\((\mathbb Z_2)^k\)-actions with trivial normal bundle of fixed point set (Q1885250)

Let \(\varphi : (\mathbb{Z}_2)^k \times M^n \rightarrow M^n\) be a smooth action of the group \((\mathbb{Z}_2)^k\) on a closed \(n\)-dimensional manifold \(M^n\). When \(k=1\), Conner and Floyd gave the complete analysis of such actions. When \(k \geq 1\), using the formula on characteristic numbers of Kosniowski and Stong, the author obtains generalizations as follows: If each component of the fixed point set \(F^l\) has a constant \(l\)-dimensional and trivial normal bundle, then the author shows that either \(M^n\) bounds equivariantly or \(F^l\) has the following property: (1) for \(l < n\), \(F^l\) possesses the linear dependence property, (2) for \(l = n\), \(F^l\) is bordant to \(M^n\). If the fixed point set \(F\) has variable dimension, i.e., \(F = \amalg_{h > 0} F^h\), and satisfies that for \(h < n\), all Stiefel-Whitney classes of the normal bundle to each component \(F^h\) vanish in positive dimension and each \(F^h\) possesses the linear independence property, then he shows that for \(h < n\) each connected component of \(F^h\) bounds and \(F^n\) is bordant to \(M^n\).