An analogue of the stabilization map for regular \(\mathbb{Z}_ p\) actions (Q1335195)

The paper is motivated by a famous classical result of \textit{P. E. Conner} and \textit{E. E. Floyd} [Differentiable periodic maps (1964; Zbl 0125.401)]: Theorem. For all \(k\in \mathbb{Z}\), \(k\geq 0\), there exists \(\varphi (k)\in \mathbb{Z}\), \(\varphi (k)\geq 0\), such that if \((T, M^n)\) is a smooth involution on a closed manifold with \(\dim (M^{\mathbb{Z}_2}) \leq k\) and \(n= \dim (M)> \varphi (k)\), then \(M^n\) bounds in \({\mathfrak N}_*\). Boardman showed that one can take \(\varphi (k)= 5k/2\), and that this is the best possible general result. An action of \(\mathbb{Z}_p\), \(p\) a prime, is said to be regular if there exists an irreducible representation \(\rho\) of \(\mathbb{Z}_p\) such that over every fixed point the fibre of the normal bundle of the fixed point set is a multiple of \(\rho\) [\textit{K. Kawakubo}, Am. J. Math. 97, 182-204 (1975; Zbl 0334.57027)]. Manifolds whose Pontrjagin numbers are \(0\bmod p\) define an ideal \(I(p)\) in the oriented bordism group \(\Omega\). The main result of the paper is: Theorem. Let \(p\) be an odd prime, \(T\) a smooth regular action of \(\mathbb{Z}_p\) on an oriented manifold \(M^n\) of dimension \(n\), and \(k\) the dimension of the highest dimensional component of the fixed point set of the action. If the class of \(M^n\) is non-zero in \(\Omega/ I(p)\), then \(k\geq n/2\). Remarks of the reviewer. The formulation of the theorem in this review is more precise than the one in the paper, and the definition of a regular action in the paper is unnecessarily rigid. The result holds for regular actions as defined by Kawakubo.