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On the homotopy type of the space of generalized Morse functions. (English) Zbl 0595.57025

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

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

Citations:

Zbl 0548.58005
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