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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.

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