Geroch’s Conjecture by Dimension Reduction

The following theorem is the form of the Geroch conjecture used in the positive mass theorem. The dimension restriction comes only from the regularity theory for area-minimizing hypersurfaces: an area-minimizing hypersurface in an \(n\)-manifold is smooth when \(n\leq7\).

Theorem 6.3.1

Let \(3\leq n\leq7\). Let \(X^n\) be a closed oriented smooth manifold. If there is a continuous map

\[ F:X\to \mathbb{T}^n \]

of nonzero degree, then \(X\) admits no metric of positive scalar curvature.

Proof. We argue by induction on \(n\). The base \(n=2\) is Gauss–Bonnet: if a closed oriented surface maps to \(\mathbb{T}^2\) with nonzero degree, then it has genus at least one, so it cannot carry a metric with positive Gaussian curvature.

Assume now \(3\leq n\leq7\), and suppose the theorem is known in dimension \(n-1\). Let \(g\) be a positive-scalar-curvature metric on \(X\). Write

\[ \theta_1,\ldots,\theta_n\in H^1(\mathbb{T}^n;\mathbb{Z}) \]

for the standard degree-one cohomology classes, and set

\[ \omega_i:=F^*\theta_i\in H^1(X;\mathbb{Z}). \]

Since \(\deg F\neq0\),

\[ \omega_1\smile\cdots\smile\omega_n\neq0 \qquad\text{in }H^n(X;\mathbb{Z}). \]

Let \(\alpha=\omega_n\). Its Poincare dual is a nonzero integral homology class in \(H_{n-1}(X;\mathbb{Z})\). Choose an area-minimizing integral current \(\Sigma\) in this class. Since \(n\leq7\), regularity theory implies that \(\Sigma\) is a smooth embedded closed hypersurface, possibly with several components and integer multiplicities. It is oriented, two-sided, minimal, and stable.

By Proposition 6.2.1, every component of \(\Sigma\) of dimension at least three admits a positive-scalar-curvature metric; in the surface case, Remark 6.2.2 says that every component is a two-sphere.

Now compute the cup product on \(\Sigma\). Since \([\Sigma]\) is Poincare dual to \(\omega_n\),

\[ \int_\Sigma \omega_1\smile\cdots\smile\omega_{n-1} = \int_X \omega_1\smile\cdots\smile\omega_n \neq0. \]

Therefore at least one connected component \(\Sigma_0\) has

\[ \int_{\Sigma_0} \omega_1\smile\cdots\smile\omega_{n-1}\neq0. \]

Equivalently, the map

\[ F_0:=(F_1,\ldots,F_{n-1})|_{\Sigma_0}: \Sigma_0\to\mathbb{T}^{n-1} \]

has nonzero degree.

If \(n-1=2\), this contradicts the fact that \(\Sigma_0\) is a two-sphere. If \(n-1\geq3\), then \(\Sigma_0\) carries a positive-scalar-curvature metric by conformal descent, contradicting the induction hypothesis in dimension \(n-1\). This proves the theorem. ◻

Corollary 6.3.2

For \(3\leq n\leq7\) and for every closed oriented \(n\)-manifold \(Y\), the connected sum

\[ \mathbb{T}^n\#Y \]

admits no metric of positive scalar curvature.

Proof. There is a degree-one map

\[ \mathbb{T}^n\#Y\to\mathbb{T}^n \]

obtained by collapsing \(Y\) minus a ball to the connected-sum point and using the identity map on the torus side. Theorem 6.3.1 applies. ◻

Geroch’s Conjecture by Dimension Reduction

Gaoming Wang

Assistant Professor