that

$$ \begin{align*} \ast a&=\ast_\Sigma a\wedge dt\\ \ast(a\wedge dt)&=-\ast_\Sigma a. \end{align*} $$

Consider a general $1$ -form

$$ \begin{align*} b=fdt+\alpha \end{align*} $$

where f is a function on Y and $\alpha $ is a $1$ -parameter family of $1$ -forms on $\Sigma $ . Let us consider the operator $\ast d$ (which squares to $\Delta $ on coclosed $1$ -forms)

$$ \begin{align*} \ast db&=\ast(d_{\Sigma}fdt+d_\Sigma\alpha-\alpha'dt)=\\ &=-\ast_\Sigma d_\Sigma f+\ast_\Sigma \alpha'+(\ast_{\Sigma}d_\Sigma \alpha)dt. \end{align*} $$

Hence the eigenvalue equation $\ast db=\mu b$ takes the form

$$ \begin{align*} \ast_{\Sigma}d_\Sigma \alpha&=\mu f\\ -\ast_\Sigma d_\Sigma f+\ast_\Sigma \alpha'&=\mu\alpha. \end{align*} $$

Notice that solutions to the eigenvalue equation are automatically coclosed, and correspond to eigenforms with eigenvalue $\mu ^2$ . Let us assume $\mu \neq 0$ (harmonic forms are spanned by $dt$ and the harmonic forms on $\Sigma $ ). Substituting the first equation in the second one we then get (as $d^*_\Sigma =-\ast _\Sigma d_\Sigma \ast _\Sigma $ on $2$ -forms on a surface)

$$ \begin{align*} d^*_\Sigma d_\Sigma {\alpha}+\mu\ast_\Sigma \alpha'=\mu^2\alpha. \end{align*} $$

We now consider a basis of $1$ -eigenforms $e_n$ , $h_k$ and $c_n$ (exact, harmonic and coexact respectively), where we can assume $c_n=\ast _\Sigma e_n$ have eigenvalue $\lambda _n\neq 0$ for $\Delta _\Sigma =d_\Sigma d^*_\Sigma +d^*_\Sigma d_\Sigma $ , and $\ast h_i=h_{i+g}$ for $i=1,\dots , g$ . If $S^1=\mathbb {R}/2\pi L \mathbb {Z}$ has coordinate t, expanding in Fourier series in the t variable

$$ \begin{align*} \alpha(t)=\sum E_n(t)e_n+\sum C_n(t)c_n+\sum_{i\leq g} H_i(t)h_i+ H_{i+g}(t)h_{i+g}, \end{align*} $$

with $E_n, C_n$ and $H_k$ periodic of period $2\pi L$ , we have then

$$ \begin{align*} d^*_\Sigma d_\Sigma\alpha(t)&=\sum \lambda_n\cdot C_n(t)c_n\\ \ast_\Sigma \alpha'&=-\sum C_n'(t)e_n+\sum E_n'(t)c_n+\sum_{i\leq g}-H_{i+g}'(t)h_{i}+H_i'(t)h_{i+g}. \end{align*} $$

Hence, looking at the $c_n$ and $e_n$ component we get the system

$$ \begin{align*} \lambda_n C_n(t)+\mu E^{\prime}_n(t)&=\mu^2C_n(t)\\ -\mu C_n'(t)&=\mu^2 E_n(t). \end{align*} $$

and looking at the $h_i$ components we obtain the system

$$ \begin{align*} H_i'&=\mu H_{i+g}\\ H_{i+g}'&=-\mu H_{i}, \end{align*} $$

which we can address separately. In the first system, differentiating the second equation, and substituting in the first, we obtain

$$ \begin{align*} C_n"(t)=(\lambda_n-\mu^2)C_n(t). \end{align*} $$

In order for this equation to have periodic solutions (as $C_n$ has to be) we necessarily need $\lambda _n\leq \mu ^2$ . Conversely, one readily sees that $c_1$ defines a coexact eigenform on $S^1\times \Sigma $ with eigenvalue $\lambda _1(\Sigma )$ .

The second system involves the equation

$$ \begin{align*} H_k"(t)=-\mu^2 H_k(t). \end{align*} $$

On $S^1=\mathbb {R}/2\pi L \mathbb {Z}$ , the Fourier eigenmodes are $\{e^{int/L}\}$ with respective eigenvalue $n^2/L^2$ . Hence by choosing L sufficiently small these corresponding coexact eigenforms will have eigenvalue strictly larger than $\lambda _1(\Sigma )$ .

Suppose now that $\Sigma $ is equipped with a hyperbolic metric (by uniformization, there is a unique one in any conformal class). Then the Ricci tensor of $S^1\times \Sigma $ for the product metric is

$$ \begin{align*} \begin{bmatrix} 0&0&0\\ 0&-1&0\\ 0&0&-1 \end{bmatrix} \end{align*} $$

so that $\tilde {s}\equiv -2$ . By Proposition 1.1 we can therefore conclude the following.

We will focus on the two following examples of low-genus hyperbolic surfaces for which $\lambda _1>1$ :

• • the Bolza surface (which corresponds to the affine equation $y^2=x^5-x$ ) has genus $2$ and $\lambda _1\approx 3.84$ [Reference Strohmaier and UskiSU13].

• • the Klein quartic (which corresponds to the homogeneous equation $x^3y+y^3z+z^3x=0$ ) has genus $3$ and $\lambda _1\approx 2.68$ [Reference CookCoo18].

To proceed further, we need to study the locus in $\mathbb {T}_{S^1\times \Sigma }$ where the perturbed Dirac equation has solutions. Let us briefly recall the situation for the unperturbed Dirac equation, see [Reference LinLin22b] for more details. Fix a square root L of the canonical line bundle K on $\Sigma $ , so that $L\otimes \bar {K}\equiv \bar {L}$ ; this is equivalent to a choice of spin structure on $\Sigma $ . We then have for C a connection obtained from the spin connection one by twisting by a flat $U(1)$ -connection on $\Sigma $ the operator

$$ \begin{align*} \bar{\partial}_C:\Gamma(L)\rightarrow\Gamma(\bar{L}) \end{align*} $$

and its adjoint $\bar {\partial }^*_C$ . The spinor bundle on $S^1\times \Sigma $ (equipped with the product metric) is then $S=L\oplus \bar {L}$ , and the Dirac operator corresponding to $B=C+i\lambda dt$ is

$$ \begin{align*} D_B= \begin{bmatrix} i\frac{d}{dt}-\lambda& \sqrt{2}\bar{\partial}^*_C\\ \sqrt{2}\bar{\partial}_C&-i\frac{d}{dt}+\lambda \end{bmatrix} \end{align*} $$

From here one concludes that if $D_B$ has non-trivial kernel, after gauge transformation $\lambda =0$ and solutions $(\alpha ,\beta )$ satisfy $\bar {\partial }_C\alpha =0$ and $\bar {\partial }_C^*\beta =0$ (cf. [Reference LinLin22b] for an exposition). In particular B defines a holomorphic line bundle of degree $h-1$ with non-trivial holomorphic sections, i.e. it belongs to the theta divisor $\Theta \subset \mathrm {Jac}(\Sigma )$ as previously mentioned. The latter is a very well-studied object in the geometry of algebraic curves. It is the image of the Abel-Jacobi map

$$ \begin{align*} \mathcal{A}:\mathrm{Sym}^{h-1}(\Sigma_{h})\rightarrow\mathrm{Jac}(\Sigma_h). \end{align*} $$

This map is generically injective (because the degree of the line bundles is less than the genus of the surface), and the inverse image of a holomorphic line bundle is the projectivization of its space of holomorphic sections. In low genus, we have the following results (see, for example, [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85]):

• • When $h=2$ , the map is always an isomorphism onto its image.

• • When $h=3$ is generic (not hyperelliptic), the map is again an isomorphism onto its image.

Notice that the Klein quartic is non-hyperelliptic, because smooth quartics are canonically embedded (see, for example, [Reference MirandaMir95, Ch. VII]). Let us recall some well-known facts about the topology of symmetric products [Reference MacdonaldMac62]:

• • When $h=2$ , the map $\mathcal {A}:\Sigma \rightarrow \mathrm {Jac}(\Sigma )$ induces an injection in homology.

• • When $h=3$ , we have

  (18) $$ \begin{align} H_i\left(\mathrm{Sym}^2(\Sigma_3)\right)= \begin{cases} \mathbb{Q}\text{ if }i=0,4\\ H_1(\Sigma)\text{ if }i=1,3\\ \Lambda^2H_1(\Sigma)\oplus\mathbb{Q}\text{ if }i=2. \end{cases} \end{align} $$
  Furthermore, the induced map $\mathcal {A}_*$ has kernel given by the extra $\mathbb {Q}$ summand in $H_2$ .

When perturbing the equations, we have the following.

Proof. The theta divisor is the zero set of the theta function, and the Riemann singularity theorem says that the multiplicity of the theta function at a point L is given by $h^0(L)$ [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85, Ch. VI] by our discussion above, this is $1$ at all points under the assumption of the lemma. In our context, the Riemann singularity theorem and its proof mean that the map

$$ \begin{align*} \mathrm{Jac}(\Sigma)&\rightarrow\mathrm{Fred}_0\\ C&\mapsto \bar{\partial}_C, \end{align*} $$

where $\mathrm {Fred}_0$ is the space of index zero Fredholm operators (in a suitable analytical setup) and is transverse to the locus of operators with kernel. This is because $\bar {\partial }_C$ , the normal bundle is identified with operators on $\mathrm {ker}(\bar {\partial }_C)$ , i.e. a copy of $\mathbb {C}$ .

We can then reduce to study the linearization of the local model in the normal directions. In particular, after a quick analysis of the Fourier expansion as in [Reference LinLin22b], we are left to understand the locus is the space of matrices

$$ \begin{align*} \begin{bmatrix} -\lambda& \bar{a}\\ a& \lambda \end{bmatrix}\text{ where }\lambda\in\mathbb{R},a\in\mathbb{C} \end{align*} $$

having eigenvalue $\delta $ . This is exactly the locus $\lambda ^2+|a|^2=\delta ^2$ (where the multiplicity is always $1$ ), i.e. a copy of $S^2$ .

Putting everything together, we have proved the following.

Using Theorem 3.3 we then obtain the following concrete computations; the analogue in Heegaard Floer homology was proved using surgery techniques in [Reference Ozsváth and SzabóOS03], [Reference Jabuka and MarkJM08].

Corollary 4.5. We have the isomorphism of absolutely graded $\mathbb {Q}[U]$ -modules

where ${\mathit {HM}}_*(S^1\times \Sigma _3,\mathfrak {s}_0)=\mathbb {Q}_{-4}$ and the indices denote the absolute grading.

Proof. We begin with the case $g=2$ , and set $\delta>0$ . Consider a basis of $H_1(\mathbb {T})=H^1(S^1)\oplus H^1(\Sigma _2)$ of the form $z,x_1,y_1,x_2,y_2$ where the latter is symplectic for the cup product on $H^1(\Sigma _2)$ . The triple cup product is then given by

$$ \begin{align*} \cup_Y^3 =z\cup x_1\cup y_1+z\cup x_2\cup y_2. \end{align*} $$

Theorem 3.3 then identifies with the homology of the following complex:

The underlined towers correspond to $H_*(\mathbb {T}_-)\otimes \mathcal {T}_+$ because by Lemma 4.3 the subset $\mathbb {T}_-$ is a disk bundle over $\Sigma _2$ , the latter being the theta divisor; the shift comes from the spectral flow. The summands which are not underlined correspond to $H_*(\mathbb {T}_Y,\mathbb {T}_-)\otimes \mathcal {T}_+$ . Here the ith column corresponds to Morse index filtration $5-i$ on each of the pieces of $\mathbb {T}_Y\setminus \mathsf {K}$ . The fact that the arrows are surjections readily follows from the description of the triple cup product and the computation in $\overline {\mathit {HM}}_*$ .

In the genus three case the situation is more complicated because the map

$$ \begin{align*} H_2(\mathbb{T}_-)\rightarrow H_2(\mathbb{T}_Y) \end{align*} $$

has one dimensional kernel corresponding to the extra $\mathbb {Q}$ summand $\mathrm {ker}\mathcal {A}_*$ in the homology of $\mathrm {Sym}^2(\Sigma _3)$ , see (18). Theorem 3.3 identifies with the homology of the following complex:

Again, the underlined terms correspond to $H_*(\mathbb {T}_-)\otimes \mathcal {T}_+$ , while the non-underlined ones to $H_*(\mathbb {T}_Y,\mathbb {T}_-)\otimes \mathcal {T}_+$ , the non-underlined term $\mathrm {ker}\mathcal {A}_*\otimes {\mathcal {T}_+}$ arising from the connecting map (16).

We used solid arrows to indicate maps that do not interact with the two $\mathrm {ker}\mathcal {A}_*\otimes {\mathcal {T}_+}$ terms; as in the genus $2$ case, these are directly computed in terms of the triple cup product and the computation in $\overline {\mathit {HM}}_*$ . The dashed map only involves the two $\mathrm {ker}\mathcal {A}_*\otimes {\mathcal {T}_+}$ terms and is of the form (17); its kernel gives rise to the one dimensional reduced homology in the statement.

Finally, absolute gradings can be determined by noticing that the manifolds admit an orientation reversing diffeomorphism and the duality isomorphism

maps elements of degree j to elements of degree $-1-b_1(Y)-j$ , see [Reference Kronheimer and MrowkaKM07, Ch. 28].

Remark 4.3. It is natural to ask whether our approach applies to higher genus surfaces; there are two aspects of the story related to spectral and algebraic geometry respectively.

On the spectral side, there are some known examples of surfaces of genus with $\lambda _1^*>1$ . For example, the Bring curve (the common zero locus of the equations ${v^k+w^k+x^k+y^k+z^k}$ for $k=1,2,3$ ) has genus $4$ and $\lambda _1\approx 1.92$ [Reference CookCoo18] while the Fricke-Macbeath surface, the unique Hurwitz surface of genus $7$ , satisfies $\lambda _1\in [1.23,1.26]$ [Reference C-hLee23]. Notice though that for genus high enough there are no hyperbolic surfaces with $\lambda _1^*>1$ (see, for example, [Reference Fortier Borque and PetriFBP23]). Indeed, from the point of view of spectral theory of the Laplacian of functions on a hyperbolic surface it is natural to divide eigenvalues in small or large, the threshold being $1/4$ (significance of the latter is that it is the spectral gap in $L^2$ of the hyperbolic plane $\mathbb {H}^2$ ).

The main complication arises on the algebro-geometric side, as the theta divisor is never smooth when $g\geq 4$ [Reference Arbarello, Cornalba, Griffiths and HarrisACGH85]. Smoothness was a key input in proving that our examples are of the simplest type in Lemma 4.3. For example, for a generic genus $4$ surface, there are exactly two singular points in $\Theta $ consisting of line bundles with $h^0=2$ ; furthermore the singular point $\Theta \subset \mathrm {Jac}(\Sigma )$ is modelled after the singular quadric

$$ \begin{align*} \{xy=tz\}\subset\mathbb{C}^4 \end{align*} $$

and the Abel-Jacobi map $\mathcal {A}$ is given by the small resolution.

5 The three-torus

We will prove our main results, Theorem 0.2 and 5.1. We begin by briefly recalling how to determine the Floer homology group

(19)

by looking at a flat metric (cf. [Reference Kronheimer and MrowkaKM07, Ch. $38$ ]). There is a unique flat spin $^c$ connection $B_0$ for which $B_0$ has kernel, corresponding to the two dimensional space of parallel spinors. By adding a $\delta $ -perturbation to the equations, this gives rise to a small $S^2\subset \mathbb {T}_{T^3}$ of operators having kernel as in Lemma 4.3. Choosing a standard Morse function on $\mathbb {T}_{T^3}$ so that minimum lies inside $S^2$ , the Floer chain complex is then directly determined to be

where the underlined summand is shifted up because of the spectral flow.

We are now ready to prove our main result.

Proof of Theorem 0.2

For a generic spinorial perturbation A, $\mathsf {K}$ is smooth for dimensional reasons, see Remark 1.4. Of course, it will in general not be of the simplest type. For a component $U\subset \mathbb {T}_Y\setminus \mathsf {K}$ , so that U is a component of $\mathrm {U}\subset \mathrm {int}\mathbb {T}_j\setminus \mathbb {T}_{j-1}$ for some j, we call its upper boundary the intersection of $\bar {U}\cap \partial \mathbb {T}_j$ and its lower boundary the intersection of $\bar {U}\cap \partial \mathbb {T}_{j-1}$ . We can then find an A-transverse Morse function $f:\mathbb {T}_Y\rightarrow \mathbb {R}$ satisfying the following property: f has maxima only in the components of $\mathbb {T}_Y\setminus \mathsf {K}$ with no upper boundary, and has exactly one maximum in each of them; similarly, f has minima only in the components of $\mathbb {T}_Y\setminus \mathsf {K}$ with no lower boundary, and has exactly one maximum in each of them. This can be achieved by suitable canceling $0$ - $1$ and $2$ - $3$ handle pairs.

Fix an A-transverse Morse function f satisfying the condition above. By Lemma 2.7 we can consider the spectral sequence induced by the Morse filtration. Let us denote

$$ \begin{align*} E^2=\mathcal{F}_3\oplus \mathcal{F}_2\oplus \mathcal{F}_1\oplus \mathcal{F}_0 \end{align*} $$

where the subscripts denote the filtration index. Notice that $\mathcal {F}_i$ lies in degree i modulo $2$ , so that $d_2=0$ and $E^2=E^3$ . In degrees high enough, $(E^3,d_3)$ is forced to look like the Floer chain complex of the flat torus described above. In particular we have

$$ \begin{align*} \mathcal{F}_i= \begin{cases} \mathcal{T}_+\langle d_i\rangle\oplus G_i \text{ if }i=0,3\\ \mathcal{T}_+^{\oplus3}\langle d_i\rangle\oplus G_i \text{ if }i=1,2 \end{cases} \end{align*} $$

for some integer $d_i$ and $\mathbb {Q}[U]$ -modules $G_i$ which are finite dimensional $\mathbb {Q}$ -vector spaces.

Consider the collection of all minima $x_i$ for $i=1,\dots ,n$ of ${f}$ . Because $\mathbb {T}$ is connected, we can find (up to reordering) critical points $y_i$ for $1\leq i\leq n-1$ of index $1$ such that $\partial y_i=x_i-x_{i+1}$ . Our choice of Morse function f assures that there is non-zero spectral flow on the trajectories between $y_i$ and $x_i$ or $x_{i+1}$ . Hence, using Lemma 2.7, these critical points determine a subcomplex in $\check {C}_*$ of the form

(20)

where the top (resp. bottom) row consists of the $y_i$ (resp $x_i$ ), and each arrow is multiplication by $U^{k}$ for some $k\geq 1$ because of the spectral flow (we are not keeping track of gradings in the figure). From here we see that in fact

(21) $$ \begin{align} \mathcal{F}_0=\mathcal{T}_+\langle d_0\rangle \end{align} $$

i.e. $G_0=0$ , and $G_1\neq 0$ as soon as there is more than one minimum (cf. the analogous computations for the case $b_1=1$ in [Reference LinLin22b]). Considering now the higher differential

(22) $$ \begin{align} d_3:\mathcal{F}_3\rightarrow \mathcal{F}_0, \end{align} $$

which is clearly surjective because of the $\mathbb {Q}[U]$ -module structure, we see that it is injective. This follows because anything in the kernel would lead to non-trivial reduced homology in , again because the U-action from towers lowers the filtration level. Hence we conclude that

(23) $$ \begin{align} \mathcal{F}_3=\mathcal{T}_+\langle d_0+1\rangle. \end{align} $$

Turning upside down the argument for critical points of Morse index $0$ and $1$ above (20), this implies that f has exactly one maximum. Furthermore, for grading reasons (22) is the only non-zero component of $d_3$ . In particular, $G_1$ survives in the $E^{\infty }$ page. Because of filtration reasons, it contributes to reduced homology, hence it vanishes, and f has exactly one minimum as well.

Furthermore, comparing the gradings in (21) and (23), we see that any path between the maximum and the minimum has spectral flow $-1$ . This implies in particular that $\mathsf {K}$ is of the simplest type, so that we can apply Theorem 3.3. Because $T^3$ has trivial reduced Floer homology, we see that $i_*:H_*(\mathbb {T}_-)\rightarrow H_*(\mathbb {T})$ is injective. By inspecting the compatibility with the $\Lambda ^* H_1(T^3)$ -action, the $E^1$ page of the simplest type spectral sequence is forced to be either that of the flat torus described above or

which is simply the chain complex corresponding to $(-\delta ,-\varepsilon )$ . Let us focus on the former as the latter case is identical. We see that $H_*(\mathbb {T}_-)=H_*(\mathrm {pt})$ . This implies that $\mathsf {K}$ is a union of spheres, because for a general three-manifold M the inclusion of the boundary $\partial M\subset M$ satisfies the ‘half dies, half lives’ property, i.e. the map induced by the inclusion

$$ \begin{align*} H_1(\partial M;\mathbb{Q})\rightarrow H_1(M;\mathbb{Q}) \end{align*} $$

has kernel of dimension $b_1(\partial M)/2$ . Finally, there is only one boundary component because otherwise $H_2(\mathbb {T}_-)=H^1(\mathbb {T_-},\mathsf {K})$ would be non-trivial; hence $\mathsf {K}$ is a two-sphere.

To conclude, we note that under some mild additional assumptions, we can say more about the location of the two-sphere $\mathsf {K}=S^2$ inside $\mathbb {T}_{T^3}$ . First of all, because $\mathbb {T}_{T^3}$ is an irreducible three-manifold, Alexander’s theorem implies that $\mathsf {K}$ separates $T^3$ in two components, one of which is diffeomorphic to the interior of an embedded ball $B^3\hookrightarrow T^3$ (see, for example, [Reference RolfsenRol90]). In $\mathbb {T}_{T^3}$ there are eight special points corresponding to the spin structures $\mathsf {s}$ on $T^3$ ; once one is fixed, the other ones are obtained by twisting by a flat line bundle with holonomy $\pm 1$ around each loop. Among these spin structures there is a special one $\mathsf {s}_0$ , which can be identified by the fact that it has Rokhlin invariant $\mu (T^3,\mathsf {s}_0)=1$ , while the other ones have $\mu (T^3,\mathsf {s})=0$ (see [Reference KirbyKir89, Ch. 5]). We then have the following.

Theorem 5.1. In the setup the Theorem 0.2, assume that under the natural decomposition

$$ \begin{align*} A=A_0+\tau\cdot I \end{align*} $$

with $A_0$ traceless and $\tau =\mathrm {tr}A/2$ a function, the $C^0$ norm of $A_0$ is small compared to that of $\tau $ . Then the component of $\mathbb {T}_{T^3}\setminus \mathsf {K}$ which is diffeomorphic to $\mathrm {int} B^3$ contains the spin structure $\mathsf {s}_0$ , while the other component contains all the remaining spin structures $\mathsf {s}\neq \mathsf {s}_0$ .

Notice that in the case of a flat metric $g_{\mathrm {flat}}$ , the spin structure $\mathfrak {s}_0$ can be identified as the unique $B_0$ for which $D_{B_0}$ has non-trivial kernel, and the conclusion of the result can be again checked explicitly for $A=\delta \neq 0$ small.