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Four-manifolds with positive isotropic curvature. (English) Zbl 0892.53018
An incompressible space form $N$ of a 4-manifold $M$ is a 3-dimensional submanifold diffeomorphic to the quotient of a 3-sphere by a group $G$ of linear isometries without fixed points, such that the fundamental group of $\pi_1(N)$ injects into $\pi_1(M)$. Such a space form is said to be essential unless $G= \{1\}$ or $G= \bbfZ_2$ and the normal bundle is non-orientable. In the paper under review, the author studies compact four-manifolds with no essential incompressible space forms. It is shown that $M$ admits a metric of positive isotropic curvature if and only if the manifold is diffeomorphic to a sphere $S^4$, the projective space $\bbfR P^4$, the product $S^3 \times S^1$, the nonoriented $S^3$ bundle over $S^1$, or a connected sum of the above. Positive isotropic curvature means that for all orthonormal vectors $\{e_1,e_2, e_3,e_4\}$, the curvature tensor satisfies $$R_{1313} +R_{1414} +R_{2323} +R_{2424} \ge 2R_{1234}.$$ The result is proved by using the Ricci flow. The essential space forms defined above are obstructions for such a flow. If non-essential space forms exist, then a surgically modified Ricci flow is used to obtain the desired metric.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 57N13 Topology of the Euclidean $4$-space, $4$-manifolds