\(\mathbb Z _2\) actions on complexes with three non-trivial cells (Q924291)

The paper under review considers \({\mathbb Z}_2\)-actions on a cell complex \(K\) having the same cohomology ring as the wedge sum \(P^2(n)\vee S^{3n}\) or \(S^n\vee S^{2n}\vee S^{3n}\). The cohomology ring of \(K\) is completely determined by a pair \((a, b)\) of integers, satisfying the following condition \[ u_1^2=au_2, u_1u_2=bu_3 \] where \(u_i\in H^{in}(K;{\mathbb Z})\cong {\mathbb Z}\) is a generator for \(i=1,2,3\). When \(b\not\equiv 0\mod p\), the fixed point set of \({\mathbb Z}_p\)-actions has been determined by \textit{G. E. Bredon} [Introduction to Compact Transformation Groups, Academic Press, New York (1972; Zbl 0246.57017)] and \textit{J. C. Su} [Trans. Am. Math. Soc. 112, 369--380 (1964; Zbl 0178.57205), ibid. 106, 305--318 (1963; Zbl 0109.41501)]. When \(b\equiv 0\mod p\), the fixed point set of \({\mathbb Z}_p\)-actions has been determined by \textit{R. M. Dotzel} and \textit{T. B. Singh} [Proc. Am. Math. Soc. 113, 875--878 (1991; Zbl 0739.57024), ibid. 123, No. 11, 3581--3585 (1995; Zbl 0849.57031)]. The author of this paper completely determines all the possible fixed point sets, and shows that if \(K\) is totally non-homologous to zero in \(K_{{\mathbb Z}_2}\), then the fixed point set \(K^{{\mathbb Z}_2}\) has at most four components; and if \(K\) is not totally non-homologous to zero in \(K_{{\mathbb Z}_2}\), then the fixed point set \(K^{{\mathbb Z}_2}\) is either empty or has the same homology of \(S^r\), where \(1\leq r\leq 3n\) is an odd integer.