Circle bundles with PSC over some four manifolds

If $k<n$, then a small $k$-Urysohn width intuitively means that the manifold is close to a $k$-dimensional space. This notion of closeness is captured in the following definition.

The notions of Urysohn width and macroscopic dimension appear in several works of Gromov, with one of the aims being to capture the appropriate notion of smallness that must be exhibited by a manifold admitting a metric of positive scalar curvature (PSC). These also appear naturally in the study of non-collapsing and thick-thin decomposition. See also some of these recent works on the topic [ABG24, Gut11, Nab22, Sab22]. The primary motivation behind this article is the following conjecture formulated by Gromov for manifolds with positive scalar curvature [Gro96, Section $2\text{\textonehalf}$].

Intuitively, the conjecture says that a manifold admitting a PSC metric, on large scales, looks like a manifold of at least two dimensions less. The primary evidence in this direction is the following simple observation: Given any closed manifold $M$, consider the manifold $M\times S^{2}_{r}$ with the product metric. Here, $S^{2}_{r}$ denotes a sphere of radius $r$. Then for $r$ small enough, the product metric on $M\times S^{2}_{r}$ has positive scalar curvature. In light of the previous remark, we pass to the universal cover of $M\times S^{2}_{r}$, which is $\widetilde{M}\times S^{2}_{r}$. Since the projection map to $\widetilde{M}$ is continuous and each fiber is $S^{2}_{r}$, we have

and therefore the macroscopic dimension $\dim_{mc}(\widetilde{M\times S^{2}_{r}})\leq n-2$. Conjecture 1 asserts that all PSC manifolds must exhibit this behavior. Gromov’s conjecture has been verified in dimension $3$ [GL83, LM23]. In higher dimensions there are several partial results under various assumptions (cf. [BD16, DD22]).

While Conjecture 1 arises from the simple observation that existence of a PSC metric is guaranteed given the presence of an $S^{2}$ factor in the manifold, it is also natural to ask about the possible interaction between existence of PSC metrics and having an $S^{1}$ factor. This has been widely investigated. For instance, PSC manifolds with a trivial $S^{1}$ factor (i.e., $M\times S^{1}$) are amenable to the Schoen-Yau conformal descent argument [SY79], as they have a non-trivial codimension-$1$ homology class given by the Poincaré dual of the $S^{1}$ factor. Such $S^{1}$ factors are also useful in quantitative geometric problems related to scalar curvature. For example, building on an observation of Fisher-Colbrie and Schoen [FCS80], Gromov and Lawson introduced the toric symmetrization procedure [GL83] which was used to estimate the Lipschitz constant of a non-zero degree map from a PSC manifold. More recently, Gromov [Gro18] observed that toric symmetrization can also be used to prove width estimates for Riemannian bands. On the same theme, Rosenberg [Ros07, Conjecture 1.24] formulated the following conjecture regarding the relationship between PSC metrics and trivial $S^{1}$ factors.

However, the situation is less clear when the $S^{1}$ factor arises as a fiber of a non-trivial circle bundle. The key difficulty in studying existence of PSC metrics on non-trivial circle bundles is that, in this case, the Schoen-Yau conformal descent argument does not apply in a straightforward way.¹¹1Indeed, the integral homology class of the $S^{1}$ fiber in nontrivial circle bundles is always torsion, with order given by the divisibility of the Euler class of the bundle. Therefore, its Poincaré dual is zero. In this direction, Gromov posed the following question [Gro18, pg. 661]:

In the negative direction we point out the recent comprehensive work [He23] on PSC metrics on non-trivial circle bundles due to He. In particular, he showed that circle bundles with homologically non-trivial fibers do not admit PSC metrics if the base space is a $1$-enlargeable²²2$1$-enlargeable means that the maps to $S^{n}$ in the definition of enlargeability are of degree $1$. manifold of dimension $n\leq 6$ [He23, Proposition 4.12].

The main goal of this article is to study and exhibit examples of interactions between the existence of a positive scalar curvature metric and macroscopic dimension on non-trivial circle bundles over some $4$-manifolds. In particular, we give a positive answer to Question 1. We construct these examples using techniques from symplectic geometry. The inspiration for our construction came from the following remark by Gromov [Gro23, pg. 8]:

The emergent picture of spaces with $Sc.curv\geq 0$, where topology and geometry are intimately intertwined, is reminiscent of the symplectic geometry, but the former has not reached yet the maturity of the latter.

Granting this dictum of similarity between scalar curvature geometry and symplectic geometry, one is led to believe that the analogue of the Schoen-Yau descent for positive scalar curvature manifolds, in the symplectic case must be Donaldson’s symplectic divisor theorem [Don96], as it can be viewed as a codimension-$2$ descent-type result for symplectic manifolds. This analogy is strengthened when seen in conjunction with a recent result of Cecchini, Räde and Zeidler [CRZ23] which establishes a certain codimension-$2$ descent for positive scalar curvature manifolds of dimension $7$. In fact, we show that circle bundles over Donaldson divisors in symplecic $6$-manifolds fit neatly into the framework of applicability of [CRZ23].

We discuss the necessary symplectic preliminaries in Section 2.

In this article, we construct infinitely many examples of $4$-manifolds $Z$ with arbitrarily large Euler number $|\chi(Z)|$, arising as Donaldson divisors (see Theorem 1) in symplectic $6$-manifolds, such that $Z$ does not admit a PSC metric and macroscopic dimension of $\widetilde{Z}$ is exactly $4$. Nevertheless, we construct circle bundles over these $Z$,

such that the $5$-dimensional total space $E_{Z}$ admits a PSC metric. An intriguing feature of these examples is that $\widetilde{Z}$ and $\widetilde{E_{Z}}$ are quasi-isometric.³³3$\pi_{1}(E_{Z})$ is a $\mathbf{Z}_{m}$-extension of $\pi_{1}(Z)$, where $m$ is the divisibility of the Euler class of $E_{Z}$ pulled back to the universal cover $\widetilde{Z}$ of $Z$. Therefore, $\pi_{1}(E_{Z})$ and $\pi_{1}(Z)$ are quasi-isometric. By the Švarc-Milnor lemma, $\widetilde{E_{Z}}$ and $\widetilde{Z}$ must be quasi-isometric as well. Therefore, since $\dim_{mc}\widetilde{Z}=4$, one may naively expect $\dim\widetilde{E_{Z}}=4$ as well (since $E_{Z}$ is PSC, that would give a counterexample to Gromov’s Conjecture 1!). However, this is not the case. Indeed, while it is straightforward to see that macroscopic dimension is invariant under quasi-isometric homeomorphisms, it is not invariant under just a quasi-isometry. See also [KR23, Theorem F]. Thus, even though $\dim_{mc}\widetilde{Z}=4$, and $\widetilde{Z}$ and $\widetilde{E_{Z}}$ are quasi-isometric, strangely the macroscopic dimension drops for $E_{Z}$, even though $E_{Z}$ is obtained from $Z$ by taking a twisted product with $S^{1}$. Precisely, it turns out that $\dim_{mc}\widetilde{E_{Z}}\leq 3$ in consonance with Gromov’s Conjecture 1. We show this using the natural deformation retraction from $\widetilde{E_{Z}}$ to a $3$-dimensional simplicial complex arising as the spine of the complement of the Donaldson divisor $Z$ in the ambient symplectic $6$-manifold.

Note that this phenomenon of the macroscopic dimension decreasing upon taking a (twisted) product with $S^{1}$ is not possible in the case of a trivial product. Indeed,

To the best of our knowledge examples exhibiting a drop in macroscopic dimension on twisting with $S^{1}$ have not appeared before in literature. On a similar theme, we were informed by Guth [Gut24] that a drop in the Urysohn width upon taking a double cover, i.e. an $S^{0}$ bundle, was observed by Balitskiy [Bal21, Theorem 2.4.4, pg. 36].

We now state the main result of this article:

We emphasize that Conclusions $(1)$ and $(2)$ make no assumptions either on the fundamental groups of $M$ and $E_{Z}$, or on the (non-)existence of spin or almost spin structures on the universal cover of $E_{Z}$. In particular, Conclusion $(2)$ does not follow from any existing results on Gromov’s conjecture (Conjecture 1). On the other hand, Conclusions $(3)$, $(4)$ and $(5)$ rest on Lemma 5, where we will show that there is a positive degree map from $Z$ to $M$. Then Conclusion $(3)$ follows from a theorem of Taubes [Tau94], Conclusion $(4)$ follows from the work of Chodosh, Li and Liokumovich [CLL23] and Conclusion $(5)$ follows from combining Proposition 6 with a result of Dranishnikov [Dra11].

By combining Theorem A with a Euler characteristic computation for Donaldson divisors carried out in Proposition 4, we obtain the following uniqueness corollary:

As a further corollary of Theorem A, we will construct examples sought after in Gromov’s Question 1.

In Section 2, we define the necessary symplectic preliminaries and make some observations regarding Donaldson divisors. In Section 3, we assemble other lemmas and propositions that we need to prove our main results. In Section 4, we prove Theorem A and Corollary C as well as provide explicit examples that are used towards proofs of Corollary B. The reader may first read Section 4, and refer to Sections 2 and 3, for the proofs of the results used in Section 4 as they arise.

The authors would like to thank Shihang He for several helpful comments on the first version of this paper. They would also like to thank Alexey Balitskiy, Alexander Dranishnikov, and Larry Guth for answering their questions and providing helpful remarks. The first author wishes to thank his advisor Yannick Sire for support and encouragement. The second author wishes to thank his advisor Mahan Mj, as well as his friends and colleagues Ritwik Chakraborty, and Sekh Kiran Ajij for several helpful conversations.

We note the following two simple but useful observations.

The following is an analogue of the Lefschetz hyperplane theorem in the symplectic setting, which we shall use in the next section:

In fact, the following lemma can be deduced from the proof of Proposition 2 in [Don96, pg. 700-701]:

We deduce two corollaries from Lemma 2.2. These are central to the proof of Part (2) of Theorem A.

In the next proposition, we show that Donaldson divisors of a $6$-dimensional symplectic manifold $(X^{6},\omega)$ with integral symplectic form $\omega$, realizing the homology classes Poincaré dual to $k[\omega]$, are topologically distinct for distinct values of $k\gg 1$. We do this by calculating their Euler characteristic.

Let $(M^{4},\eta_{0})$ be an integral symplectic $4$-manifold. We equip the manifold $M\times S^{2}$ with the integral symplectic form $\omega=\eta_{0}\oplus\omega_{0}$, where $\omega_{0}$ denotes the standard area form on $S^{2}$. We now make the following observation, which will be crucial to the proof of Parts (4) and (5) of Theorem A.

This class of manifolds was originally introduced by Floer in the context of the Arnol’d conjecture in symplectic geometry. We refer the reader to the survery article [KRT08] for detailed information. The key property of symplectically aspherical manifolds that we shall be using is the following:

We first prove a general lemma relating macroscopic dimension of a manifold with the homotopy type of a nullcobordism of the manifold. This will be used to prove of Part (2) of Theorem A.

We begin with the following result due to Cecchini, Räde and Zeidler in [CRZ23], which establishes a codimension-$2$ obstruction to existence of a PSC metric on a $7$-manifold.

Let $(M^{6},\omega)$ be an integral symplectic $6$-manifold. Let $Z\subset(M^{6},\omega)$ be a Donaldson divisor. Recall from Lemma 2.1 that the normal bundle of $Z$ in $M$ is isomorphic to $L^{\otimes k}$, which is a nontrivial complex line bundle over $Z$. However, we will now consider the unit circle bundle of $L^{\otimes k}$, denoted by $\mathbb{S}(L^{\otimes k})$. In conjunction with it, we will consider the restriction of the unit circle bundle of $L^{\otimes k}$ to $Z$, denoted by $\mathbb{S}(L^{\otimes k}|_{Z})$. Note that $\mathbb{S}(L^{\otimes k})$ is a $7$-manifold and $\mathbb{S}(L^{\otimes k}|_{Z})$ is a $5$-manifold.

In the following proposition, we shall verify that the pair $(\mathbb{S}(L^{\otimes k}),\mathbb{S}(L^{\otimes k}|_{Z}))$ satisfies the hypotheses of Theorem 2.

Before the proof of Proposition 7, we state and prove an elementary topology lemma.

In this section, we prove our main result. Let $(M,\eta)$ be an integral symplectic $4$-manifold. Let $\omega_{0}$ be the standard area form on $S^{2}$. We equip $M\times S^{2}$ with the integral symplectic form $\omega=\eta\oplus\omega_{0}$. Let $L$ be the pre-quantum line bundle on the symplectic manifold $(M\times S^{2},\omega)$. Let $Z\subset(M\times S^{2},\omega)$ be a Donaldson divisor, with $[Z]=\mathrm{PD}(k\cdot[\omega])$, for $k\gg 1$ sufficiently large. We set $E_{Z}:=\mathbb{S}(L^{\otimes k}|_{Z})$ as the unit circle bundle of $L^{\otimes k}$ restricted to $Z$ (as introduced in Lemma 2.1 and Proposition 7).

1. Proof of (1).

   As $M\times S^{2}$ admits a PSC metric, and $\mathbb{S}(L^{\otimes k})$ is a circle bundle on $M\times S^{2}$, $\mathbb{S}(L^{\otimes k})$ also admits a PSC metric [BB83, Theorem C]. By Proposition 7, this implies that $E_{Z}=\mathbb{S}(L^{\otimes k}|_{Z})$ also admits a PSC metric.

2. Proof of (2).

   Let $\nu(Z)$ denote a tubular neighbourhood of $Z\subset M\times S^{2}$. Consider the manifold with boundary $W:=(M\times S^{2})\setminus\nu(Z)$. Note that $\partial W=\partial\nu(Z)$. By Lemma 2.1, $\partial\nu(Z)\cong\mathbb{S}(L^{\otimes k}|_{Z})=E_{Z}$. Therefore $\partial W\cong E_{Z}$. In Corollary 3, we deduced that $(W^{6},E_{Z})$ satisfies the hypothesis of Lemma 3.1 with $n=6$ and $k=3$. Therefore, $\dim_{mc}\widetilde{E_{Z}}\leq 3$.

3. Proof of (3).

   By Proposition 5, the map $f:Z\to M$ is of positive degree. Moreover, by hypothesis $b_{2}^{+}(M)>1$. Since $f$ is of positive degree, the induced map

   $$f^{*}:H^{2}(M;\mathbf{Q})\to H^{2}(Z;\mathbf{Q})$$

   is an injective map of quadratic spaces, where both domain and range are equipped with the quadratic form given by the cup product $\smile$. Therefore, $b_{2}^{+}(Z)>1$. In [Tau94], Taubes proved that symplectic $4$-manifolds with $b_{2}^{+}>1$ have non-zero Seiberg-Witten invariant. Since having non-zero Seiberg-Witten invariant obstructs the existence of PSC metrics, this in particular implies $Z$ does not admit a PSC metric.

4. Proof of (4).

   By Proposition 5, the map $f:Z\to M$ is of positive degree. Moreover, by hypothesis $M$ admits a map of positive degree to an aspherical manifold. Composing these maps, we get a positive degree map from $Z$ to an aspherical manifold. Then by [CLL23, Corollary 3], $Z$ does not admit a PSC metric.

5. Proof of (5).

   Since $M$ is symplectically aspherical, by Proposition 6, $M$ is a rationally essential manifold. Since the map $f:Z\to M$ if of positive degree by Proposition 5, $Z$ must be rationally essential as well. Now, by Lemma 2,

   $$\pi_{1}(Z)\cong\pi_{1}(M\times S^{2})\cong\pi_{1}(M).$$

   By hypothesis, $\pi_{1}(M)$ is amenable. Therefore, $Z$ is a rationally essential manifold with amenable fundamental group. Hence, by [Dra11], $\dim_{mc}Z=4$.

Before we move to the proof of Corollary B we give several examples of $4$-manifolds $M$ satisfying the hypotheses of Theorem A.