Homotopy groups of highly connected manifolds

TL;DR

This paper derives formulas for the homotopy groups of highly connected manifolds, revealing their structure in terms of spheres and confirming a conjecture of J. C. Moore.

Contribution

It provides explicit formulas for homotopy groups of certain manifolds and demonstrates when these groups decompose into sums of sphere homotopy groups.

Findings

Homotopy groups expressed as sums of sphere homotopy groups for manifolds with Betti number > 1

Decomposition may not hold when Betti number is 1

Confirmed J. C. Moore's conjecture for these spaces

Abstract

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 $n^{th}$ Betti number is larger than $1$. We demonstrate that when the $n^{th}$ Betti number is $1$ the homotopy groups might not have such a decomposition. The techniques used in this computation also yield formulae 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.