3

The 3-torus bundle $E$ is a fiber bundle with 3-torus $T^3=S^1\times S^1\times S^1$ as the fiber, and $S^1$ as the base space. Then what are the cohomology rings of $E$: $H^*(E;\mathbb{Z}_n)=?$ and $H^*(E;\mathbb{Z})=?$ (in terms of 3-by-3 integer matrices in the mapping class group $GL(3,\mathbb{Z})$ of $T^3$).

The following paper addressed a similar problem, but for 2-torus bundle: https://arxiv.org/abs/1307.0518

There is a related question Cohomology of fibrations over the circle: how to compute the ring structure?, but $H^*(E;\mathbb{Z}_n)$ and $H^*(E;\mathbb{Z})$ were not computed.

Community
  • 1
Xiao-Gang Wen
  • 4,716
  • 21
  • 43
  • Just for clarification: what would you want the answer to be? The arXiv paper you linked suggests that the full answer will be a long list of case distinctions, based on invariants of the monodromy action on $H^\bullet(T^3;\mathbb{Z})$. If you have a specific case in mind, that could probably be done by computer algebra (I think HAP has an implementation of Wall's resolution for group extensions and it also supports computation of the ring structure). – Matthias Wendt Dec 20 '16 at 10:32
  • The group elements of $SL(3,\mathbb{Z})$ is a 3-by-3 integer matrix. For each integer matrix there will be a ring $H^*(E,\mathbb{Z}_n)$. I like to obtain the ring in terms of the 3-by-3 integer matrix. If there is a computer algorithm, it will be good enough for me. I wonder if there is a reference for such a algorithm. – Xiao-Gang Wen Dec 20 '16 at 19:22

0 Answers0