Circle actions on homotopy spheres with codimension 4 fixed point set
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 correspondence is realized by associating to the \(S^ 1\) action the structured orbit space and viewing \(\Delta_ k\) and \(\Delta_ m\) as the images in the orbit space of the fixed point sets of \({\mathbb{Z}}_ k\) and \({\mathbb{Z}}_ m\). The authors point out a relationship with a generalization of the twist-spinning process for knots.
classification of smooth circle actions on homotopy n-spheres with codimension 4 fixed point set
diffeomorphism classes of homotopy (n-1)- sphere
scientific article; zbMATH DE number 3877996