Curvature, diameter, homotopy groups, and cohomology rings
This paper reveals some new topological constraints imposed by some Riemannian invariants such as sectional curvature, diameter and volume. The first main result of the paper (theorem A) states that for a simply connected \(n\)-dimensional manifold \(M\), the minimal number of generators of \(\pi_q(M)\), \(q\geq 2\), is bounded above by a number depending solely on the minimal number of generators of \(H_*(M, \mathbb{Z})\) and \(q\). As a consequence, using the Cheeger-Gromoll soul theorem and Gromov's Betti number theorem, it is proved that if \(M\) is complete, with nonnegative curvature, then the minimal number of generators of \(\pi_q(M)\), \(q\geq 2\), is bounded above by a constant depending only on \(n\) and \(q\), a result conjectured by X. Rong for compact manifolds of positive curvature. To state the second main result, let \(\mathcal{M}(n,Y)\) be the set of all compact orientable \(n\)-manifolds such that the oriented frame bundle admits \(SO(n)\)-invariant fibrations over \(Y\), with fiber a compact nilpotent manifold such that the induced \(SO(n)\)-actions on \(Y\) are equivalent. Then theorem C states that there exists only a finite number, depending on \(n, Y, q\), of isomorphism classes of finitely generated \(\pi_q(M)\), \(M\in \mathcal(n, Y)\). Using results from \textit{J. Cheeger} [Am. J. Math. 92, 61-74 (1970; Zbl 0194.52902)] and \textit{J. Cheeger, K. Fukaya} and \textit{M. Gromov} [J. Am. Math. Soc. 5, 327-372 (1992; Zbl 0758.53022)], this leads to the existence of only a finite number \(N(n,\lambda, \Lambda, d, q\geq 2)\) of isomorphism classes of \(\pi_q(M)\) for a compact \(n\)-dimensional \(M\) with sectional curvature in the interval \([\lambda, \Lambda]\) and diameter \(\leq d\). In particular, for \(\delta\)-pinched \(n\)-manifolds, \(N\) depends only on \(n,\delta\) and \(q\geq 2\); if \(M\) is also simply connected, then also \(H_2(M)\) has only finitely many isomorphism classes depending solely on \(n\) and \(\delta\).
scientific article; zbMATH DE number 1820797
scientific article