Group of covering transformations

of a regular covering $p:X \rightarrow Y$

The group $\Gamma(p)$ of those homeomorphisms $\gamma$ of the space $X$ onto itself such that $\gamma p = p$. ($X$ and $Y$ are understood to be connected, locally path-connected, Hausdorff spaces.)

The group of covering transformations of the covering of the circle by the real line $\mathbf{R}$ given by $t \mapsto (\cos t,\sin t)$ is thus the group of translations $t \mapsto t + 2\pi n$, $n \in \mathbf{Z}$.

$\Gamma(p)$ is a discrete group of transformations of $X$ acting freely (that is, $\gamma(x) = x$ for some $x \in X$ implies $\gamma=1$), and $Y$ is naturally isomorphic to the quotient space $X/\Gamma(p)$. The group $\Gamma(p)$ is isomorphic to the quotient group of the fundamental group $\pi_1(Y,y_0)$, where $y_0 \in Y$, by the image of the group $\pi_1(X,x_0)$, where $p(x_0)=y_0$, under the homomorphism induced by the mapping $p$. In particular, if $p$ is the universal covering, then $\Gamma(p)$ is isomorphic to the fundamental group of $Y$.

References

[1]	S.-T. Hu, "Homotopy theory" , Acad. Press (1959)

Comments

See also Covering; Universal covering.

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. 2