Let $X$ and $Y$ be two path-connected $n$-dimensional manifolds. Then one can construct their connected sum $X\# Y$ which can be visualised as removing a disc from $X$, removing a disc from $Y$, and gluing the boundaries together using a cylinder (a copy of $S^{n-1}\times[0,1]$). 


There is a natural $(n-1)$-sphere in $X\# Y$ due to the cylinder. If $X$ or $Y$ is a sphere, then this $(n-1)$-sphere will bound. Is this the only case in which it bounds?



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> If $X\# Y$ is a non-trivial connected sum (i.e. neither $X$ nor $Y$ is a sphere), is the map $S^{n-1} \to X\# Y$ essential (i.e. not nullhomotopic)? 
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Oscar Randal-Williams' answer to [this question](https://mathoverflow.net/q/282355/21564) shows that the inclusions of the sphere into $X\setminus D^{\circ}$ and $Y\setminus D^{\circ}$ are essential. 


If the answer is yes, then $\pi\_{n-1}(X\# Y) \neq 0$. I don't even know if this is true.



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> If $X \# Y$ is a non-trivial connected sum of $n$-dimensional manifolds, is $\pi\_{n-1}(X\# Y)$ necessarily non-zero?
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I believe the answer to both of these questions is yes, but I have not been able to completely convince myself.



Is $\pi_{n-1}$ of a non-trivial connected sum of $n$-manifolds always non-zero?
The answer to this is no, in a stronger sense than Ryan's comment above. There are two sources, Kahn, Peter J. Codimension-one imbedded spheres (Invent. Math. 10 1970 44–56), and an appendix Null-homotopic, codimension-one embedded spheres, to a paper of Terng-Thorbergsson (Taut Immersions into Complete Riemannian Manifolds in Tight and taut submanifolds. MSRI Publications, 32, 1997) that I wrote, not knowing Kahn's result. The main theorem of the appendix states:


Suppose that
$\imath:S^{n-1} \to M^n$ is a smooth embedding which is null-homotopic.
Then either $S$ bounds a homotopy ball on one side, or {1.} M is a rational homology sphere, and therefore X and Y are as
well. {2.} The fundamental groups of both X and Y are finite, and at
least one of them is trivial. 


In the appendix, there are constructions of connected sums of rational homology spheres where the summing sphere is null-homotopic that show that this result is essentially best possible.


