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.

中文翻译：

---

由射影空间稳定的流形的同伦

我们研究射影空间流形连通和的同伦，将其视为稳定流形的典型方法。特别是，我们展示了射影空间稳定后流形的循环同伦分解，并提供了具体的例子。为此，我们通过显示远离经典图像顺序的本地化后的循环同伦分解来追踪手术同伦理论对某些乘积流形的影响$J$-同态。