Constructing a Solid Klein Bottle

[lavinia]

can one construct a solid Klein bottle - a 3 manifold whose boundary is a Klein bottle as follows.

- Start with a solid cylinder and identify the two bounding disks by a reflection.

- The boundary becomes a Klein bottle but is this a smooth manifold whose boundary is this Klein bottle?

- If so does this manifold deform onto its central circle just as a solid torus would?

- Since reflection is an isometry of the disk, can one give this manifold a flat metric?
In general if the boundaries of two Riemannian manifolds are identified by an isometry do their metrics extend?

 

[homeomorphic]

It's a disk bundle over a circle, so it does deform onto the central circle. I think the rest is true, but not completely sure.

 

[homeomorphic]

The boundary becomes a Klein bottle but is this a smooth manifold whose boundary is this Klein bottle?

I forgot to mention, 3-manifolds always have a unique smooth structure, so yes.

 

[lavinia]

Thanks homeomorphic. I have a worry that the gluing of the two solid Klein bottles can not have a flat metric. Your answer justifies the worry because it makes the computation of the homology of this manifold easy to do. The homology to me seems impossible for a manifold that is covered by a torus.

Split the 3 manifold into two solid Klein bottles with a small collar around them. Their intersection is a collar neighborhood of the bounding Klein bottle where they are glued together.

Since the solid Klein bottles deform onto a circle they have the homology of a circle so with Z2 coefficients the Meyer Vietoris sequence is

0 -> Z2 -> Z2 -> 0 -> H2(Solid K u Solid K )-> Z2 + Z2 -> Z2 + Z2 -> H2(Solid K u Solid K ) -> 0

The last H2 is by Poincare Duality.

So the Z2 homology of Solid K u Solid K is either zero of Z2 in dimensions 1 and 2.