On the homotopy type of the space of generalized Morse functions
type: Item
instance of: scholarly article
title: On the homotopy type of the space of generalized Morse functions
zbMATH Open document ID: 0595.57025
DOI: 10.1016/0040-9383(84)90043-0
author: Kiyoshi Igusa
published in: Topology
publication date: 1984-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57R45
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 58D15
zbMATH DE Number: 3958256
zbMATH Keywords: compact smooth n-manifold
zbMATH Keywords: Morse function
zbMATH Keywords: space \({\mathcal H}(N,\partial N)\) of all smooth real-valued functions
zbMATH Keywords: critical points
zbMATH Keywords: birth-death singularities
zbMATH Keywords: Thom-Boardman singularities
reviewed by: R. J. Stern
MaRDI profile type: Publication
full work available at URL: 0040-9383(84)90043-0
OpenAlex ID: W2025596829
description: scientific article; zbMATH DE number 3958256