The cup-length of the oriented Grassmannians vs a new bound for zero-cobordant manifolds
Let \(X\) be a compact path connected space. The \({\mathbb Z}_2\)-cup-length, \(\text{cup} (X)\), is the maximum of all numbers \(c\) such that there exist cohomology classes \(a_1, \dots , a_n \in H^*(X;\mathbb Z_2)\) of positive degrees with their cup product \(a_1\cup \dots \cup a_n\) not equal to zero. Firstly, the author proves the following result. Theorem 1.1. Let \(M\) be a closed connected smooth \(d\)-dimensional manifold cobordant to zero in the unoriented sense. Let \({\tilde H}^r(M,{\mathbb Z}_2), r<d,\) be the first non-zero reduced cohomology group of \(M\). Let \(z<d-1\) be an integer such that each element of \( H^j(M,{\mathbb Z}_2), j\leq z,\) can be written as a polynomial in the Stiefel-Whitney classes of \(M\). Then \[ \text{{cup}} (M)\leq 1+{ {d-z-1}\over {r}}. \] Motivated by Theorem 1.1, the author introduces an invariant of a closed connected smooth manifold of dimension \(d\) called \textit{the characteristic rank} which he denotes by charrank\((M)\). It is the largest integer \(k\leq d\) such that each element of \( H^j(M,{\mathbb Z}_2), j\leq k,\) can be written as a polynomial in the Stiefel-Whitney classes of \(M\). The author uses his inequality to obtain upper bounds for the cup-length of the oriented Grassmann manifolds \(\tilde G_{n,k}\) if \(n\) is odd; and, in a different way, if \(n\) is even. He also determines lower bounds for the \({\mathbb Z}_2\)-cup-length of \(\tilde G_{n,k}\). If \(n=2^t-1, k=3, t\geq 3,\) the upper and lower bounds coincide, the \({\mathbb Z}_2\)-cup-length is \(2^t-3\) and the characteristic rank is \(2^t-5\). The author applies his results to the Lyusternik-Schnirelmann category.
scientific article; zbMATH DE number 5689251