Involution number sequence and dimensions of the components in fixed point set of an involution. II
[For Part I see ibid. 34, No. 11, 1302-1306 (1991; Zbl 0784.57021).] The authors prove: Given any smooth unfree involution on \((S^ 1)^ n\) the fixed-point set of the involution must be constant dimensional. In order to show this they compute the total Stiefel-Whitney classes \(W(\mathbb{R}^ m (\tau))\) by comparing them with \(W(\mathbb{R}^ m (\tau_ i))\), where \(\tau\) is the given involution and \(\tau_ i\), \(i=1,\dots,n\), are particular involutions on \((S^ 1)^ n\). Finally the computation of \(W(\overline T_ m (S^ 1)^ n)\), where \(\overline T_ m (S^ 1)^ n\) is the tangent bundle along the fibre of \(\mathbb{R}^ m (\tau)\), leads to the result.
smooth unfree involution on \((S^ 1)^ n\)
scientific article; zbMATH DE number 98076