Morse function and attaching map
type: Item
instance of: scholarly article
title: Morse function and attaching map
zbMATH Open document ID: 0831.58013
author: Kouyemon Iriye
author: Akira Kono
published in: Journal of Mathematics of Kyoto University
publication date: 1996-02-13T00:00:00Z
review text: Let \(M\) be a closed manifold and \(f\) be a Morse function, a differentiable function on \(M\) with isolated, non-degenerate, critical points. Associated to a Morse function there is a cell complex \(L\) which is homotopy equivalent to \(M\) [\textit{J. Milnor}, Morse theory, Princeton University Press (1963; Zbl 0108.104)]. It is assumed that the Morse function \(f\) on the manifold \(M\) has no critical points of indices \(j+1, \dots, j+\ell-1\), where \(\ell\) is a pointwise integer. The purpose of the reviewing article is to prove the following theorem and to give its application to the projective spaces. Theorem. The attaching map of the \(j+\ell\)-cell \(\alpha\) to the \(J\)-cell \(\beta\) is described as the (transversal) intersection manifold of the unstable manifold of \(- \text{grad } f\) at \(\alpha\) with the stable manifold \(-\text{grad } f\) at \(\beta\) considered as a framed manifold embedded in the unstable manifold.
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 784736
zbMATH Keywords: Morse function
zbMATH Keywords: critical points
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250518842
OpenAlex ID: W1599286232
DOI: 10.1215/KJM/1250518842
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description: scientific article; zbMATH DE number 784736