The homotopy type of the loops on $(n-1)$-connected $(2n+1)$-manifolds

TL;DR

This paper computes the homotopy groups of certain high-dimensional, highly connected manifolds, revealing their structure in relation to homology and expressing loop space homotopy types as sums of sphere homotopy groups.

Contribution

It provides explicit calculations of homotopy groups for $(n-1)$-connected $(2n+1)$-manifolds, linking them to homology and sphere homotopy groups, and determines the integral homotopy type of their loop spaces.

Findings

Homotopy groups determined by homology rank and torsion away from certain primes.

Homotopy groups expressed as sums of sphere homotopy groups.

Loop space homotopy type depends only on homology rank and torsion subgroup.

Abstract

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.