3.1 Review of the three-manifold case

Let Y be a three-manifold. As in [Reference Kronheimer and MrowkaKM07, Section $35.1$ ], given a metric on $\mathbb T_Y$ and a Morse function $f: \mathbb T_Y \to \mathbb R$ , one may choose an appropriate perturbation of the Seiberg–Witten equations so that all reducible critical points and trajectories are cut out transversely. In this situation, Kronheimer and Mrowka give a description of $\bar {C}_*(Y,\mathfrak {s})$ in terms of “coupled Morse theory”. The structure of this complex is determined in [Reference Kronheimer and MrowkaKM07, Section 33.3], which we record.

Lemma 3.1. In the situation above, one has an isomorphism of relatively $\Bbb Z$ -graded complexes over $\mathbb Z[U]$

$$ \begin{align*}\bar{C}_*(Y, \mathfrak s; \Gamma_\eta) \cong C_*(\mathbb T_Y, f; \Gamma_\eta)[U, U^{-1}].\end{align*} $$

With respect to this isomorphism, the differential $\overline \partial $ is sent to

$$ \begin{align*}\partial_f + \partial_3 U^{-1}+\partial_5 U^{-2}+\dots\end{align*} $$

with $\partial _f$ the Morse differential on $C_*(\mathbb T_Y, f; \Gamma _\eta )$ .

In [Reference Kronheimer and MrowkaKM07, Section $34.2$ ] the authors further show that one can deform the family of operators (in a suitable space of operators on a Hilbert space) so that the for the resulting coupled Morse complex the terms $\partial _{2i+1}$ with $i\geq 2$ are zero. On the other hand, in the present paper we will work with small perturbations of the family (of geometric nature) so in general one should expect the higher terms to be nonvanishing.

It is worth being more precise about the identifications in this isomorphism. For each critical point $q \in \text {Crit}(f)$ , choose a labeling of the eigenvalues of the Dirac operator $D_q$ by

$$ \begin{align*}\cdots < \lambda_{-1}(q) < \lambda_0(q) < \lambda_1(q) < \cdots\end{align*} $$

For some i we have $\lambda _{i-1}(q) < 0 < \lambda _i(q)$ ; we record $d(q) = i$ as the shift of this labeling from the natural labeling. Finally, we demand that these labelings respect the spectral flow, in the sense that

$$ \begin{align*}\text{sf}(q_1, q_2) = d(q_2) - d(q_1);\end{align*} $$

this is possible because $c_1(\mathfrak {s})$ torsion implies that there is no spectral flow around loops. Writing

$$ \begin{align*}\tilde u_2: \bar C(Y, \mathfrak s; \Gamma_\eta) \to \bar C(Y, \mathfrak s; \Gamma_\eta)\end{align*} $$

for the chain map whose induced map on homology is the U-action, [Reference Kronheimer and MrowkaKM07, Lemma 33.3.9] guarantees $\tilde u_2$ is an isomorphism. The inverse of our identification sends $qU^i$ to $\tilde u_2^i(q, \lambda _0(q))$ , which is equal to the sum of $(q, \lambda _{-i}(q))$ and critical points of strictly smaller Morse index.

For the purposes of the paper, we will not be interested in the actual U-action on $\bar {C}_*(Y, \mathfrak s; \Gamma _\eta )$ , but only on the natural filtration by powers of U, with

(3.1) $$ \begin{align} \mathcal F_k \bar{C}_* \cong C_*(\mathbb T_Y, f)\otimes U^{\lfloor k/2\rfloor} \mathbb{Z}[U^{-1}], \end{align} $$

that is, the terms with U-power $\leq k/2$ . While this is not a filtration of U-modules, multiplication by U sends each $\mathcal F_k$ isomorphically onto $\mathcal F_{k+2}$ , and does induce the structure of a $\Bbb Z[U, U^{-1}]$ -module on the pages of the associated spectral sequence. The $E^1$ page of the associated spectral sequence is precisely the Laurent polynomial ring over the Morse complex $C_*(\mathbb T_Y, f; \Gamma _\eta )[U, U^{-1}]$ .

Remark 3.2. The U-filtration is seen to be essentially equivalent to the filtration by Morse index,

$$ \begin{align*}\mathcal F^{Morse}_k = C_{* \le k}(\mathbb T_Y, f) \otimes \mathbb Z[U, U^{-1}];\end{align*} $$

we have $\mathcal F_k \bar C_d = \mathcal F^{Morse}_{d+k} \bar C_d.$

In [Reference Kronheimer and MrowkaKM07, Section 35], the authors determine the differential $\partial _3$ on the $E^3$ page of this spectral sequence in terms of the triple cup product

$$ \begin{align*} \cup_Y: \Lambda^3 H^1(Y;\mathbb{Z})&\rightarrow \mathbb{Z}\\ \alpha\wedge\beta\wedge\gamma&\rightarrow \langle\alpha\cup\beta\cup\gamma,[Y]\rangle \end{align*} $$

and show that the higher differentials in this spectral sequence vanish over $\Bbb Q$ . As a consequence, they establish a canonical isomorphism

(3.2) $$ \begin{align} \text{gr } \overline{\mathit{HM}}_*(Y,\mathfrak{s};\mathbb{Q})\cong {HC}_*(Y;\mathbb{Q}) \end{align} $$

where the former is the associated graded group of the U-filtration on homology, and the latter is the cup homology [Reference MarkMar08]. The cup homology is the homology of the chain complex whose underlying module is

$$ \begin{align*} CC_*(Y)=\Lambda^* H^1(Y;\mathbb{Z})\otimes \mathbb{Z}[U^{-1},U] \end{align*} $$

and whose differential is given by

(3.3) $$ \begin{align} d^3(\omega\otimes U^n)= \iota_{\cup_Y^3}\omega\otimes U^{n-1}, \end{align} $$

where $\iota _{\cup _Y^3}$ is the contraction with the triple cup product $\cup _Y^3$ sending $\alpha _1\wedge \dots \wedge \alpha _k$ to

$$ \begin{align*} \sum_{i_1<i_2<i_3} (-1)^{i_1+i_2+i_3}\langle \alpha_{i_1}\cup \alpha_{i_2}\cup \alpha_{i_3},[Y]\rangle\cdot \alpha_1 \wedge\dots \wedge\hat{\alpha}_{i_1} \wedge\dots \wedge\hat{\alpha}_{i_2} \wedge\dots \wedge\hat{\alpha}_{i_3}\wedge\dots\wedge\alpha_k. \end{align*} $$

3.2 The formula

Whereas [Reference Kronheimer and MrowkaKM07, Section 35] determined the homology groups $\overline {\mathit {HM}}$ , we will now identify the cobordism maps (up to filtration).

In the present situation, we have a correspondence

$$ \begin{align*}(\mathbb T_{Y_-}, f_-) \xleftarrow{i_-^*} \mathbb T_W \xrightarrow{i_+^*} (\mathbb T_{Y_+}, f_+),\end{align*} $$

with $f_\pm $ Morse functions on the tori $\mathbb T_{Y_\pm }$ , and the maps $i_\pm ^*$ affine. So long as the maps $i_\pm ^*$ are generic with respect to the Morse functions $f_\pm $ – which can achieved by a translation – there is a well-defined induced map on Morse complexes, given by counting intersection points of $\mathbb T_W$ with $U_a \times S_b$ in $\mathbb T_{Y_-} \times \mathbb T_{Y_+}$ [Reference Kronheimer and MrowkaKM07, Section 2.8]. This map will be denoted $m_W$ .

In the case that $Y_-$ and $Y_+$ are equipped with real 1-cycles $\eta _\pm \in C_1(Y_\pm; \Bbb R)$ , a real 2-chain $\nu \in C_2(W; \Bbb R)$ whose boundary is $\partial \nu = i_+ \eta _+ - i_- \eta _-$ provides an isomorphism of local systems

$$ \begin{align*}\Gamma_\nu: i_-^* \Gamma_{\eta_-} \cong i_+^* \Gamma_{\eta_+}.\end{align*} $$

In this situation, we have an induced map on Morse complexes

$$ \begin{align*}m_{W,\nu}: C_*(Y_-, f_-; \Gamma_{\eta_-}) \to C_*(Y_+, f_+; \Gamma_{\eta_+}),\end{align*} $$

also described in [Reference Kronheimer and MrowkaKM07, Section 2.8], and defined in a similar fashion.

Theorem 3.2. Suppose $(W,\mathfrak {s}_W): (Y_-, \mathfrak {s}_-) \to (Y_+, \mathfrak {s}_+)$ is a cobordism with $b^+(W) = 0$ and $\mathfrak {s}_\pm $ torsion. With respect to the isomorphisms of Lemma 3.1, the map $\bar {m}_*$ inducing $\overline {\mathit {HM}}_*(W, \mathfrak {s}_W; \Gamma _\nu )$ is filtered, and takes the form

$$ \begin{align*}\bar{m}_* = m_{W, \nu} U^d + m_2 U^{d-1} + m_4 U^{d-2} + \cdots\end{align*} $$

for an appropriate integer d and appropriate maps $m_{2i}$ for $i> 0$ .

That is, if we consider the filtration $\mathcal {F}_k\bar {C}_*$ in (3.1), the map $\overline {m}_*(W, \mathfrak {s}_W; \Gamma _\nu )$ is a filtered map with associated graded map equal to the induced map on the associated Morse complex. As a direct consequence, we obtain the following computation, which might be of independent interest.

Remark 3.4. Notice that the latter is actually a chain map on the cup chain complex; this follows because $i_+^*(\cup _{Y_+})$ is the same as $i_-^*(\cup _{Y_-})$ as maps from $\Lambda ^3H^1(W)$ to $\mathbb {Z}$ . More generally, for a manifold W with possibly disconnected boundary $\partial W$ , let

$$ \begin{align*} r:H^1(W)\rightarrow H^1(\partial W) \end{align*} $$

denote the restriction map; we denote the induced map on the exterior algebras by r as well. Then the pullback of the triple cup product of $\partial W$ under r vanishes. This can be seen geometrically as follows: the map r corresponds via Poincaré duality to the boundary map

$$ \begin{align*} \partial: H_3(W,\partial W)\rightarrow H_2(\partial W). \end{align*} $$

Given three elements $\alpha _i$ in $H^1(\partial W)$ , represent their duals by surfaces $S_i$ in $\partial W$ . If the $\alpha _i$ are in the image of r, then $S_i$ is the boundary of three-manifolds $T_i$ in W. After moving everything to a transverse position, $S_1\cap S_2\cap S_3$ (which is dual to $\alpha _1\wedge \alpha _2\wedge \alpha _3$ ) is a $0$ -manifold which is the boundary of $T_1\cap T_2\cap T_3$ , hence is zero in homology.

These results deserve some further comment. First of all, the isomorphism in Lemma 3.1 is only defined up to multiplication by powers of U. This can be pinned down by using the absolute $\Bbb Q$ -grading of [Reference Kronheimer and MrowkaKM07, Section 28.3], according to which the degree of $\overline {\mathit {HM}}_*(W;\mathfrak {s}_W)$ is given by

(3.4) $$ \begin{align} \frac{1}{4}c_1^2(\mathfrak{s}_W)-\iota(W)-\frac{1}{4}\sigma(W) = d(\mathfrak{s}_W) - \frac 12 \big(b_1(Y_-) - b_1(Y_+)\big) \end{align} $$

where $d(\mathfrak {s}_W)$ is the expression (1.3) and

$$ \begin{align*} \iota(W)=\frac{1}{2}\big(\chi(W)+\sigma(W)+b_1(Y_-)-b_1(Y_+)\big). \end{align*} $$

As U has grading $-2$ , and the degree of $m_{W, \nu }$ is $b_1(W) - b_1(Y_-)$ , we may thus compute the integer d from Theorem 3.2 as

$$ \begin{align*}d = -\frac 12 d(\mathfrak{s}_W) - \frac 14 \left(2b_1(W) - b_1(Y_-) - b_1(Y_+)\right).\end{align*} $$

In the case of most interest to us in the current paper, we have $Y_- = Y_+$ and $b_1(W) = b_1(Y_-)$ so $d = -\frac 12 d(\mathfrak {s}_W).$

Secondly, even though it will not be needed for this paper, for some applications one would like to compute the map itself rather than the associated graded map. One might hope to compute the higher terms up to (filtered) homotopy, and expect them to be related to the kernel of the Dirac operators on W coupled to the spectral decomposition associated to $D_{Y_\pm }$ . An important technical complication when trying to write down an explicit formula in homology is that while there is a canonical identification of the associated graded module of $\overline {\mathit {HM}}_*$ (3.2), there is no known canonical lift to $\overline {\mathit {HM}}_*$ itself.

3.3 Proof of Theorem 3.2

Recall from [Reference Kronheimer and MrowkaKM07, Chapter $25$ ] that the map induced by a cobordism $(W,\mathfrak {s}_W)$ on $\overline {\mathit {HM}}_*$ (possibly with local coefficient systems) is obtained by counting solely reducible solutions to the Seiberg–Witten equations in the blow-up on the cobordism $W^*$ with cylindrical ends attached. The unperturbed equations for a reducible configuration $(A,0,\varphi )$ in the blow-up are

$$ \begin{align*} \begin{cases} D_A^+\varphi=0\\ F_{A^t}^+=0, \end{cases} \end{align*} $$

which one needs to study with the right asymptotic growth conditions for $\varphi $ on the cylindrical ends; different asymptotic conditions change the index of the Dirac operator $D_A^+$ . We are counting solutions to this equation of index zero. We may break this count up into a sum

$$ \begin{align*}\bar{m}_*=m_0 + m_2 + \cdots\end{align*} $$

as follows. The map $m_{2i}$ , by definition, counts solutions $(A, 0, \phi )$ to a perturbed version of these equations for which A lies in a $2i$ -dimensional moduli space (or equivalently, is asymptotic to flat connections whose Morse index has difference $2i-(b_1(W)-b_1(Y_+))$ , see the formula on p. 47 of [Reference Kronheimer and MrowkaKM07, Section 2.8]) and the asymptotic conditions are chosen so that $\text {ind}_{\mathbb C}(D_A^+) = 1-i$ ; after projectivizing one obtains a finite number of points. The highest-filtration term in $m_0$ then corresponds to the count of index-zero solutions to a perturbed version of the second equation and asymptotic conditions for which $\text {ind}_{\mathbb {C}}(D_A^+) = 1$ .

The second equation is simply the ASD equation, and its solutions up to gauge with $L^2$ -curvature form the $b_1(W)$ -dimensional torus

$$ \begin{align*} \mathbb{T}_W=H^1(W;i\mathbb{R})/H^1(W;2\pi i\mathbb{Z}). \end{align*} $$

Here, following [Reference Atiyah, Patodi and SingerAPS75, Proposition 4.9], we can interpret the numerator as a space $L^2_{\text {ext}}$ of extended $L^2$ -harmonic $1$ -forms $\alpha $ , that is, the harmonic $1$ -forms that exponentially decay on each end to a time-independent harmonic $1$ -form $\alpha _{\pm }$ . Sending each such form to the limiting value is the Hodge theoretic incarnation of the maps

$$ \begin{align*} i^*_{\pm}:H^1(W)\rightarrow H^1(Y_\pm). \end{align*} $$

In the isomorphism of Lemma 3.1, we use a perturbation of the form $f_\pm \circ p_\pm $ where $f_\pm $ are our chosen Morse functions and

$$ \begin{align*} p_\pm:\mathcal{B}(Y_\pm,\mathfrak{s})\rightarrow \mathbb{T}_{Y_\pm} \end{align*} $$

is the retraction obtained by mapping $(B,\Psi )$ via the $L^2$ -orthogonal projection of the $1$ -form $B-B_0$ to its harmonic part $(B-B_0)^{\mathrm {harm}}$ given by the Hodge decompositionFootnote ² . We use these to perturb the Seiberg–Witten equations as in [Reference Kronheimer and MrowkaKM07, Chapter $24$ ], using on $[0,\infty )\times Y_+$ a smooth cutoff function

$$ \begin{align*}\beta: [0, \infty)\to [0,1]\end{align*} $$

which is nondecreasing, equal to zero near zero and equal to $1$ for $t \ge 1$ , and similarly on the other end.

We then consider solutions $(A,\varphi )$ to a perturbed Seiberg–Witten equation for which the gauge equivalence classes $A\lvert _{(-\infty ,0]\times Y_-}$ and $A\lvert _{[0,+\infty )\times Y_+}$ converge to limits $q_\pm $ , reducible solutions on $Y_{\pm }$ which are critical points of $f_{\pm }$ . We demand $\varphi $ satisfies some asymptotic growth conditions at $\pm \infty $ in terms of the eigenvalues and eigenspinors of the limit Dirac operators $D_{q_{\pm }}$ (which can also be rephrased in terms of convergence in the $\tau $ -model).

Again let us focus on the perturbed ASD equation. The torus $\mathbb T_W$ is equipped with a map to $\mathbb T_{Y_-} \times \mathbb T_{Y_+}$ , and inside the latter we can consider the unstable and stable submanifolds $U_{q_-},S_{q_+}$ of critical points $q_\pm $ of $f_\pm $ .

Proof. Identifying $1$ -forms on the end $[0,\infty ) \times Y_+$ with time-dependent elements $\omega (t)\in \Omega ^0(Y_+) \oplus \Omega ^1(Y_+)$ , the unperturbed ASD equation together with the Coulomb gauge fixing condition

$$ \begin{align*} (-d^*)\oplus d^+:\Omega^1\rightarrow \Omega^0\oplus \Omega^+ \end{align*} $$

on the cylinder can be written as

$$ \begin{align*}\frac{d}{dt}\omega(t) + L\omega(t)=0\end{align*} $$

where

$$ \begin{align*}L = \begin{pmatrix} 0 & -d^* \\ -d & *d \end{pmatrix}.\end{align*} $$

The perturbations we consider are defined via the $L^2$ -orthogonal projection to harmonic $1$ -forms, and the corresponding perturbed ASD equations together with Coulomb gauge fixing then take the form

$$ \begin{align*}\frac{d}{dt}\omega(t) + L\omega(t) + \beta(t) (\mathrm{grad}\tilde f_+)(\omega^{\text{harm}}(t)) = 0,\end{align*} $$

where $\tilde f_+: H^1(Y_+; i\mathbb R) \to \mathbb R$ is the periodic lift of the chosen Morse function on $\mathbb T_{Y_\pm },$ which is then applied to the harmonic part of the time-dependent form in $\Omega ^1(Y_+)$ .

The torus $\mathbb T_W$ can be understood as the space of unperturbed ASD connections in $L^2_k$ modulo gauge on the compact manifold W such that the coclosed parts of the boundary values

$$ \begin{align*}\Pi: L^2_k(\Omega^1(W)) \to \ker(d^*_{Y_-}) \oplus \ker(d^*_{Y_+}) \subset L^2_{k-1/2}(\Omega^1(Y_-)) \oplus L^2_{k-1/2}(\Omega^1(Y_+))\end{align*} $$

lie in the spectral subspace

$$ \begin{align*}H_-^{\le 0} \oplus H_+^{\ge 0} \subset \ker(d^*_{Y_-}) \oplus \ker(d^*_{Y_+})\end{align*} $$

spanned by nonnegative (respectively nonpositive) eigenspaces of the signature operator $*d_{Y_\pm }$ . Because the unperturbed ASD connections solves a first-order ODE on the ends, each boundary value lying in this subspace extends to a unique solution to the ASD equation on the cylinder (up to gauge); the spectral condition guarantees that the solution is in $L^2_{\text {ext}}$ up to gauge. Notice that $H_+^{\ge 0}$ splits into the sum $H^1(Y_+) \oplus H_+^{>0}$ of the space of harmonic forms and the span of strictly positive eigenspaces, and similarly for $H_-^{\geq 0}$ .

Because a perturbed ASD connection on $[0, \infty ) \times Y_+$ again solves a first-order ODE, it is determined by its boundary value in $\ker (d^*_{Y_+})$ (up to gauge). Further, because the operator is only perturbed on the harmonic part, it is still true that for the ASD connection to be $L^2_{\text {ext}}$ this boundary value must lie in the spectral subspace $H^{\ge 0}_+$ . A similar argument applies for $Y_-$ . Because the equation is unchanged on W, the perturbed ASD solutions are identified with a subset of $\mathbb T_W$ , which was earlier identified with ASD solutions on W whose boundary values lie in $H^{\le 0}_- \times H^{\ge 0}_+$ .

Finally, examining the harmonic part, we see that a boundary value extends to a solution to the perturbed equation with the correct asymptotics if and only if the harmonic part $[\omega ^{\text {harm}}]|_{\{0\} \times Y_\pm } \in \mathbb T_{Y_\pm }$ lies in the unstable and stable manifold of $q_-$ and $q_+$ , respectively.

The claim about orientations follows from compatibility between orientations and gluing, as in [Reference Kronheimer and MrowkaKM07, Section 20.3]. The claim about the local system follows immediately from the definition: both maps are weighted by a factor of $e^{\int _\nu F^+_{A^t}}$ , the only difference being that in the Morse theory case we integrate over the intersection of $\nu $ with the compact part W of the cobordism while in the gauge theory case we integrate over the noncompact surface $\nu ^* \subset W^*$ . However, because $A^t$ is flat on the ends, the additional contribution is zero.

This identifies the “top-degree term” in $\bar m_*$ , which corresponds to the case where the relevant Dirac operator has index $1$ : the kernel of this operator is a one-dimensional complex vector space, so that after projectivizing it contributes a single positively oriented point, up to gauge. Hence the relevant moduli spaces of solutions simply count solutions to the perturbed ASD equations.

To conclude the proof, following [Reference Kronheimer and MrowkaKM07, Section $2.8$ ] (and dropping local systems for simplicity), the map induced on the Morse complexes

(3.5) $$ \begin{align} m_{W}: C_*(\mathbb T_{Y_-})\rightarrow C_*(\mathbb T_{Y_+}) \end{align} $$

by the correspondence $\mathbb T_W \to \mathbb T_{Y_-} \times \mathbb T_{Y_+}$ is obtained (under suitable transversality conditions) by considering triples $(\gamma _-,\gamma _+,w)$ where

• • $\gamma _-:(-\infty ,0]\rightarrow \mathbb T_{Y_-}$ is a finite-energy half trajectory with $\gamma _-(-\infty ) = q_-$

• • $\gamma _+:[0,\infty )\rightarrow \mathbb T_{Y_+}$ is a finite-energy half trajectory with $\gamma _+(+\infty ) = q_+$

• • $w \in \mathbb T_W$ and $i^*_\pm (w)=(\gamma _-(0),\gamma _+(0))$ .

Because the space of such half-trajectories can be identified with the unstable and stable manifolds $U_{q_-}$ and $S_{q_+}$ , we see that $m_W$ is defined by exactly the same counts as the top-filtration term of $\bar m_*$ .

4.1 Definitions.

To proceed, suppose we have a metric and perturbation on Y (for the spin $^c$ structure $\mathfrak {s}$ ) such that there are no irreducible solutions, transversality is achieved in the sense of [Reference Kronheimer and MrowkaKM07], and the bar complex is computed by coupled Morse theory as in the previous section. The relatively graded $\Bbb Q$ -vector space $\hat {C}_*(Y,\mathfrak {s})$ can then be identified as a subspace

(4.1) $$ \begin{align}\hat{C}_*(Y,\mathfrak{s}) = \bigoplus_{q \in \text{Crit}(f)} U^{d(q)} \Bbb Q[U] \subset \bigoplus_{q \in \text{Crit}(f)} \Bbb Q[U, U^{-1}] = \bar{C}_*(Y,\mathfrak{s}), \end{align} $$

where the shift $d(q) \in \Bbb Z$ and the identification of $\bar {C}_*(Y,\mathfrak {s})$ (as a $\Bbb Q[U]$ -module) are discussed after Lemma 3.1. It is often convenient to normalize d so that $d(q) \le 0$ , with largest value equal to zero, and we will do so in what follows.

There are four closely related subtleties. The first is that $\hat C$ is generally not a subcomplex: the differential on $\hat C$ is

$$ \begin{align*}\hat{\partial} = -\overline \partial^u_u - \overline \partial^s_u \partial^u_s,\end{align*} $$

where the term $\partial ^u_s$ is defined in terms of irreducible Seiberg–Witten solutions on $\mathbb R \times Y$ . Secondly, even though $\hat {C}(Y,\mathfrak {s})$ is naturally a filtered module, with filtration given by (3.1), the differential $\hat \partial $ does not necessarily preserve it. Thirdly, the comparison (anti-)chain map

(4.2) $$ \begin{align} p = 1 + \partial_s^u : \hat{C}_*(Y,\mathfrak{s}) \to \bar{C}_*(Y,\mathfrak{s}) \end{align} $$

also involves this term and is not necessarily filtration-preserving. Finally, $\hat C$ is generally not a U-submodule, because p only commutes with the action of U up to homotopy.Footnote ³

Because of this, we make the following definition.

In this case, $\hat {C}_*(Y,\mathfrak {s})$ is a filtered complex for which p is a filtered map with associated graded map the natural inclusion; the associated graded complexes are

$$ \begin{align*} \text{gr}_{2i+1} \hat C &= 0,\\ \text{gr}_{2i} \hat C &= \text{span} \{q \in \text{Crit}(f) \mid d(q) \le i\} \subset C_*(\mathbb T, f; \Gamma_\eta).\\[-17pt] \end{align*} $$

Moving to the cobordism maps, there are two distinct cases to analyze:

• • if $b^+(W)> 0$ , after choosing a perturbation by a self-dual form as in [Reference Kronheimer and MrowkaKM07, Section $27.2$ ], there are no reducible solutions on the cobordism so that $\bar {m}_*(W)$ vanishes and therefore $\hat {m}_*(W)$ vanishes as well because p is injective.

• • if $b^+(W) =0$ , working in the setup of Section 3 it follows from the equation (4.2) that the associated graded map of $\bar {m}_*(W)$ determines the associated graded map of $\hat {m}_*(W)$ , and we determined the former in Theorem 3.2.

We record these observations as a lemma in the specific case of interest, where the two ends coincide.

Lemma 4.2. If $(Y, \mathfrak {s})$ is an $RSF$ -space, the complex $\hat {C}_*(Y, \mathfrak {s}_Y; \Gamma _\eta )$ is a filtered complex. If $(W, \mathfrak {s}_W, \nu )$ is a cobordism from $(Y, \mathfrak {s}, \eta )$ to itself, the map $\hat {m}_*(W)$ satisfies

$$ \begin{align*}\text{gr } \hat{m}_*(W, \mathfrak{s}_W, \nu) = U^d m_{W, \nu}\end{align*} $$

for $b^+(W) = 0$ , and is zero on the nose otherwise.

4.2 Examples of $RSF$ -spaces

The simplest example of an $RSF$ -space is given by a torsion spin $^c$ three-manifold $(Y,\mathfrak {s})$ admitting a metric of positive scalar curvature, as follows from the discussion in [Reference Kronheimer and MrowkaKM07, Chapter $36$ ]. These examples are somewhat trivial because the family of Dirac operators parametrized by $\mathbb {T}_Y$ never has kernel; in the setup of Theorem C, which we prove below, this leads to vanishing results. We now discuss some nontrivial examples where we obtain instead rigidity results.

All of our nontrivial examples come from the following procedure:

1. (i) Find a metric g so that $(Y, \mathfrak {s})$ supports no irreducible Seiberg–Witten solutions, so that the critical locus of the Chern–Simons–Dirac functional $\mathcal {L}$ is exactly $\mathbb {T}$ . Notice that $\mathcal {L}$ fails to be Morse–Bott exactly along the locus $\mathsf {K}$ where the corresponding Dirac operator has kernel.

2. (ii) After a suitable perturbation without introducing irreducible solutions, choose an additional Morse function $f: \mathbb T_Y \to \mathbb R$ so that the critical points of f are Dirac operators with no kernel. Labeling the eigenvalues of the Dirac operator at each critical point q as following Lemma 3.1 and writing $d(q)$ for the index of the first positive eigenvalue, we demand that for each pair of critical points $x,y$ with $d(x)> d(y)$ , we have $f(x) < f(y)$ . (This includes as a special case the “A-adapted perturbations” of [Reference LinLin24b]).

3. (iii) Even if the unperturbed equations did not admit irreducible solutions, it is challenging to carry out the perturbation process without introducing such solutions. One uses the geometry at hand to argue that the perturbed Seiberg–Witten equations are now regular, but still have no irreducible solutions. After this, one can add additional perturbations as in [Reference Kronheimer and MrowkaKM07, Ch. 12] to achieve transversality (in particular by making the spectrum of the perturbed Dirac operators at reducible critical points simple).

We now list some examples of $RSF$ -spaces obtained from this procedure.

Example 4.1. The simplest nontrivial example is that of the torus endowed with a flat metric as discussed [Reference Kronheimer and MrowkaKM07, Section 38]. In this setup, the quantity $d(x)$ in (4.1) is given by

$$ \begin{align*}d(x) = \begin{cases} 0 & \mathrm{ind}(x)>0 \\ -1 & \mathrm{ind}(x) = 0, \end{cases}\end{align*} $$

and the complex $\hat {C}_*(T^3,\mathfrak {s}_0)$ is given by

where the ith column corresponds to the points with Morse index $3-i$ , and the arrow is an isomorphism.

4.3 Proof of Theorem C

In this section and what follows, we will frequently use the Mayer–Vietoris type sequence

(4.3) $$ \begin{align} 0 \to H_3(W) \xrightarrow{p_3} H_3(X) \xrightarrow{\delta_3} H_2(Y) \xrightarrow{i_2 - j_2} H_2(W)\xrightarrow{p_2} H_2(X)\xrightarrow{\delta_2} \cdots \end{align} $$

as well as the sequences which are related to this one by Poincaré duality and chain-level duality

$$ \begin{align*} \cdots &\to H_3(W, \partial W) \to H_2(Y) \to H_2(X) \to H_2(W, \partial W) \to H_1(Y) \to \cdots\\ 0 &\to H^1(W, \partial W) \to H^1(X) \to H^1(Y) \to H^2(W, \partial W) \to H^2(X) \to \cdots \\ \Bbb Z &\to H^1(X) \to H^1(W) \to H^1(Y) \to H^2(X) \to H^2(W) \to \cdots. \end{align*} $$

It is useful to keep in mind the intersection-theoretic definition of $\delta _k$ : one takes a generic k-cycle in X and intersects it with Y to give a $(k-1)$ -cycle in Y. This is Poincaré dual to the map

$$ \begin{align*}f^*_{4-k} : H^{4-k}(X) \to H^{4-k}(Y)\end{align*} $$

given by pullback along the inclusion.

Observe now that because $b^+(X)> 0$ and $b^+(W) = 0$ , the map $p_2: H_2(W) \to H_2(X)$ is not surjective, so the exactness of (4.3) implies that there exists some $[\bar \nu ] \in H_2(X;\Bbb R)$ for which

$$ \begin{align*}[\eta] = [\bar \nu \cap Y] \ne 0 \in H_1(Y; \Bbb R),\end{align*} $$

and we choose chain-level representatives for these classes. Applying Proposition 2.1, we find

$$ \begin{align*}\mathfrak m(X, \mathfrak{s}_W) = \lim_{t \to 0} \mathfrak m(X, \mathfrak{s}_W, t[\bar \nu]) = \lim_{t \to 0} \text{Tr}\big({\mathit{HM}}_*(W, \mathfrak{s}_W; \Gamma_{t\nu})\big).\end{align*} $$

If $b^+(W)> 0$ it follows from Lemma 4.2 that this quantity is zero. When $b^+(W) = 0$ , we will relate this to the associated graded map discussed above by way of the classic Hopf trace formula: if C is a finite-dimensional $\Bbb Z/2$ -graded chain complex, then the alternating trace of a map $\phi : C \to C$ coincides with the alternating trace of the induced map on homology $\phi _*$ . We will also use the elementary observation that the trace of a filtered map is the same as the trace of its associated graded map, because the trace of an upper-triangular matrix is independent of the entries above the diagonal.

Proof of Theorem C.

In our setting the differential $\hat {\partial }$ and the map p are filtered by definition, and this can be applied as follows. Observe that when $t\eta \ne 0$ , the Morse homology $H_*(\mathbb T_Y, f; \Gamma _{t\eta })$ is trivial. Recall that we normalize the quantity $d(q)$ from (4.1) so that $d(q) \le 0$ ; say $d(q) = k \le 0$ is the minimum value. To have a concrete example in mind, in the case of the torus $T^3$ in Example 4.1, we have

$$ \begin{align*}\text{gr}_i \hat{C}(T^3, \mathfrak{s}_0; \Gamma_\eta) = \begin{cases} C_*(T^3, f; \Gamma_\eta) & i \ge 0 \text{ and is even} \\ \langle x_0\rangle & i = -2 \\ 0 & \text{otherwise} \end{cases}\end{align*} $$

Here $k = d(x_0) = -1$ is the minimum value.

In particular, the associated graded complex is $C_*(\mathbb T_Y, f; \Gamma _{t\eta })$ for $i \ge 0$ and zero for $i < 2k$ , so we have $E^2_i = 0$ in the associated spectral sequence converging to ${\mathit {HM}}_*(Y, \mathfrak {s}; \Gamma _{t \eta })=\widehat {\mathit {HM}}_*(Y, \mathfrak {s}; \Gamma _{t \eta })$ except possibly when i is an even integer satisfying $2k \le i \le -2$ . In particular, $E^2$ and all successive pages of the associated spectral sequence are finite-dimensional. Applying Hopf’s trace formula to the successive pages of the spectral sequence, we find

$$ \begin{align*} \mathfrak m(X, \mathfrak{s}_W) &= \lim_{t \to 0} \text{Tr}\left(HM_*(W, \mathfrak{s}_W; \Gamma_{t\nu})\right) = \lim_{t \to 0} \sum_{i=k}^{-1} \text{Tr}(E^\infty_{2i}(\hat{m}(W, \mathfrak{s}_W; \Gamma_{t \nu})\big) \\ &= \lim_{t \to 0} \sum_{i=k}^{-1} \text{Tr}(E^r_{2i}(\hat{m}(W, \mathfrak{s}_W; \Gamma_{t \nu})\big) \end{align*} $$

for all $r \ge 2$ , where

$$ \begin{align*}E^r_{2i}(\hat{m}(W, \mathfrak{s}_W; \Gamma_{t \nu})):E^r_{2i}(\hat{C}(Y, \mathfrak{s}; \Gamma_{t \eta}))\to E^r_{2i}(\hat{C}(Y, \mathfrak{s}; \Gamma_{t \eta}))\end{align*} $$

is the map induced on the $E^r$ page of the spectral sequence.

The $E^2$ page of the spectral sequence coincides with the homology of the associated graded complex $\text {gr}_{2i} \hat {C}(Y, \mathfrak {s}_W; \Gamma _{t \eta })$ ; notice that for each i this can be naturally identified with a subcomplex of $C_*(\mathbb T_Y, f; \Gamma _{t \eta })$ which, as an R-module, is independent of t. Applying Hopf’s formula once more, we see

$$ \begin{align*}\mathfrak m(X, \mathfrak{s}_W) = \lim_{t \to 0} \sum_{i=k}^{-1} \text{Tr}\left(U^d m_{W, t\nu} \lvert_{\text{gr}_{2i} \hat{C}(Y, \mathfrak{s}; \Gamma_{t \eta})}\right)\end{align*} $$

for d as in Theorem 3.2. Because the domain is independent of t and the map is continuous in t, taking the limit as $t \to 0$ gives

(4.4) $$ \begin{align}\mathfrak m(X, \mathfrak{s}_W) = \sum_{i=k}^{-1} \text{Tr}\left(U^d m_W \lvert_{\text{gr}_{2i} \hat{C}(Y, \mathfrak{s})}\right).\end{align} $$

From this we immediately conclude our rigidity result that $\mathfrak m(X, \mathfrak {s}_W)$ depends only on $m_W, d$ , and $\text {gr}_{2i} \hat {C}(Y, \mathfrak {s})$ , because the map $m_W$ depends only on the diagram (1.5), the integer d depends only on the characteristic numbers of W and the spin $^c$ structure $\mathfrak {s}_W$ , and the complex depends only on $(Y, \mathfrak {s})$ .

We have proved much of Theorem C; all that remains is to show that this trace vanishes when the three listed conditions are not met. That the Seiberg–Witten invariant $\mathfrak m(X, \mathfrak {s}_W)$ vanishes for $b^+(W)> 0$ was discussed above. Because $m_W$ has degree $b_1(W) - b_1(Y)$ , we see that even when $b^+(W) = 0$ the trace can only be nonzero if $b_1(W) = b_1(Y)$ . Similarly, this trace can only be nonzero if $d = 0$ , which when $b_1(W) = b_1(Y)$ is equivalent to $d(\mathfrak {s}_W) = 0$ .

It is worth naming the dependency of the constant in (4.4).

Definition 4.4. Suppose $(X, Y, \mathfrak s_Y)$ satisfy the hypotheses of Theorem C. We write

$$\begin{align*}c(W, Y, \mathfrak{s}_Y) = \sum_{i=k}^{-1} \text{Tr}\left(m_W \lvert_{\text{gr}_{2i} \hat{C}(Y, \mathfrak{s})}\right)\end{align*}$$

for the quantity arising in (4.4) when $d = 0$ .

6.1 Proof of Theorem B

Our first observation restricts attention to the case we can actually calculate, namely that of spin $^c$ structures on X which are torsion on Y.

Proof. Let $\Sigma \subset Y$ be a torus, so in particular $[\Sigma ] \cdot [\Sigma ] = 0$ and $g(\Sigma ) = 1$ . First, the adjunction inequality [Reference Kronheimer and MrowkaKM07, Theorem 40.2.3] implies that if $\mathfrak m(X, \mathfrak {s}_X) \ne 0$ , we have

$$ \begin{align*}0 = 2g - 2 \ge |\langle c_1(\mathfrak{s}_X), [\Sigma]\rangle| + [\Sigma] \cdot [\Sigma] = |\langle c_1(\mathfrak{s}_X), [\Sigma]\rangle|.\end{align*} $$

It follows that

$$ \begin{align*}0 = \langle c_1(\mathfrak{s}_X), [\Sigma]\rangle = \langle c_1(\mathfrak{s}_X)|_Y, [\Sigma]\rangle\end{align*} $$

for all tori $\Sigma $ in Y. Because Y has trivial Thurston norm, the real homology $H_2(Y; \Bbb R)$ is generated by embedded tori, and it follows that $c_1(\mathfrak {s}_X)|_Y$ pairs trivially with every real homology class. By Poincaré duality, $c_1(\mathfrak {s}_X)|_Y$ is torsion.

Now to prove Theorem B, assume $Y = T^3$ . For a spin $^c$ structure on W with $i^* \mathfrak {s}_W \cong j^* \mathfrak {s}_W \cong \mathfrak {s}_0$ the torsion spin $^c$ structure on Y, we established in Proposition 5.1 that

$$ \begin{align*}c(W,Y,\mathfrak{s}_0) = \begin{cases} \pm \text{disc}(W) & \text{if }b^+(W) = 0, \; b_1(W) = b_1(Y), \; \text{and}\; d(\mathfrak{s}_W) = 0, \\ 0 & \text{otherwise}. \end{cases}\end{align*} $$

Applying Lemma 6.1, reindexing the sum, and applying Theorem C gives

$$ \begin{align*} \mathfrak m(X) &= \sum_{\mathfrak{s}_X} \mathfrak m(X, \mathfrak{s}_X) = \sum_{\substack{\mathfrak{s}_X \\ \mathfrak{s}_X|_Y \text{ torsion}}} \mathfrak m(X, \mathfrak{s}_X) = \sum_{\substack{\mathfrak{s}_W \\ i^* \mathfrak{s}_W \cong j^* \mathfrak{s}_W \\ i^* \mathfrak{s}_W \cong \mathfrak{s}_0}} \sum_{\substack{\mathfrak{s}_X \\ p^* \mathfrak{s}_X \cong \mathfrak{s}_W}} \mathfrak m(X, \mathfrak{s}_X) \\ &= \sum_{\substack{\mathfrak{s}_W \\ i^* \mathfrak{s}_W \cong j^* \mathfrak{s}_W \\ i^* \mathfrak{s}_W \text{ torsion} \\ d(\mathfrak{s}_W) = 0}} c(W, Y, \mathfrak{s}_0). \end{align*} $$

By Proposition 5.1, these summands simplify to $\pm D(W) \text {disc}(W)$ when $b^+(W) = b_1(W) - b_1(Y) = 0$ , and is otherwise zero; further, as discussed in Remark 4.1, the equality $c(W, Y, \mathfrak {s}_Y) = D(W) \text {disc}(W)$ holds up to an overall sign ambiguity.

6.2 Homology 4-tori

Let X be a homology 4-torus. The goal of this subsection is to deduce Theorem A from Theorem B via computations in algebraic topology. Recall the definition of the determinant of X in (1.1); this is the class of the 4-form $\cup ^4_X \in \Lambda ^4 H^1(X;\Bbb Z) \cong \Bbb Z$ up to sign. We begin with the following lemma.

Lemma 6.2. Suppose X is a rational homology $4$ -torus and $Y \subset X$ is a nonseparating $3$ -torus. Then we have $\operatorname {det}(X) \ne 0$ if and only if the intersection map

$$ \begin{align*}\delta_3: H_3(X) \to H_2(Y)\end{align*} $$

has rank three, in which case $\operatorname {det}(X)$ is the index of $\text {im}(\delta _3)$ in $H_2(Y)$ .

Proof. Because the map $\delta _3$ is given by intersection with Y, if we choose a basis for $H_3(X)$ given by $[Y], \Sigma _1, \Sigma _2, \Sigma _3$ , then

$$ \begin{align*}\operatorname{det}(X) = [Y] \cap \Sigma_1 \cap \Sigma_2 \cap \Sigma_3 = \delta_3(\Sigma_1) \cap \delta_3(\Sigma_2) \cap \delta_3(\Sigma_3).\end{align*} $$

Choose $\Sigma _1, \Sigma _2, \Sigma _3$ , and basis vectors $S_i$ of $H_2(Y)$ so that the map $\delta _3: H_3(X) \to H_2(Y)$ is in Smith normal form, with $\delta _3(\Sigma _i) = a_i S_i$ . Then because $\operatorname {det}(T^3) = 1$ we have $\operatorname {det}(X) = a_1 a_2 a_3$ . This is zero if and only if $\text {rank}(\delta _3) < 3$ ; if $\text {rank}(\delta _3) = 3$ , this quantity coincides with the index of the image of $\delta _3$ .

We will also be interested in the behavior of $\delta _2: H_2(X) \to H_1(Y)$ . Write $c_i = 3 - \text {rank}(\delta _i)$ . It is straightforward to verify using the Mayer–Vietoris type sequence (4.3), Poincaré duality, and the fact that the rank of homology and cohomology coincide, that

$$ \begin{align*}b_4(W) = 0, \quad b_3(W) = 1 + c_3, \quad b_2(W) = 3 + c_2 + c_3, \quad b_1(W) = 3 + c_2, \quad b_0(W) = 1.\end{align*} $$

We also have $b_i(W, \partial W) = b_{4-i}(W).$

Proof. Consider the relative long exact sequence of the pair $(W, \partial W)$ , we have

$$ \begin{align*}0 \to \Bbb Z \to H_3(W) \to H_3(W, \partial W) \to H_2(Y) \oplus H_2(Y) \to H_2(W) \xrightarrow{\varphi} H_2(W, \partial W).\end{align*} $$

We computed the rank of each group in the sequence above in terms of $c_2$ and $c_3$ ; starting on the left, we can compute the rank of each map in the sequence, concluding that $\text {rank}(\varphi ) = 2c_2$ .

As remarked in the introduction, the intersection form on $\hat H(W) \cong \text {im}(\varphi )/\text {tors}$ is well-defined and nondegenerate, and is the intersection form associated to W. On the other hand, by additivity of signature under boundary-gluing, we have $\sigma (W) = \sigma (X) = 0$ . Because $b^+(W) = b^-(W)$ it follows that

$$ \begin{align*}b^+(W) = \frac 12 \dim \hat H(W) = \frac 12 \text{rank}(\varphi) = c_2.\\[-42pt]\end{align*} $$

Recall the definition of discriminant in (1.2).

Proof. Suppose first that $\operatorname {det}(X) \ne 0$ . Then $\delta _3$ has rank three. Using the commutative diagram

and the fact that $\Lambda ^2(\delta _3)$ also has rank three, it follows that $\delta _2$ has rank three as well, so that $b_1(W) = 3$ .

Next, suppose $b_1(W) = 3$ and consider the composition

$$ \begin{align*}H^1(X) \xrightarrow{p^*} H^1(W) \xrightarrow{i^*} H^1(Y).\end{align*} $$

The composite $(pi)^* = f^*$ is Poincaré dual to the map $\delta _3$ . The first map has rank three, as

$$ \begin{align*}0 \to \Bbb Z \to H^1(X) \xrightarrow{p^*} H^1(W) \to H^1(Y) \to \cdots\end{align*} $$

is exact and $b_1(X) = 4$ , so the composite $PD(\delta _3) = i^* p^*$ has rank three if and only if $i^*$ has rank three. This establishes the claim that $\operatorname {det}(X) \ne 0$ if and only if $b_1(W) = 3$ and $\operatorname {det}(i^*) \ne 0$ .

Supposing now that $b_1(W) = 3$ and $\operatorname {det}(i^*) \ne 0$ , because $\operatorname {det}(X) \ne 0$ we find that W has the Betti numbers of the 3-torus; as $i^*: H^1(W; \Bbb Q) \to H^1(Y; \Bbb Q)$ is an isomorphism on rational cohomology, taking the cup-square and cup-cube of this map we find the natural map $i^*: H^k(W; \Bbb Q) \to H^k(Y; \Bbb Q)$ is an isomorphism for all k. The same argument holds for $j^*$ .

For the final claim, observe that exactness of

$$ \begin{align*}0 \to \Bbb Z \to H^1(X) \to H^1(W) \xrightarrow{i^* - j^*} H^1(Y) \to \cdots\end{align*} $$

implies by rank considerations that $i^* - j^*$ has rank zero, hence (being a map between free abelian groups) $i^* = j^*$ , and in particular the map $p^*: H^1(X) \to H^1(W)$ is surjective. It now follows from the equation $\delta _3 = i^* p^*$ that the index of $\text {im}(\delta _3)$ coincides with $\operatorname {det}(i^*) = \text {disc}(W)$ .

We will now prove Theorem A. This is the first place we use that X is an integer homology torus; one can extend the result to rational homology tori, with $\operatorname {det}(X)$ replaced by $\operatorname {det}(X) |H_1(X)|$ .

Proof of Theorem A.

If $\operatorname {det}(X) = 0$ , then by Lemma 6.4 we have either $b_1(W) \ne 3$ or $b_1(W) = 3$ but $\text {disc}(W) = 0$ . In either case, Theorem B gives $\mathfrak m(X) = 0$ , as desired.

More interestingly, if $\operatorname {det}(X) \ne 0$ , then by Lemma 6.4 we have $b_1(W) = 3$ , $b^+(W) = 0$ , and $\text {disc}(W) = \operatorname {det}(X)$ . What remains is to show that

$$ \begin{align*}D(W) = \begin{cases} |H^2(Y)/f^*H^2(X)| & \text{if } W \text{ supports a torsion spin}^c \text{ structure} \\ 0 & \text{otherwise.} \end{cases}\end{align*} $$

By rank considerations and the fact that $H_1(Y)$ is torsion-free, in the long exact sequence

$$ \begin{align*}\cdots \to H_2(Y) \to H_2(X) \to H_2(W, \partial W) \to H_1(Y)\end{align*} $$

the last map is zero. Thus we have an isomorphism

$$ \begin{align*}H^2(W) \cong H_2(W, \partial W) \cong H_2(X)/f_* H_2(Y).\end{align*} $$

This group is of the form $\Bbb Z^3 \oplus T$ for a torsion abelian group T; using that $H_i(X)$ is torsion-free and appealing to Smith normal form, one finds that $H^2(Y)/f^* H^2(X) \cong T$ , so $|H^2(Y)/f^* H^2(X)|$ coincides with the number of elements in $\text {Tors} H^2(W)$ ; and so long as one exists, this set is in bijection with the set of torsion spin $^c$ structures on W. Thus Theorem A is reduced to showing that $D(W)$ counts the number of torsion spin $^c$ structures on W.

$D(W)$ counts spin $^c$ structures with $i^* \mathfrak {s}_W \cong j^* \mathfrak {s}_W$ torsion and $d(\mathfrak {s}_W) = 0$ . Because $T^3$ carries a unique torsion spin $^c$ structure, the first condition is equivalent to $\mathfrak {s}_W$ being torsion on the ends (so that in particular $d(\mathfrak {s}_W)$ is defined). Because X is a homology torus we have $\chi (W) = \sigma (W) = 0$ and thus $d(\mathfrak {s}_W) = \frac 14 c_1(\mathfrak {s}_W)^2$ . Because W is negative-definite and

$$ \begin{align*}[c_1(\mathfrak{s}_W)] \in \hat H(W) = \text{im}(H^2(W, \partial W; \Bbb R) \to H^2(W; \Bbb R)),\end{align*} $$

we see that $d(\mathfrak {s}_W) = 0$ is equivalent to $[c_1(\mathfrak {s}_W)] = 0$ in real cohomology, which is equivalent to $\mathfrak {s}_W$ being torsion. Thus $D(W)$ counts torsion spin $^c$ structures.

6.3 Some examples

Write $t_i(W) = |\text {Tors} H_i(W)|$ and similarly for variations, so that our formula can be rewritten $\mathfrak m(X) = \pm \operatorname {det}(X) t^2(W)$ . It follows from the universal coefficient theorem and Poincaré duality that there are essentially two torsion coefficients associated to W:

$$ \begin{align*} t_1(W) = t^2(W) = t_2(W, \partial W) &= t^3(W, \partial W),\\ t_1(W, \partial W) = t^2(W, \partial W) &= t_2(W) = t^3(W). \end{align*} $$

Lemma 6.2 computes that $t_2(W) = \operatorname {det}(X),$ when X is an integer homology torus (and more generally for rational homology tori, $t_2(W) = \operatorname {det}(X) t_1(X)$ ). It is interesting that $t^2(W) = t_1(W)$ does not seem to have such a simple description. In this section we will discuss two concrete examples.

Example 6.1. Suppose M is a rational homology 3-torus containing a nonseparating two-torus $T \hookrightarrow M$ , and let $X = S^1 \times M$ ; notice that X is spin and $\operatorname {det}(X) = \operatorname {det}(M)$ . Denoting by C the complement, we have $W \cong S^1 \times C$ . Now by the Künneth theorem $t_1(W) = t_1(C) = t_2(W)$ , and as discussed above $t_2(W) = \operatorname {det}(X) t_1(X) = \operatorname {det}(M) t_1(M)$ . Therefore

$$ \begin{align*}\pm \mathfrak m(X) = \operatorname{det}(X) t^2(W) = \operatorname{det}(M)^2 t_1(M).\end{align*} $$

When M is an integer homology torus, this is consistent with the result of Meng-Taubes [Reference Ruberman and StrleRS00, Section 5].

Example 6.2. Fix nonzero integers $n_1,n_2, n_3$ . Consider the cobordism Z obtained from $I\times T^3$ first by adding three $1$ -handles, so that the resulting boundary is $T^3\#(S^1\times S^2)^3$ , and then three $2$ -handles along curves with homology class

$$ \begin{align*} (1,0,0,n_1,0,0),\quad (0,1,0,0,n_2,0),\quad (0,0,1,0,0,n_3). \end{align*} $$

By choosing the framings appropriately, we can arrange that Z is spin. Call Y the other boundary component. Cellular homology computations give that $H_1(Z)=\mathbb {Z}^3$ , that the image of

$$ \begin{align*} H_1(T^3)\rightarrow H_1(Z) \end{align*} $$

is a sublattice of index $n_1n_2n_3$ , and that

$$ \begin{align*} H_1(Z,T^3)&=\mathbb{Z}/n_1\oplus\mathbb{Z}/n_2\oplus\mathbb{Z}/n_3\\ H_2(Z,T^3)&=H_3(Z,T^3)=0. \end{align*} $$

By Poincaré–Lefschetz duality

$$ \begin{align*} H^1(Z,Y)=H^2(Z,Y)=0 \end{align*} $$

so that

$$ \begin{align*} H^1(Z)\rightarrow H^1(Y) \end{align*} $$

is an isomophism. We now take $W=Z\cup _Y \tilde {Z}$ , the double of Z along Y. The isomorphism just mentioned implies that the map in the Mayer-Vietoris sequence

$$ \begin{align*} H^2(W)\rightarrow H^2(Z)\oplus H^2(\tilde{Z}) \end{align*} $$

is injective, hence $H^2(W)$ is torsion-free.

To sum up, W is a cobordism from $T^3$ to itself such that the two inclusion map are the same map in $H_1$ ; gluing together the boundary components we obtain an integral homology torus which has determinant $n_1n_2n_3$ by Lemma 6.2, but $H^2(W)$ is torsion-free. Therefore, for this family of spin integer homology tori X, we have $\pm \mathfrak m(X) = \operatorname {det}(X).$