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Suppose $p:E\longrightarrow B$ is a covering projection. I have a general question, on how to find the group of all deck transformations $\Delta(p)$. Is there a common way to do this, or what could be a good approach?

Thanks in advance.

Mr.Topology
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  • Your question is very unclear. What is $p$? – Lee Mosher Jun 22 '16 at 21:02
  • $p$ is supposed to be a function, a covering space to be more specific. Should I give an example? – Mr.Topology Jun 22 '16 at 21:06
  • You should define $\Delta p$. – Pedro Jun 22 '16 at 21:07
  • $\Delta(p)$ is the group of all deck transformations, of a given covering space $p$. – Mr.Topology Jun 22 '16 at 21:08
  • So $p$ is a covering projection, and $\Delta(p)$ is the group of all deck transformations. – Pedro Jun 22 '16 at 21:09
  • Yes, it is. I edited this important detail. – Mr.Topology Jun 22 '16 at 21:11
  • You really should roll back the question to its original form and ask another question with your specific example. More over, you should show us your efforts in doing that exercise. We want to help you, not doing the work for you:) – Riccardo Jun 23 '16 at 09:15
  • I removed the example and asked it here: http://math.stackexchange.com/questions/1836762/deck-transformation-covering-space I really do not want, that you do the work for me. I would like to solve this on my own, but I am not able to. – Mr.Topology Jun 23 '16 at 09:18

1 Answers1

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Recall what is the subgroup of $\pi_1(B)$ associated to the covering $p$. To this end, recall this prop:

Thm 1.38 [Hatcher page 67] Let $B$ be path-connected, locally path-connected, and semilocally simply-connected. Then there is a bijection between the set of basepoint-preserving isomorphism classes of path-connected covering spaces $p \colon (\tilde{B} , \tilde{b_0})\to (B, b_0)$ and the set of subgroups of $\pi_1(B, b_0)$, obtained by associating the subgroup $p_* \pi_1(\tilde{B}, \tilde{b_0})$ to the covering space $(\tilde{B}, \tilde{b_0})$. If basepoints are ignored, this correspondence gives a bijection between isomorphism classes of path-connected covering spaces and conjugacy classes of subgroups of $\pi_1$.

With that in mind, let's try to solve your doubt:

The key-point is Prop 1.39 on Hatcher which briefly tells you that if $p_* \pi_1(\tilde{B}, \tilde{b_0})$ is normal in $\pi_1(B, b_0)$, then the associated covering space is normal, i.e. the for each $b\in B$, and each pair of lifts $\tilde{b},\tilde{b}'$ of $b$, there is a deck transformation taking $\tilde{b}\to \tilde{b}'$. (and therefore it's uniquely determined by this). Moreover the proposition tells you that the group $\Delta(p)$ you are interested in, is isomorphic to the quotient $N(H)/H$, where $N(H)$ is the normaliser of $H$ in $\pi_1(B,b_0)$.

tl;dr: Your first step should be to determine the groups $H$ and $\pi_1(B,b_0)$, and then apply prop 1.39.

Riccardo
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