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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 \(G= \mathbb{Z}_2\) 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 \(\mathbb{R} P^4\), 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 \[ R_{1313} +R_{1414} +R_{2323} +R_{2424} \geq 2R_{1234}. \] 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.

53C20 Global Riemannian geometry, including pinching
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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