Homotopy groups of highly connected manifolds

In this paper we give a formula for the homotopy groups of (n - 1)-connected 2n-manifolds as a direct sum of homotopy groups of spheres in the case the nth Betti number is larger than 1. We demonstrate that when the nth Betti number is 1 the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulas for homotopy groups of connected sums of sphere products and CW complexes of a similar type. In all the families of spaces considered here, we establish a conjecture of J. C. Moore. (C) 2018 Elsevier Inc. All rights reserved.