Mapping torus of Klein bottle

Question

This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO:

The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold.

The mapping class group of the Klein bottle $ K $ is the Klein four group $ C_2 \times C_2 $. See proposition 20 of Paris - Mapping class groups of non-orientable surfaces for beginners.

There are exactly four compact flat non-orientable 3 manifolds, one of which is $ K \times S^1 $, the mapping torus of the trivial mapping class of $ K $.

Now for the four compact flat non orientable 3 manifolds. These are distinguished by their first homology $ H_1(M;\mathbb{Z}) $ (see Wolf) $$ \mathbb{Z}^2 \times C_2, \mathbb{Z}^2, \mathbb{Z}\times C_2 \times C_2, \mathbb{Z} \times C_4. $$ They correspond to the trivial mapping torus $ S^1 \times K $, the mapping torus of the Dehn twist, the mapping torus of the Y homeomorphism and the mapping torus of the Dehn twist plus Y homeomorphism, respectively.

See the Wikipedia page for Seifert fibration with positive orbifold Euler characteristic.

I'm curious how these homology groups are computed.

I understand the vote to close because I agree this isn't MO level. But it was asked on MSE two months ago with no answer so I feel like it is reasonable to cross post. I honestly won't be mad if you close this post. I just love MO and MSE and all the generous people here who have helped me learn so many interesting things!

Answer 1

Here is a sketch of an approach. What one needs to do is compute the induced action of the homeomorphism $f : K \to K$ on the generators of a presentation of the fundamental group of the Klein bottle $K$. One can describe the fundamental group of the mapping torus of $f$ as $\pi_1(K) \rtimes_{f_*} \mathbb{Z}$, and once one knows the action of $f_{*}$ on generators, a presentation for the fundamental group of the mapping torus follows.

The first homology of the mapping torus is the Abelianization of $\pi_1$ by Hurwitz's theorem, and this can be computed from a presentation of the fundamental group by allowing letters in the relations to commute.