On handle theory for Morse-Bott critical manifolds. (English) Zbl 1431.37010
This paper studies flows induced by Morse-Bott functions on \(n\)-dimensional manifolds. The main results of the paper present some conditions for an abstract Morse-Bott graph to be realized as the Morse-Bott graph associated to a flow induced by a Morse-Bott function. A necessary condition for this realization is that the Morse-Bott graph must satisfy the generalized Morse-Bott inequalities for isolating blocks (Corollary 5.1). In addition, this condition turns out to be sufficient in the 2-dimensional and 3-dimensional cases to realize an abstract Morse-Bott semigraph with only one vertex as the Morse-Bott semigraph of a Morse-Bott flow on an isolating block (Theorem 6.1 and Theorem 6.3) and to realize an abstract Morse-Bott graph by a Morse-Bott flow on a closed 3-dimensional manifold (Corollary 6.1). Theorem 6.2 establishes some necessary and sufficient conditions for an abstract Morse-Bott graph to be associated to a Morse-Bott flow defined on a closed 2-dimensional manifold. Notice that the 2-manifold is completely determined by the data on the Morse-Bott graph. The authors point out that in contrast with the 2-dimensional case, a realizable 3-abstract Morse-Bott graph may be associated to flows in infinitely many 3-manifolds. The last section of the paper deals with the \(n\)-dimensional case with \(n>3\).
Reviewer: Héctor Barge (Madrid)
MSC:
| 37B30 | Index theory for dynamical systems, Morse-Conley indices |
| 37B35 | Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems |
| 57R65 | Surgery and handlebodies |
| 57R70 | Critical points and critical submanifolds in differential topology |
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