  According to Kiyoshi Igusa a generalized Morse function on an n-dimensional
manifold M is a smooth function with only Morse and birth-death singularities
and a framed function is a generalized Morse function with an additional
structure: a framing of the negative eigenspace at each critical point of the
function f. In his paper "The space of framed functions" (Trans. of Amer. Math.
Soc., 301(1987), 431-477) Igusa proved that the space of framed generalized
Morse functions is (n-1)-connected. In the paper "On the Classification of
Topological Field Theories" (arXiv:0905.0465) Jacob Lurie gave an algebraic
topological proof that the space of framed functions is contractible. In this
paper we give a geometric proof of Igusa-Lurie's theorem in the spirit of our
paper "Wrinkling of smooth mappings - II. Wrinkling of embeddings and K.Igusa's
theorem" (Topology, 39(2000), 711-732.
