Are there examples that
An un-oriented manifold is glued from pieces of oriented manifolds [with boundaries], separated by interfaces [where boundaries are glued]? I suppose a Mobius strip is one example, but do we have any concrete 4-dimensional example [glued from 3-dimensional interfaces] and 3-dimensional manifold examples [glued from 2-dimensional interfaces]?
An un-oriented closed manifold is glued from pieces of oriented manifolds [with boundaries], separated by interfaces [where boundaries are glued]? Do we have any concrete 4-dimensional manifold examples [glued from 3-dimensional interfaces] and 3-dimensional manifold examples [glued from 2-dimensional interfaces]?
An oriented closed manifold is glued from pieces of un-oriented manifolds [with boundaries], separated by interfaces [where boundaries are glued]?
Please give examples. Are interfaces oriented or unoriented in these cases? Thanks a lot.