A Generalization of the Geroch Conjecture with Arbitrary Ends

Notice the space $S^{1}$ is a $K(\mathbb{Z},1)$, so $H^{1}(M;\mathbb{Z})=[M,S^{1}]$, where $[M,S^{1}]$ are homotopy classes of maps from $M$ to $S^{1}$. Thus we can choose a non-constant smooth map $f:M\to S^{1}$ representing the Poincaré dual of $\alpha$ in $H^{1}(M;\mathbb{Z})$. By Sard’s theorem we can take the preimage $\Sigma$ of a regular value as a representative of $\alpha$. Then $\Sigma$ is a closed embedded orientable hypersurface by the regular value theorem.

$\blacksquare$

Suppose $\Sigma$ is non-separating, then $M\setminus\Sigma$ is connected, so there exists a simple loop $S$ in $M$ which crosses $\Sigma$ transversally in exactly one point. Orient $S$ so that this intersection is positive. Then the oriented intersection number $I([\Sigma],[S])$ equals $1$. Since the oriented intersection number is independent of the representative of the homology class, it follows that $\Sigma$ is homologically nontrivial.

Conversely, suppose that $\Sigma$ separates. Let $U$ be a tubular neighborhood of $\Sigma$, and let $V=M\setminus\Sigma=M_{+}\cup M_{-}$. Then $\partial V=\partial M_{+}\cup\partial M_{-}=\Sigma_{+}\cup\Sigma_{-}$, where $\Sigma_{+}$ has the same orientation as $\Sigma$ while $\Sigma_{-}$ has the opposite orientation.

Let $i:\Sigma\to M$, $i_{1}:U\cap V\to U$, $i_{2}:U\cap V\to V$, $j_{1}:U\to M$, $j_{2}:V\to M$ be inclusion maps. Consider the Mayer-Vietoris sequence in singular homology with $\mathbb{Z}$ coefficients:

Since $U\cap V$ is homotopy equivalent to a disjoint union $\Sigma_{+}\cup\Sigma_{-}$, we have $H_{n-1}(U\cap V)\cong H_{n-1}(\Sigma_{+})\oplus H_{n-1}(\Sigma_{-})$, and the map $(i_{1})_{*}:H_{n-1}(U\cap V)\to H_{n-1}(U)$ is given by $H_{n-1}(\Sigma_{+})\oplus H_{n-1}(\Sigma_{-})\to H_{n-1}(\Sigma),(a[\Sigma_{+}],b[\Sigma_{-}])\mapsto(a-b)[\Sigma]$. Since $\Sigma$ is the boundary of $M_{+}$, $\Sigma$ is null-homologous in $M_{+}$ hence also in $V$, showing the map $H_{n-1}(\Sigma)\xrightarrow{\cong}H_{n-1}(U\cap V)\to H_{n-1}(V)$ is the zero map. Thus the Mayer-Vietoris sequence becomes

Exactness at $H_{n-1}(\Sigma)\oplus H_{n-1}(V)$ shows that $H_{n-1}(\Sigma)\oplus\{0\}=\operatorname{Im}\,((i_{1})_{*},0)=\operatorname{Ker}\,(i_{*}-(j_{2})_{*})$, so $H_{n-1}(\Sigma)=\operatorname{Ker}i_{*}$. That is, $i_{*}:H_{n-1}(\Sigma)\to H_{n-1}(M)$ is the zero map, which means $[\Sigma]$ is trivial in $H_{n-1}(M)$. $\blacksquare$

Let $\Sigma_{t}:=\partial\Omega_{t}$. By the variation formulas for hypersurfaces (see e.g. [HP99, Theorem 3.2]), we have

Differentiating the first variation and using $H_{\Sigma}=h$, we thus have

$\blacksquare$

We follow the approach of [CL20, Section 6 and 7]. Namely, we pass to an appropriate covering space of $Y$, construct a weight function $h$, and apply the $\mu$-bubble technique. The main difference in our case is how to find the covering space and how to modify the construction of the weight function $h$. An illustration of the construction is in Figure 1.

Let $Y=M\#X$ be as in the assumption. By taking the orientation double cover of $X$ we can assume $X$ is orientable. Let $p\in M$ be a point such that $B(p)\cap\Sigma=\emptyset$, where $B(p)$ is a small $n$-ball around $p$. Let $M^{\prime}=M\setminus B(p)$ and $X^{\prime}=X\setminus B$, where $B$ is a small $n$-ball in $X$. Then we can take $Y=M\#X=M^{\prime}\cup X^{\prime}$, where $M^{\prime}$ and $X^{\prime}$ are glued on the boundary sphere.

Suppose $Y$ is endowed with a complete metric of positive $m$-intermediate curvature. By scaling and compactness of $M^{\prime}$ we can assume $\mathcal{C}_{m}>1$ on $M^{\prime}$. Let $a>0$ be any number.

Step 1: pass to an infinite cyclic cover by cutting and pasting $M$ along $\Sigma$. Cut $M$ along $\Sigma$. Let $\tilde{M}=M\setminus\Sigma$ and $\tilde{M}^{\prime}=M^{\prime}\setminus\Sigma=(M\setminus B(p))\setminus\Sigma$. Then $\tilde{M}$ is a connected manifold with boundary, and $\partial\tilde{M}$ has two components, both diffeomorphic to $\Sigma$. Denote $\partial\tilde{M}=\Sigma_{-}\cup\Sigma_{+}$. Let $\tilde{M}_{k}$, $k\in\mathbb{Z}$ be $\mathbb{Z}$ copies of $\tilde{M}$, and let $\tilde{M}^{\prime}_{k}$ be the corresponding $\tilde{M}^{\prime}\subset\tilde{M}$. Glue together $\tilde{M}_{k}$ along the boundary by gluing the $\Sigma_{+}$ boundary component of $\tilde{M}_{k}$ with the $\Sigma_{-}$ boundary component of $\tilde{M}_{k+1}$. Denote the resulting manifolds by

where the equivalence relation $\sim$ is the gluing we just described. Then $\hat{M}$ is an infinite cyclic covering of $M$. Denote the closed hypersurface in $\hat{M}$ coming from the $\Sigma_{-}$ boundary component of $\tilde{M}_{k}$ (equivalently, the $\Sigma_{+}$ boundary component of $\tilde{M}_{k-1}$) by $\Sigma_{k}$. Orient $\Sigma_{k}$ so that its normal is pointing towards $\tilde{M}_{k}$.

Let $X_{k}^{\prime}$, $k\in\mathbb{Z}$ be $\mathbb{Z}$ copies of $X^{\prime}$. Then we have

where $\hat{M}^{\prime}$ and each $X_{k}^{\prime}$ are glued on the boundary spheres. The manifold $Y$ is an infinite cyclic cover of $M\#X$.

We endow $\hat{Y}$ with the pullback Riemannian metric such that $\hat{Y}\to Y$ is a Riemannian covering map. Then by our assumption that $\mathcal{C}_{m}>1$ on $M^{\prime}$, we also have $\mathcal{C}_{m}>1$ on $\hat{M}^{\prime}$.

Step 2: construct the weight function $h$.

We now define $\rho_{0}:\hat{Y}\to\mathbb{R}$ as the signed distance function to the hypersurface $\Sigma_{0}$. Then $\rho_{0}$ is Lipschitz. We then take $\rho_{1}$ to be a smoothing of $\rho_{0}$ such that for each $k$, $\rho_{1}\equiv A_{k}$ for some constant $A_{k}$ in a small neighborhood of $\partial X_{k}^{\prime}$ (i.e., where $\hat{M}^{\prime}$ and $X_{k}^{\prime}$ are glued together), and $A_{k}>0$ if $k\geq 0$ and $A_{k}<0$ if $k<0$. We can further assume that $\rho_{1}\geq A_{k}$ on $X_{k}^{\prime}$ if $k\geq 0$ and $\rho_{1}\leq A_{k}$ on $X_{k}^{\prime}$ if $k<0$.

Then there is $L>0$ so that

We now define a function $h\in C(\hat{Y},[-\infty,\infty])$ as follows. On $\hat{M}^{\prime}\cap\{-\tfrac{\pi L}{2}\leq\rho_{1}\leq\tfrac{\pi L}{2}\}$, we define

On the rest of $\hat{M}^{\prime}$ we set $h=\pm\infty$ such that it is continuous to $[-\infty,\infty]$. We then define $h$ on $X_{k}^{\prime}$. When $A_{k}\geq\tfrac{\pi L}{2}$, set $h=-\infty$ on $X_{k}^{\prime}$. When $A_{k}\leq-\tfrac{\pi L}{2}$, set $h=\infty$ on $X_{k}^{\prime}$. Now assume $|A_{k}|\leq\tfrac{\pi L}{2}$.

For $k\geq 0$ and

or for $k<0$ and

we set

Otherwise we set $h(p)=\pm\infty$ such that $h$ is continuous. Observe that by definition, $h$ is finite on only finitely many $X_{k}^{\prime}$.

Notice that for $x\in\partial X_{k}^{\prime}$, we have that

and thus $h$ is Lipschitz across $\partial X_{k}^{\prime}$. If $0\leq A_{k}<\tfrac{\pi L}{2}$, $x\in X_{k}^{\prime}$ and

we have that $h(x)\to-\infty$. Similarly, if $-\tfrac{\pi L}{2}<A_{k}<0$, $x\in X_{k}^{\prime}$ and

we have that $h(x)\to\infty$. Thus $h$ is continuous on $X_{k}^{\prime}$.

Note that the set $\{|h|<\infty\}$ is bounded. This is because this region is bounded in $\hat{M}^{\prime}$, only finitely many ends $X_{k}^{\prime}$ are included in this set, and in each $X_{k}^{\prime}$, the region where $\{|h|<\infty\}$ is bounded.

Similar to [CL20], we have

Step 3: apply the $\mu$-bubble technique.

We consider $\mu$-bubbles with respect to the smooth function $h$ we have just defined. We fix

We can minimize

among all Cacioppoli sets $\Omega$ such that $\Omega\Delta\Omega_{0}$ is compactly contained in $\{|h|<\infty\}$ by Proposition 3.1. Denote by $\Omega$ the connected component of the minimizer containing $\{\rho_{1}=-\frac{\pi L}{2}\}$. Since $n\leq 7$, each component of $\partial\Omega$ is compact and regular. By the first variation formula from Lemma 3.2 and the stability inequality for $\mathcal{A}$ from Lemma 3.3, we see that $\Lambda=\partial\Omega$ satisfies $H=h$ and

for all $\psi\in C^{\infty}(\Lambda)$.

We can find a compact region $Y^{\prime}\subset Y$ with smooth boundary so that $\partial\Omega\subset Y^{\prime}$. Furthermore, we can arrange that $\partial Y^{\prime}\cap\hat{M}=\Sigma_{I}\cup\Sigma_{-I}$, for some large $I\in\mathbb{N}$. Note that the other boundary components of $Y^{\prime}$ thus lie completely in some $X_{k}^{\prime}$.

In particular, $\partial Y^{\prime}\setminus\hat{M}$ bounds some compact manifold with boundary. Cap these components off and then glue the hypersurfaces $\Sigma_{I}$ and $\Sigma_{-I}$ to each other. We thus obtain a manifold $\tilde{Y}$ diffeomorphic to $M^{\prime}\#_{i}\tilde{X}_{i}$, where $M^{\prime}$ is a $2I$-cyclic covering of $M$ obtained by cutting and pasting along $\Sigma$, and each $\tilde{X}_{i}$ is closed and we have finitely many of them. We also have a hypersurface $\Lambda^{n-1}\subset\tilde{Y}$ homologous to $[\Sigma^{n-1}\times\{*\}]\in H_{n-1}(\tilde{Y})$ that satisfies $H=h$ and (3). We can make $h$ to be a smooth function on $\tilde{Y}$ that agrees with our old $h$ in a neighborhood of $\Lambda$, so that (2) is satisfied. We can also construct a metric on $\tilde{Y}$ such that it is isometric to the original metric on $Y$ in a neighborhood of $\Lambda$. Since $Y$ has positive $m$-intermediate curvature, this means $\tilde{Y}$ has positive $m$-intermediate curvature in a neighborhood of $\Lambda$. This also means that on $\Lambda$, we have

and

for all $\psi\in C^{\infty}(\Lambda)$. $\blacksquare$

Since the space $S^{1}$ is a $K(\mathbb{Z},1)$, we have $[M,S^{1}]=H^{1}(M;\mathbb{Z})$, and the bijection is given by $f\mapsto(f_{*}:H_{1}(M)\to H_{1}(S^{1})\cong\mathbb{Z})$. Thus for any $\beta\in H^{1}(M;\mathbb{Z})$ we can get a smooth map $f:M\to S^{1}$ and vice versa. Further, the preimage of any regular value of $f$ represents the Poincaré dual of $\beta$. Thus given $\beta_{1},\dots\beta_{n-2}\in H^{1}(M;\mathbb{Z})$ we can get a smooth map $F=(F_{1},\dots,F_{n-2}):M\to\mathbb{T}^{n-2}$ and vice versa. Since the cup product is the Poincaré dual to intersection, we have

where $t=(t_{1},\dots,t_{n-2})$ is any regular value of $F$. Then the assertion follows.

$\blacksquare$

Since $M$ is an SYS manifold, there are nonzero cohomology classes $\beta_{1},\dots,\beta_{n-2}$ in $H^{1}(M;\mathbb{Z})$ such that the homology class $[M]\frown(\beta_{1}\smile\dots\smile\beta_{n-2})\in H_{2}(M;\mathbb{Z})$ is non-spherical.

Then we get pullbacks $f^{*}\beta_{1},\dots,f^{*}\beta_{n-2}$ in $H^{1}(\hat{M};\mathbb{Z})$. Claim the class $f^{*}\beta_{1}\smile\dots\smile f^{*}\beta_{n-2}\in H_{2}(\hat{M};\mathbb{Z})$ is non-spherical. Suppose not. Then there exists a map $\phi:S^{2}\to\hat{M}$ such that