Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the 1-skeleton of the boundary (the regular neighborhood of this loop inside the boundary is the Möbius band and the regular neighborhood of this loop considered in $M$ is the solid Klein bottle). Let me also assume that the resulting space $M/\ell$ after contracting the loop $\ell$ is a contractible space. (Equivalently, after gluing a disk along $\ell$, we get a contractible space.)
Let $M'$ be the orientation double cover of $M$ and $\ell'$ be the loop which covers $\ell$ (this is a single loop by the assumptions on $\ell$). Is it true that after contracting $\ell'$ in $M'$, we get a contractible space?
Remarks and background. This question is kind of easy to formulate special case of something more general that I would like to be true. It regards certain analysis of some contractible singular 3-manifolds. (I am interested in the special cases when the contractibility can be recongized algorithmically.)
In this special case, the only example of $M$ and $\ell$ satisfying the assumptions, I am aware of is the following: $M$ is the solid Klein bottle and $\ell$ is the loop which is homotopic to the core curve of this solid Klein bottle.
Homology of $M'/\ell'$ should be OK. The difficulty, in my opinion, is the fundamental group $\pi(M'/\ell')$. There is a well established theory how to compute the $\pi(M')$ out of the knowledge of $\pi(M)$. However, the trouble is that $\pi(M)$ is not completely known.
