I am reading "Non-orientable 3-manifolds of small complexity" by Amendola, Martinelli. In this work $\mathbb P^2$-irreducible complexity 6 manifolds are listed. There are 5 of them. I wonder about the following non-orientable manifolds.
- Take $S^2\times I$ and glue its top sphere to its bottom sphere with the antipodal homeomorphism or with a reflection in plane homeomorphism. Let's denote it by $S^2\widetilde\times S^1$.
- $\mathbb P^2\times S^1$
I assume that those two manifolds are not $\mathbb P^2$-irreducible. I don't know how to embed $\mathbb P^2$ into the first one. The preposition 1.3 on page 5 of the abovementioned work says that a Stiefel-Whitney surface cannot be a sphere. It seems to me that it is sphere for the first manifold.
Both of them have double cover $S^2\times S^1$, and the fundamental groups are $\mathbb Z$ and $\mathbb Z + \mathbb Z_2$. What are the fundamental groups of the 5 manifolds of complexity 6 in the above work?
At the same time I have the following additional questions about non-orientable 3-manifolds.
A. In case of surfaces we obtain a non-orientable one by the connected sum of an orientable one with $\mathbb RP^2$. Is there an analog in 3-manifolds? We could use the first manifold above $S^2\widetilde\times S^1$ since $\mathbb RP^3$ is unfortunately orientable.
B. A non-orientable surface with a removed disk is embeddable in $\mathbb R^3$. Can we embed a non-orientable $M^3$ with removed ball into $\mathbb R^4$ ?
C. Is the regular neighborhood of a loop changing orientation in a 3-manifold homeomorphic to a full Klein bottle ?
