Which Grassmannians bound?
Let \(G_{n,k}\) denote the Grassmann manifold of k-planes in \({\mathbb{R}}^ n\). It is shown that: Theorem 1.1: (i) \(G_{n,k}\) bounds if, for some \(r\geq 1\), \(n\equiv 0\) (mod \(2^ r)\) and \(k\not\equiv 0\) (mod \(2^ r)\). (ii) \(G_{n,k}\) does not bound if (a) n is odd or (b) if n is even and \(\left( \begin{matrix} n\\ k\end{matrix} \right)\) is odd. Proof in case (i) uses the Clifford module structure on \({\mathbb{R}}^ n\) to obtain a \(({\mathbb{Z}}/2)^ p\)-action on \(G_{n,k}\) without stationary points, and the result of \textit{P. E. Conner} [Differentiable periodic maps (Lect. Notes Math. 738) (1979; Zbl 0417.57019), Theorem 31.1], that manifolds admitting such actions bound. If n is odd, it is shown that the Stiefel-Whitney classes \(W_ j(G_{n,k})\), \(1\leq j\leq k\), generate the ring \(H^ *(G_{n,k}; {\mathbb{Z}}/2)\), from which case (ii) (a) follows. Case (ii) (b) uses a familiar argument involving the Euler characteristic.
\(({\mathbb Z}/2)^ p\)-action without stationary points
scientific article; zbMATH DE number 4020023