Igusa, Kiyoshi On the homotopy type of the space of generalized Morse functions. (English) Zbl 0595.57025 Topology 23, 245-256 (1984). 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. Reviewer: R.Stern Cited in 1 ReviewCited in 11 Documents MSC: 57R45 Singularities of differentiable mappings in differential topology 58C25 Differentiable maps on manifolds 58K99 Theory of singularities and catastrophe theory 55Q52 Homotopy groups of special spaces 58D15 Manifolds of mappings Keywords:compact smooth n-manifold; Morse function; space \({\mathcal H}(N,\partial N)\) of all smooth real-valued functions; critical points; birth-death singularities; Thom-Boardman singularities Citations:Zbl 0548.58005 PDFBibTeX XMLCite \textit{K. Igusa}, Topology 23, 245--256 (1984; Zbl 0595.57025) Full Text: DOI