This is my "geometric freshman explanation":

the problem is to put something instead of the "?" doing the job of $s$ in


[![enter image description here][1]][1]

...but the composition $\eta\circ\epsilon$ is disconnected, and there is no way to obtain a disconnected manifold starting gluing something to that. :( such a pity.


  [1]: https://i.sstatic.net/GAo1X.png