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On closed 4-manifolds admitting a Morse function with 4 critical points. (English) Zbl 0907.58007
Let $$M$$ be a closed connected $$C^\infty$$ 4-manifold with non-zero Euler characteristic which admits a Morse function with exactly 4 critical points. The author proves the following alternative: either $$M$$ is topologically equivalent with $$S^4$$, or $$M$$ has the integral homology groups of $$S^2\times S^2$$. Furthermore, if the second situation occurs, then the following holds: if $$M$$ is of a $$C^\infty$$ product structure, then $$M$$ is diffeomorphic with $$S^2\times S^2$$; if $$M$$ admits a Riemannian metric of positive curvature and $$M$$ is isometrically embedded into $$E^6$$, then $$M$$ is topologically equivalent with $$S^4$$; if $$M$$ admits a Riemannian structure of non-negative curvature and with this structure $$M$$ can be isometrically embedded into $$E^6$$, then either $$M$$ is a topological $$S^4$$ or $$M$$ is diffeomorphic with $$S^2\times S^2$$.
##### MSC:
 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010)
##### Keywords:
Morse function; critical point; 4-manifold; Euler characteristic
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##### References:
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