Vector bundles over Dold manifolds (Q2717761)

A Dold manifold is the space NEWLINE\[NEWLINEP(m,n)={{S^m \times \mathbb C}^n \over {-1\times (conjugation)}}NEWLINE\]NEWLINE Dold used these manifolds to determine odd-dimensional generators of the unoriented cobordism ring. The mod 2 cohomology of \(P(m,n)\) is given by \(H^*(P(m,n);{\mathbb {Z}}_2) \cong {\mathbb {Z}}_2 /(c^{m+1}=c^{m+1}=0)\), where \(c \in H^1(P(m,n);{\mathbb {Z}}_2)\) and \(d \in H^2(P(m,n);{\mathbb {Z}}_2)\). In the paper under review the author determines the possible Stiefel-Whitney classes of the vector bundles over \(P(m,n)\). The following proposition is proved. NEWLINENEWLINENEWLINEProposition. There are vector bundles over \(P(m,n)\) with Stiefel-Whitney classes: NEWLINENEWLINENEWLINE(1) \(1+c+(d+c^2)\), for \(m=2, n \geq 1,\) NEWLINENEWLINENEWLINE(2) \((1+c+(d+c^2))^2\), for \(m=4\) or \(5, n \geq 2,\) NEWLINENEWLINENEWLINE(3) \((1+c+(d+c^2))^2(1+c+d)+c^6\), for \(m=6, n \geq 1,\) and NEWLINENEWLINENEWLINE(4) \(1+c^2d^3\), for \(n=3\). The Stiefel-Whitney class of every bundle is a product of these classes and the classes \(1+c\) and \(1+c+d\).