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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 \(S^2\times S^2\).
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)
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