Positive scalar curvature and homology cobordism invariants

Manolescu’s Seiberg–Witten Floer stable homotopy type [Ma03] is a space-valued Floer theoretic invariant of a rational homology $3$-sphere equipped with a spin^c structure, and recovers the monopole Floer homology defined by Kronheimer and Mrowka [KM07] for this class of $3$-manifolds [LM18]. Therefore, in principle, the Seiberg–Witten Floer stable homotopy type contains all Floer-theoretic information from Seiberg–Witten theory for rational homology $3$-spheres.

In this paper, we will consider a spin rational homology $3$-sphere $(Y,\mathfrak{t})$ embedded into a spin $4$-manifold $(X,\mathfrak{s})$ with the rational homology of $S^{1}\times S^{3}$ so that the fundamental class of $Y$ generates $H_{3}(X;\mathbb{Z})$. The main theorem of this paper states that, if $X$ admits a metric with positive scalar curvature (PSC), we can determine the Seiberg–Witten Floer stable homotopy type of such $(Y,\mathfrak{t})$, denoted by $\mathrm{SWF}(Y,\mathfrak{t})$, up to the local equivalence relation explained below. This result gives a strong obstruction to PSC metrics of spin rational homology $S^{1}\times S^{3}$, and this is the authors’ original motivation for this study. To the same class of $4$-manifolds, there are two known obstructions based on Seiberg–Witten theory, Jianfeng Lin’s obstruction [Lin19] and the authors’ obstruction [KT20] explained later, and the main theorem of this paper recovers both of them.

To motivate to consider the local equivalence relation, let us recall several homology cobordism invariants from Seiberg–Witten theory. Applying various equivariant ordinary/generalized cohomologies to the Seiberg–Witten Floer stable homotopy type, many numerical homology cobordism invariants can be extracted, such as, the Frøyshov invariant [Fr96, Fr10], which we denote by $\delta$ following [Ma16], Manolescu’s invariants $\alpha,\beta,\gamma,\kappa$ [Ma14, Ma16], and Stoffregen’s invariants $\overline{\delta},\underline{\delta}$ [Sto171]. These invariants have different applications, for example: The Frøyshov invariant $\delta$ was used to extend Donaldson’s diagonalization theorem [Do83] to negative-definite $4$-manifolds with boundary [Fr96, Fr10, Ma03]. Manolescu used the invariant $\beta$ to disprove the triangulation conjecture [Ma16], and used $\kappa$ to extend Furuta’s 10/8-inequality [Fu01] to spin $4$-manifolds with boundary [Ma14]. Stoffregen’s invariants $\overline{\delta},\underline{\delta}$ should correspond, respectively, to $\overline{d},\underline{d}$ in involutive Heegaard Floer homology [HM17], using $\mathbb{Z}/4$-equivariant ordinary cohomology.

These invariants $\alpha,\beta,\gamma,\delta,\overline{\delta},\underline{\delta},\kappa$ are spin rational homology cobordism invariants, and obtained from $\mathrm{SWF}(Y,\mathfrak{t})$ described above. However, these invariants factor through a weaker invariant than $\mathrm{SWF}(Y,\mathfrak{t})$, the local equivalence class of $\mathrm{SWF}(Y,\mathfrak{t})$, defined by Stoffregen [Sto20]. The local equivalence is an equivalence relation on a certain class of spaces including $\mathrm{SWF}(Y,\mathfrak{t})$ for rational homology $3$-spheres $Y$, and this is an abstraction of a relation between $\mathrm{SWF}(Y_{0},\mathfrak{t}_{0})$ and $\mathrm{SWF}(Y_{1},\mathfrak{t}_{1})$ for $(Y_{0},\mathfrak{t}_{0})$ and $(Y_{1},\mathfrak{t}_{1})$ which are spin rational homology cobordant to each other. To summarize this situation, let us denote by $\Theta_{\mathbb{Z}}^{3}$ the $3$-dimensional homology cobordism group, and denote by $\Theta_{\mathbb{Q},\rm spin}^{3}$ the $3$-dimensional spin rational homology cobordism group. Namely, an element of $\Theta_{\mathbb{Q},\rm spin}^{3}$ is the equivalence class $[(Y,\mathfrak{t})]$ of a spin rational homology $3$-sphere, and the equivalence relation is given by a spin rational homology cobordism. Stoffregen [Sto20] introduced the local equivalence group $\mathcal{LE}$, which consists of the local equivalence classes of certain spaces modeled on $\mathrm{SWF}(Y,\mathfrak{t})$. Then one has group homomorphisms

For a spin rational homology $3$-sphere $(Y,\mathfrak{t})$, the local equivalence class $[\mathrm{SWF}(Y,\mathfrak{t})]$ is valued in $\mathcal{LE}$, and the above numerical invariants $\alpha,\beta,\gamma,\delta,\overline{\delta},\underline{\delta},\kappa$ factor through $\mathcal{LE}$, such as $\alpha(Y,\mathfrak{t})=\alpha([(Y,\mathfrak{t})])=\alpha([\mathrm{SWF}(Y,\mathfrak{t})])$:

As described, the local equivalence class $[\mathrm{SWF}(Y,\mathfrak{t})]\in\mathcal{LE}$ of $\mathrm{SWF}(Y,\mathfrak{t})$ is, so far at least, a candidate of the ‘universal’ Seiberg–Witten theoretic homology cobordism invariant of $(Y,\mathfrak{t})$: it contains information of all known homology cobordism invariants obtained from Seiberg–Witten theory. The main theorem of this paper determines $[\mathrm{SWF}(Y,\mathfrak{t})]$ when $Y$ is embedded into a spin rational homology $S^{1}\times S^{3}$ admitting a PSC metric so that $Y$ generates $H_{3}(X;\mathbb{Z})$:

Here $\lambda_{SW}(X,\mathfrak{s})$ is the Casson-type invariant defined by the Mrowka–Ruberman–Saveliev [MRS11] for an integral homology $S^{1}\times S^{3}$, which was later generalized for a rational homology $S^{1}\times S^{3}$ by J. Lin–Ruberman–Saveliev [LRS17]. Recall that an element of $\mathcal{LE}$ is expressed as the class of a triple $(Z,m,n)$, where $Z$ is a space of type SWF [Ma16, Definition 2.7], and $m\in\mathbb{Z}$, $n\in\mathbb{Q}$.

Now we regard Theorem 1.1 as an obstruction to PSC metrics on homology $S^{1}\times S^{3}$, and compare this with known obstructions on PSC metrics for the same class of $4$-manifolds. We can extract from Theorem 1.1 convenient obstructions to PSC metrics, and moreover that Theorem 1.1 provides a systematic way to recover prior results.

Recall that it is well-understood which rational homology $3$-spheres admit PSC metrics: only connected sums of spherical $3$-manifolds. Rational homology $S^{1}\times S^{3}$ is a class of $4$-manifold that may be seen to be closed to rational homology $3$-sphere, but it is not easy to rule out the existence of PSC metrics on such $4$-manifolds. In dimension $4$, the Seiberg–Witten invariant is known as a powerful obstruction to PSC metric, but it cannot be used to rational homology $S^{1}\times S^{3}$, since the Seiberg–Witten invariant is not well-defined for such $4$-manifolds. J. Lin recently made a breakthrough in this situation: he gave the first obstruction to PSC metric based on Seiberg–Witten theory for integral homology $S^{1}\times S^{3}$ in [Lin19], and later this result was generalized by himself with Ruberman and Saveliev to any rational homology $S^{1}\times S^{3}$ in [LRS17]. J. Lin’s obstruction is described as follows: under the same assumption with Theorem 1.1, one has the equality

Using Theorem 1.1, we can give an alternative proof of J. Lin’s formula (2), and further generalize it to various Frøyshov-type invariants:

Note that, by Corollary 1.3, we can replace $\lambda_{SW}(X,\mathfrak{s})$ in the right-hand side of (1) with various invariants of $(Y,\mathfrak{t})$.

An obvious consequence of Corollary 1.3 is:

Here we drop the unique spin structure from our notation for (integral) homology $3$-spheres.

J. Lin [Lin19] and J. Lin–Ruberman–Saveliev [LRS17] used monopole Floer homology to establish the obstruction (2). Morally, our argument in this paper can be thought of as a stable cohomotopy version of J. Lin’s argument in [Lin19].

After J. Lin’s work, the authors [KT20] gave another obstruction based on a 10/8-type inequality, described in Corollary 1.5. Using Corollary 1.3 combined with Manolescu’s relative 10/8-inequality [Ma14], we can give an alternative proof of the authors’ previous result (with a minor change):

Here is an explanation of an outline of the proof of Theorem 1.1. The heart of this paper is, under the assumption of the existence of PSC metric on $X$, to consider finite-dimensional approximations of the Seiberg–Witten equations on a periodic-end $4$-manifold. More precisely, we shall construct a relative Bauer–Furuta-type invariant over a half-periodic-end $4$-manifold

along the spirit of Furuta [Fu01], Bauer–Furuta [BF04], and Manolescu [Ma03]. Here $W$ is the $4$-manifold defined by cutting $X$ open along $Y$, and the ‘left side’ end is equipped with a periodic PSC metric and a neighborhood of the ‘right side’ boundary is equipped with a product metric of the form $[0,1]\times Y$. Technically, the relative Bauer–Furuta invariant over such a non-compact 4-manifold is defined using the similar method given in [IT20] which defines the relative Bauer–Furuta invariant for a certain class of 4-manifolds with conical end.

The key observation is that $W[-\infty,0]$ with such a periodic PSC metric on the end looks like a homology cobordism from $S^{3}$ to $Y$ from Seiberg–Witten theoretic point of view. The relative Bauer–Furuta invariant over $W[-\infty,0]$ gives a local map from $\left[\left(S^{0},0,-\lambda_{SW}(X,\mathfrak{s})/2\right)\right]$ to $\mathrm{SWF}(Y,\mathfrak{t})$. The quantity $\lambda_{SW}(X,\mathfrak{s})$ emerges from the spin Dirac index over $W[-\infty,0]$, discussed in Subsection 3.3.

Similarly, by considering the relative Bauer–Furuta invariant over

we get a local map from $\mathrm{SWF}(Y,\mathfrak{t})$ to $\left[\left(S^{0},0,-\lambda_{SW}(X,\mathfrak{s})/2\right)\right]$, and we can conclude that $\mathrm{SWF}(Y,\mathfrak{t})$ is locally equivalent to $\left[\left(S^{0},0,-\lambda_{SW}(X,\mathfrak{s})/2\right)\right]$.

In Section 8 we shall give examples of concrete 3-manifolds $Y$ to which we can apply the obstructions given in Subsection 1.3. Here let us exhibit a part of those examples.

As a consequence of his formula (2), J. Lin proved in [Lin19, Corollary 1.3] that a homology $S^{1}\times S^{3}$ having a cross-section $Y$ with $\mu(Y)\neq\delta(Y)\mod 2$ does not admit a PSC metric. Here $\mu(Y)\in\mathbb{Z}/2\mathbb{Z}$ denotes the Rohlin invariant. For Seifert homology 3-spheres, we can get an ‘integer-valued lift’ of this result by J. Lin. Moreover, also for linear combinations of Seifert homology 3-spheres of certain type, we can get an obstructions described in terms of some integer-valued invariants of certain 3-manifolds:

For the definition of projective Seifert homology 3-spheres, see Section 8.

We finish off this introduction with an outline of the contents of this paper. The contents until Section 5 are devoted to construct the relative Bauer–Furuta invariant on the periodic-end 4-manifold $W[-\infty,0]$. In Section 2 we give several notations related to infinite cyclic covering spaces of a $4$-manifold. We also review Fredholm theory for infinite cyclic covering spaces, Seiberg–Witten Floer homotopy types and notion of local equivalence. In Section 3 we ensure Fredholm properties of elliptic operators on certain 4-manifolds with periodic end and boundary. We calculate cohomologies of the Atiyah–Hitchin–Singer operator on such non-compact 4-manifolds. We also calculate the Dirac index on $W[-\infty,0]$ in Subsection 3.3. In Section 4 we show a boundedness result which is needed to construct the relative Bauer–Furuta invariant. In Section 5 we construct the relative Bauer–Furuta invariant for the 4-manifolds $W[-\infty,0]$ with periodic end and boundary. In Section 6 we prove Theorem 1.1 along the idea explained in Subsection 1.4. In Section 7 we give a generalization of Theorem 1.1, which is stated as an obstruction of embeddings of $3$-manifolds into $4$-manifolds admitting PSC metrics. In Section 8 we provide several families of examples of homology $S^{1}\times S^{3}$’s which cannot admit PSC metrics using Theorem 1.1.

In this Subsection we introduce several notations on periodic $4$-manifolds. Let $(X,\mathfrak{s})$ be an oriented spin rational homology $S^{1}\times S^{3}$, i.e. a spin $4$-manifold whose rational homology is isomorphic to that of $S^{1}\times S^{3}$. Fix a Riemannian metric $g_{X}$ on $X$ and a generator of $H_{3}(X;\mathbb{Z})$, denoted by $1\in H_{3}(X;\mathbb{Z})$. Note that $H_{3}(X;\mathbb{Z})$ is isomorphic to $H^{1}(X;\mathbb{Z})$, and hence to $\mathbb{Z}$. Let $Y$ be an oriented rational homology $3$-sphere, and assume that $Y$ is embedded into $X$ so that $[Y]=1$. We call such $Y$ a cross-section of $X$. Let $W_{0}$ be the rational homology cobordism from $Y$ to itself obtained by cutting $X$ open along $Y$. The manifold $W_{0}$ is equipped with an orientation and a spin structure induced by those of $X$. For $(m,n)\in(\{-\infty\}\cup\mathbb{Z})\times(\mathbb{Z}\cup\{\infty\})$ with $m<n$, we define the periodic $4$-manifold

where $W_{i}$ is a copy of $W_{0}$ for each $i\in\mathbb{Z}$. This $4$-manifold $W[m,n]$ is also equipped with an orientation and a spin structure as well as $W_{0}$. The element of $H^{1}(X;\mathbb{Z})$ corresponding to $1\in H_{3}(X;\mathbb{Z})$ via the Poincaré duality gives the isomorphism class of an infinite cyclic covering

and an identification

Via the identification (6), let us think of $p$ as a map from $W[-\infty,\infty]$ to $X$. Define the map $p_{-}:W[-\infty,0]\to X$ as the restriction of $p$. We call an object defined on $W[-\infty,0]$, such as connection, metric, bundle, and differential operator, a periodic object if the restriction of the object to $W[-\infty,0]$ can be identified with the pull-back of an object on $X$ under $p_{-}$. Considering the pull-back under $p_{-}$, the Riemannian metric $g_{X}$ on $X$ induces a Riemannian metric, denoted by $g_{W[-\infty,0]}$, on $W[-\infty,0]$. Let $S^{+},S^{-}$ be the positive/negative spinor bundles respectively over $W[-\infty,0]$ with respect to the metric and the spin structure above. Fixing a trivialization of the determinant line bundle of the spin structure on $W[-\infty,0]$, we obtain the canonical reference connection $A_{0}$ on $W[-\infty,0]$ corresponding to the trivial connection.

To consider the weighted Sobolev norms on $W[-\infty,0]$, fix a function

with $T^{*}\tau=\tau+1$, where $T:\widetilde{X}\to\widetilde{X}$ is the deck transform determined by $T(W_{i})=W_{i-1}$. Note that $d\tau$ defined a cohomology class $[d\tau]\in H^{1}(X;\mathbb{Z})$ which is equal to $1\in H^{1}(X;\mathbb{Z})$ corresponding to $1\in H_{3}(X;\mathbb{Z})$ via the Poincaré duality.

Note that the equivalence class of norms $\|-\|_{L^{2}_{k,\delta}(W[-\infty,0])}$ does not depend on the choices of a periodic metric and a periodic connection on $E$.

In this Subsection we review the Fredholm property of periodic elliptic operators on the infinite cyclic covering $\widetilde{X}$ developed by C. Taubes [T87]. He showed that a periodic elliptic operator is Fredholm under some condition with respect to $L^{2}_{k,\delta}$-norms for generic $\delta\in\mathbb{R}$. For the details, see [T87], or [KT20, Subsection 2.1].

Let $\mathbb{D}=(D_{i},E_{i})$ be a periodic elliptic complex on $\widetilde{X}$, i.e. the complex

consisting of first order periodic linear differential operators $D_{i}$ between periodic vector bundles $E_{i}$ on $\widetilde{X}$ with exact symbol sequence. Here, for a vector bundle $E$, the notation $\Gamma(\widetilde{X},E)$ denotes the set of compactly supported smooth sections of $E$. As well as Definition 2.1, define the weighted Sobolev norm on $\widetilde{X}$ by

using a periodic connection and a periodic metric. The complex (7) gives rise to the complex of bounded operators

for each $k>0$ and $\delta\in\mathbb{R}$.

Note that, since the operators in (8) are periodic differential operators, there exist differential operators $\hat{\mathbb{D}}=(\hat{D}_{i},\hat{E}_{i})_{i=0,\cdots,N}$ on $X$ such that $\mathbb{D}$ is given as the pull-back $p_{-}^{*}\hat{\mathbb{D}}$.

The admissibility condition does not depend on the choice of classifying map $f$. One can show that every PSC metric on $X$ is an admissible metric (See (2) in [RS07]).

In [KT20], we confirmed that Theorem 2.3 can be used for differential operators appearing as the linearization of the Seiberg–Witten equations:

In the proof of Theorem 1.1, we use a variant of the relative Bauer–Furuta invariant for 4-manifolds with periodic end. In this subsection we review several notions of Manolescu’s Seiberg–Witten Floer stable homotopy type, which is necessary to describe the relative Bauer–Furuta invariant. The main references of this subsection are Manolescu [Ma03] and Khandhawit [Kha15].

Let $Y$ be an oriented rational homology $3$-sphere with a Riemannian metric $g_{Y}$. Let $\mathfrak{t}$ be a spin^c structure on $Y$, and $S$ be the spinor bundle of $\mathfrak{t}$. We fix a flat spin^c reference connection $a_{0}$ of the determinant line bundle of $S$.

The gauge group $\mathcal{G}_{k}(Y)$ naturally acts on $\mathcal{C}_{k}(Y,\mathfrak{t})$ and the functional $CSD$ is invariant under the action. The global slice of the action of $\widetilde{\mathcal{G}}_{k}(Y)$ on $\mathcal{C}_{k}(Y,\mathfrak{t})$ is given by

on which we still have the remaining $S^{1}$-action. We often drop $k$ and/or $(Y,\mathfrak{s})$ from our notation to denote $V_{k}(Y,\mathfrak{s})$. The $S^{1}$-equivariant formal gradient flow on $V(Y,\mathfrak{s})$ of $CSD$ with respect to the Coulomb projection of the $L^{2}$-metric can be written as the sum of the linear term

and some quadratic term, denoted by $c:V_{k}(Y,\mathfrak{s})\to V_{k-1}(Y,\mathfrak{s})$.

For $\lambda<0<\mu$, we define $V_{\lambda}^{\mu}(Y)$ as the direct sum of eigenspaces of $l$, regarded as an unbounded operator on $V_{1/2}(Y,\mathfrak{s})$, whose eigenvalues belong to $(\lambda,\mu]$. Here we think of $V_{\lambda}^{\mu}(Y)$ as a subspace of $V_{k}(Y,\mathfrak{s})$. We denote by

the $L^{2}$-projection of $V_{k}(Y,\mathfrak{s})$ onto $V_{\lambda}^{\mu}(Y)$. We often abbreviate $V_{\lambda}^{\mu}(Y)$ as $V_{\lambda}^{\mu}$. Since $l$ is the sum of a real operator and a complex operator, $V_{\lambda}^{\mu}$ decomposes into a real vector space and a complex vector space, denoted by

Let us use basic terms of Conley index theory following [Ma03, Section 5]. Manolescu proved some compactness result [Ma03, Proposition 3], and as a consequence, it turns out that a closed ball in $V_{\lambda}^{\mu}$ of sufficiently large radius centered at the origin is an isolating neighborhood of the invariant part of the ball. Precisely, the flow on $V_{\lambda}^{\mu}$ considered here is a flow obtained from $(l+p_{\lambda}^{\mu}c)$ by cutting off outside a larger ball (see [Ma03, page 907]). We denote by $I_{\lambda}^{\mu}$ the $S^{1}$-equivariant Conley index of the invariant part. The Seiberg–Witten Floer homotopy type $\mathrm{SWF}(Y,\mathfrak{t})$ is defined as the triple $(\Sigma^{-V^{0}_{\lambda}}I_{\lambda}^{\mu},0,n(Y,\mathfrak{t},g_{Y}))$, which is symbolically denoted by

The triple is regarded as an object a certain suspension category $\mathfrak{C}$. In general an object of $\mathfrak{C}$ is given as a triple $(Z,m,n)$, where $Z$ is a pointed topological $S^{1}$-space, $m\in\mathbb{Z}$, and $n\in\mathbb{Q}$. The quantity $n(Y,\mathfrak{t},g_{Y})\in\mathbb{Q}$ is defined to be

where $(W,\mathfrak{t}^{\prime})$ is a compact $\text{spin}^{c}$ 4-manifold satisfying $\partial(W,\mathfrak{t}^{\prime})=(Y,\mathfrak{t})$ and $\mathop{\mathrm{ind}}\nolimits_{\mathbb{C}}D^{+}$ means the index of the Dirac operator with APS boundary condition. For the meaning of formal desuspensions, see [Ma03].

Here let us consider the case when the spin^c structure $\mathfrak{t}$ comes from a spin structure. In this case, the formal gradient flow of $CSD$ admits a larger symmetry of the group $\operatorname{Pin}(2)$ defined by

This group $\operatorname{Pin}(2)$ acts on $V_{k}(Y,\mathfrak{s})$ for any non-negative integer $k$ as follows: the $\operatorname{Pin}(2)$-action on spinors given as the restriction of the natural $Sp(1)$-action on spinor bundles, and the $\operatorname{Pin}(2)$-action on $\Omega^{1}_{Y}$ is given via the non-trivial homomorphism $\operatorname{Pin}(2)\to O(1)$. We denote by $\widetilde{\mathbb{R}}$ the real $1$-dimensional representation of $\operatorname{Pin}(2)$, and by $\mathbb{H}$ the space of quaternions, on which $\operatorname{Pin}(2)$ naturally acts. Thus we have decompositions

and

Considering $\operatorname{Pin}(2)$-equivariant Conley index instead, we obtain a stable homotopy type of a pointed $\operatorname{Pin}(2)$-space

which lies in a suspension category $\mathfrak{C}^{\prime}$. An object of $\mathfrak{C}^{\prime}$ is given as a triple $(Z,m,n)$, where $Z$ is a pointed topological $\operatorname{Pin}(2)$-space, $m\in\mathbb{Z}$, and $n\in\mathbb{Q}$.

Let us recall the definition of local equivalence.

Typical examples of local maps are obtained as the relative Bauer–Furuta invariants for negative definite spin cobordisms between rational homology 3-spheres.

In this subsection we describe the Seiberg–Witten equations on $W[-\infty,0]$, mainly to fix notations. We use the double Cloumb gauge condition introduced in [Kha15].

We will prove that $\mathcal{U}_{k,\delta}(W[-\infty,0])$ gives a global slice with respect to the action of $\mathcal{G}_{k+1,\delta}(W[-\infty,0])$ on $\mathcal{C}_{k,\delta}(W[-\infty,0])$. Note that, on $\mathcal{C}_{k,\delta}(W[-\infty,0])$, the ‘full gauge group’

also acts. Thus we have an additional $S^{1}$-symmetry on $\mathcal{U}_{k,\delta}(W[-\infty,0])$ coming from the limits with respect to the end of gauge transformations.

Based on the Sobolev embedding $\mathcal{G}_{k+1,\delta}(W[-\infty,0])\to C^{0}(W[-\infty,0],S^{1})$, we can naturally define the group structure on $\mathcal{G}_{k+1,\delta}(W[-\infty,0])$ by pointwise multiplication.

On $W[-\infty,0]$, one can define the Seiberg–Witten map

by

When we write $(a,\phi)=(A,\Phi)-(A_{0},0)$, we often decompose the Seiberg–Witten map $\mathcal{F}_{W[-\infty,0]}$ as the sum of the linear part