Circle actions on homotopy spheres with codimension 4 fixed point set
type: Item
instance of: scholarly article
title: Circle actions on homotopy spheres with codimension 4 fixed point set
zbMATH Open document ID: 0551.57020
author: Ronald Fintushel
author: Peter Sie Pao
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: The paper contains a complete classification of smooth \(S^ 1\)-actions on homotopy n-spheres with codimension 4 fixed point set. Semifree actions of this type have been classified by \textit{J. Levine} [Invent. Math. 22, 161-186 (1973; Zbl 0266.57010)]. In the general case exceptional orbit types may exist different from those of semifree actions. There are the following possibilities: (i) no exceptional orbit type (semifree action), (ii) one exceptional orbit type, (iii) two exceptional orbit types and (iv) three exceptional orbit types. According to the possibilities (ii), (iii), and (iv) the equivalence classes of \(S^ 1\)-actions are in 1-1 correspondence to the sets \({\mathcal S}^ n_ k\), \({\mathcal S}^ n_{k,m}\), and \({\mathcal T}^ n_{k,m}\). Here \({\mathcal S}^ n_ k\) denotes the set of diffeomorphism classes of pairs \((\Sigma^{n-1},\Delta_ k)\), where \(\Sigma^{n-1}\) is a homotopy (n- 1)-sphere and \(\Delta_ k\) is a smooth orientable (n-3)-dimensional \({\mathbb{Z}}_ k\)-acyclic submanifold with boundary an integral homotopy sphere. \({\mathcal S}^ n_{k,m}\) with k, m relatively prime positive integers is the set of diffeomorphism classes of triads \((\Sigma^{n- 1},\Delta_ k,\Delta_ m)\), where \(\Sigma^{n-1}\) is a homotopy (n-1)- sphere, \(\Delta_ k\) a smooth orientable \({\mathbb{Z}}_ k\)-acyclic (n-3)- submanifold, \(\Delta_ m\) a smooth orientable \({\mathbb{Z}}_ m\)-acyclic (n- 3)-submanifold of \(\Sigma^{n-1}\), \(\Delta_ k\cap \Delta_ m=\partial \Delta_ k=\partial \Delta_ m\) is an integral homology sphere, and \(\Delta_ k\cup \Delta_ m\) is a smooth submanifold of \(\Sigma^{n- 1}\). The set \({\mathcal T}^ n_{k,m}\) is defined as the set of diffeomorphism classes of triads \((\Sigma^{n-1},\Delta_ k,\Delta_ m)\) as above except that Int \(\Delta\) \({}_ k\) and Int \(\Delta\) \({}_ m\) meet transversely in a (n-5)-dimensional smooth submanifold Q, and \(\Delta_ k\cup \Delta_ m\setminus Q\) is a smooth submanifold of \(\Sigma^{n-1}\). The 1-1…
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 57S25
zbMATH DE Number: 3877996
zbMATH Keywords: classification of smooth circle actions on homotopy n-spheres with codimension 4 fixed point set
zbMATH Keywords: diffeomorphism classes of homotopy (n-1)- sphere
zbMATH Keywords: acyclic submanifold
zbMATH Keywords: semifree action
zbMATH Keywords: structured orbit space
zbMATH Keywords: twist-spinning process for knots
reviewed by: K. H. Mayer
MaRDI profile type: Publication
full work available at URL: pjm.1983.109.349
OpenAlex ID: W2041384371
DOI: 10.2140/PJM.1983.109.349
description: scientific article; zbMATH DE number 3877996