Homotopy of manifolds stabilized by projective spaces

TL;DR

This paper investigates how the homotopy type of manifolds changes when stabilized by connected sums with projective spaces, providing a decomposition and concrete examples through homotopy theory and surgery techniques.

Contribution

It introduces a loop homotopy decomposition for stabilized manifolds and analyzes the effects of surgery using localization away from the J-homomorphism's image.

Findings

Loop homotopy decomposition after stabilization

Concrete examples illustrating the decomposition

Analysis of surgery effects via localization

Abstract

We study the homotopy of the connected sum of a manifold with a projective space, viewed as a typical way to stabilize manifolds. In particular, we show a loop homotopy decomposition of a manifold after stabilization by a projective space, and provide concrete examples. To do this, we trace the effect in homotopy theory of surgery on certain product manifolds by showing a loop homotopy decomposition after localization away from the order of the image of the classical $J$-homomorphism.