Homotopy groups of certain highly connected manifolds via loop space homology

TL;DR

This paper investigates the homotopy groups of highly connected manifolds by relating them to loop space homology, showing they can be described in terms of spheres and cohomology structures.

Contribution

It extends previous results by linking the $p$-local homotopy groups of certain manifolds to cohomology and sphere homotopy groups, under prime restrictions.

Findings

Homotopy groups determined by indecomposable cohomology elements

Homotopy groups expressed as sums of sphere homotopy groups

Results hold away from a finite set of primes

Abstract

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M)$. Moreover, we show that these $p$-local homotopy groups can be expressed as direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work.