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 following non-orientable manifolds.
- Take $S^2\times I$ and glue top with bottom sphere by antipodal homeomorphism or by 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 embed $\mathbb P^2$ into the first one. The preposition 1.3 on page 5 of mentioned work say that 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 fundamental group is $\mathbb Z$ and $\mathbb Z + \mathbb Z_2$. What is the fundamental group of 5 manifolds complexity 6 in above work ?
In the same time I have following additional questions about non-orientable 3-manifolds.
A. In case of surfaces we obtain non-orientable one by connected sum of orientable one with $\mathbb RP^2$. Is analog possible in 3-manifolds ? We could use the first manifold above $S^2\widetilde\times S^1$ since $\mathbb RP^3$ is unfortunately orientable.
B. Non orientable surface with removed disk is embeddable in $\mathbb R^3$. Can we embed non-orientable $M^3$ with removed ball into $\mathbb R^4$ ?
C. Is regular neighborhood of loop changing orientation in 3-manifold homeomorphic to full Klein bottle ?