\(Z_2^k\)-actions fixing \(K_dP^{2^s}\cup K_dP^{even}\) (Q1004048)

Let \(M\) be a smooth and closed manifold, equipped with a smooth involution \(T:M \rightarrow M\). In this case, the fixed point set, \(F\), is a finite and disjoint union of closed submanifolds of \(M\). In this setting, for a given \(F\), a natural question is the classification, up to equivariant cobordism, of the pairs \((M, T)\) for which the fixed point set is \(F\). For \(F=\mathbb{R} P^n\) = the real and \(n\)-dimensional projective space, the classification was established by \textit{P. E. Conner} and \textit{E. E. Floyd} in [Differentiable periodic maps. Berlin-Göttingen-Heidelberg: Springer-Verlag (1964; Zbl 0125.40103)] and by \textit{R. E. Stong} in [Mich. Math. J. 13, 445--447 (1966; Zbl 0148.17402)]. \textit{D. C. Royster} then obtained this classification for \(F=\mathbb{R} P^m \cup \mathbb{R} P^n\) in [Indiana Univ. Math. J. 29, 267--276 (1980; Zbl 0406.57027)], with the exception of the case in which \(m > 0\) and \(n >0\) are even. In the recent paper [Algebr. Geom. Topol. 7, 29--45 (2007; Zbl 1133.57021)], the present authors together with \textit{R. de Oliveira} solved the case \(F=\mathbb{R} P^2 \cup \mathbb{R} P^n\), where \(n>2\) is even. They also defined and computed the number \(h_{m,n}\), which constitutes an upper bound for the dimension of manifolds with involution fixing \(F=\mathbb{R} P^m \cup \mathbb{R} P^n\). Finally, by putting together the obtained results with the material of [\textit{P. L. Q. Pergher} and \textit{R. de Oliveira}, Fundam. Math. 201, No. 3, 241--259 (2008; Zbl 1160.57030)], this classification was automatically extended for \((\mathbb{Z}_2)^k\)-actions. In the present paper, the authors extend their previous results in the following way: they denote by \(K_dP^n\) the real (\(d=1\)), complex (\(d=2\)) and quaternionic (\(d=4\)) \(n\)-dimensional projective space, with real dimension \(dn\), and establish the classification in question for \(F = K_dP^{2^s} \cup K_dP^n\), \(d=1, 2\) or \(4\), \(n \geq 2^{s+1}\) is even and \(s \geq 1\). In other words, they get an extension for the previous \(d=1\) and \( s=1\) case. They also obtain the classification for \(d=2\) and \(4\) in the cases \(F=\{\text{point}\} \cup K_dP^n\), where \(n \geq 2\) is even, and \(F=K_dP^m \cup K_dP^n\), where \(m\) is odd and \(n \geq 0\) is even: this means extensions for the corresponding \(d=1\) cases, due to D. C. Royster. The complex and quaternionic version of the number \(h_{m,n}\), denoted by \(h_{m,n}^d\), is also defined and computed. In the same way, the automatic extension for \((\mathbb{Z}_2)^k\)-actions is also established. The paper was dedicated to Professor Robert E. Stong (in memoriam).