Generators of the unoriented bordism ring which are fibered over products of projective spaces \(RP(2^ r-1)\)
The author constructs simple generators of the unoriented cobordism ring \(M O_*\). Fix an integer \(0<k\). Suppose \(n\) is a positive integer such that \(n+1\) is not a power of 2. Write \(n=a 2^k+m\) where \(0\leq a\), \(0\leq m<2^k\). The author defines an integer \(g_k(n)\) \((g_k (n) \leq n-a)\). (When \(k=1\), \(a= [n/2]\), \(g_1(n)\) is either \(a\) or \(a-1)\). The author proves there is an \(n\)-dimensional manifold \(X^n\) representing an indecomposable class in \(MO_n\) and \(X^n\) is fibered over a \(g_k(n)\)-fold product of projective spaces, \(\Pi_i {\mathbf R} P(2^{r_i}-1)\), where \(1\leq r_i\leq k\). When \(k=1\), all \(r_i=1\), and all \({\mathbf R} P(2^{r_i} -1)\) are circles. In this case, the author's result gives representing manifolds fibered over \(g_1(n)\)-dimensional tori. This enhances a quarter-century old result of Stong.
scientific article; zbMATH DE number 980693
scientific article