A generic 1-parameter family of Morse functions on a compact manifold with boundary has exactly three possible non-Morse events: interior birth/death, boundary birth/death, and boundary-interior collision.
Monopoles and three-manifolds , volume 10 of New Mathematical Monographs
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Families of Morse functions for manifolds with boundary
A generic 1-parameter family of Morse functions on a compact manifold with boundary has exactly three possible non-Morse events: interior birth/death, boundary birth/death, and boundary-interior collision.