We define the notion of a “framed function” on a compact smooth manifold N N and we show that the space of all framed functions on N N is ( dim N − 1 ) (dim \,N - 1) -connected. A framed function on N N is essentially a smooth function N → R N → R with only Morse and birth-death singularities together with certain additional structure.
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Kiyoshi Igusa (1987) studied this question.