The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations

TL;DR

This paper determines the loop space homotopy types of certain simply-connected four-manifolds and related complexes, providing new decompositions based on their topological and algebraic properties.

Contribution

It introduces a general method for decomposing loop spaces of specific torsion-free CW-complexes with Poincaré duality, extending previous results to broader classes of manifolds.

Findings

Loop space decompositions for simply-connected four-manifolds

Decompositions for (n-1)-connected 2n-dimensional manifolds with n≠4,8

Results for connected sums of products of two spheres

Abstract

We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with well-behaved skeleta and some Poincar\'{e} duality features.