A Homotopy Theoretic Analogue to a Theorem of Wall

TL;DR

This paper extends homotopy theory techniques to prove a higher-dimensional analogue of Wall's theorem on decomposing simply connected 6-manifolds, advancing understanding of manifold structures.

Contribution

It introduces a homotopy theoretic approach to decompose high-dimensional manifolds, building upon recent work on loop space decompositions.

Findings

Established a homotopy theoretic analogue to Wall's decomposition theorem.

Developed new methods for analyzing the homotopy types of high-dimensional manifolds.

Applied these methods to prove the decomposition result for certain classes of manifolds.

Abstract

It is a well-known result of C.T.C. Wall's that one may decompose a simply connected 6-manifold as a connected sum of two simpler manifolds. Recent work of Beben and Theriault on decomposing based loop spaces of highly connected Poincar\'e Duality complexes has yielded new methods for analysing the homotopy theory of manifolds. In this paper we will expand upon these methods, which we will then apply to prove a higher dimensional homotopy theoretic analogue to Wall's Theorem.