\((Z_ 2)^ k\)-actions with fixed point set of constant codimension
\textit{R. E. Stong} [Trans. Am. Math. Soc. 178, 431-447 (1973; Zbl 0267.57025)], and others, considered the ideal \({\mathfrak I}^ r_ *\) of cobordism classes in \({\mathfrak N}_ *\) which contain a representative \(M^ n\) admitting an involution whose fixed point set is \((n-r)\) dimensional. For \(r\leq 8\) these ideals have been calculated. Generalizing the problem, the author considers ideals \({\mathfrak J}^ r_{*,k}\) of cobordism classes represented by manifolds \(M^ n\) which admit \(k\) commuting involutions such that the fixed point set of the \((\mathbb{Z}_ 2)^ k\) action is of dimension \((n-r)\). This problem is more difficult because the fixed point set for the individual involutions may vary, i.e., there are actions with different fixed point data. The author shows that \({\mathfrak I}^ 1_{*,k}=(0)\), \({\mathfrak I}^ 2_{*,k}=\oplus^ \infty_{i=2} {\mathfrak N}_ i\), and determines for a specific set of fixed point data the corresponding ideal \({\mathfrak I}\subset {\mathfrak I}^ 3_{*,2}\).
manifolds which admit \(k\) commuting involutions
scientific article; zbMATH DE number 62953
scientific article