# Mapping torus of Klein bottle

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.

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!

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.