@MikeMiller Here's the definition of connected sum I have: Let $M,M^{\prime}$ be two manifolds of dimension $n$ and take two charts $\varphi\colon B^n\rightarrow M$, $\varphi^{\prime}\colon B^n\rightarrow M^{\prime}$. We define the connected sum $M\# M^{\prime}$ as the identification space obtained by gluing together $M\setminus\varphi(0)$ and $M^{\prime}\setminus\varphi^{\prime}(0)$ via identifying $\varphi(x)$ with $\varphi^{\prime}((1-||x||)x)$.
Ted told me that the reason for this $(1-||x||)$ factor (which turns the ball inside out, essentially) is to ensure that the connected sum has a…