The Covering Transformation Theorem for Finding Fundamental Groups

# The Covering Transformation Theorem for Finding Fundamental Groups

Recall from the Covering Transformations page that if $X$ is a topological space and $(\tilde{X}, p)$ is a cover of $X$ then a homeomorphism $f : \tilde{X} \to \tilde{X}$ is said to be a covering transformation of $\tilde{X}$ if the following diagram commutes:

And we denote the set of covering transformations of $\tilde{X}$ by $A(\tilde{X})$ or $A(\tilde{X}, p)$.

We now state a very important theorem regarding covering transformations which will give us a method to compute certain fundamental groups.

Theorem 1 (The Covering Transformation Theorem for Finding Fundamental Groups): Let $X$ be a topological space and let $(\tilde{X}, p)$ be a covering space of $X$. Let $x \in X$ and let $\tilde{x} \in p^{-1}(x)$.a) If $p_*(\pi_1(\tilde{X}, \tilde{x}))$ is a normal subgroup of $\pi_1(X, x)$ then $\displaystyle{A(\tilde{X}) \cong \frac{\pi_1(X, x)}{p_*(\pi_1(\tilde{X}, \tilde{x}))}}$.b) If $(\tilde{X}, p)$ is the universal covering of $X$ then $A(\tilde{X}) \cong \pi_1(X, x)$. |