On the homotopy type of the space of generalized Morse functions (Q1077765)

Let N be a compact smooth n-manifold and let \(g: N\to {\mathbb{R}}\) be a fixed Morse function without critical points on \(\partial N\). This paper computes the (n-1)-homotopy type of the space \({\mathcal H}(N,\partial N)\) of all smooth real-valued functions on N that agree with g near \(\partial N\) and whose critical points are either Morse or birth-death singularities. The main theorem states that there is an n-connected map \({\mathcal H}(N,\partial N)\to \Omega^{\infty}S^{\infty}(BO\bigwedge N_+))\). The (n-1)-connectivity is due to the author, and an argument of C. Ogle is given that extends this to n-connectivity. The key step in the proof is the result of the author [Ann. Math., II. Ser. 119, 1-58 (1984; Zbl 0548.58005)]. He then goes on to prove the analogue of this theorem for spaces of functions with certain Thom-Boardman singularities.