Confusion over wording of classification of covering spaces

algebraic-topology

I'm a little confused interpreting the statement of the classification of covering spaces in Hatcher's text.

Let $X$ be path connected, locally path connected, and semi locally simply connected. Then there is a bijection between the base point preserving isomorphism classes of path-connected covering spaces $p:(\tilde{X},\tilde{x_0})\to (X,x_0)$ and the set of subgroups of $\pi_1(X,x_0)$. If basepoints are ignored, the correspondence gives a bijection between the isomorphism classes of path connected covering spaces and conjugacy classes of subgroups of $\pi_1(X,x_0)$.

The way I'm reading "the base point preserving isomorphism classes of path-connected covering spaces" is that you fix a base point $x_1$ in the covering space $p_1:(X_1,x_1)\to (X,x_0)$. We identify $Q(X_1,x_1)$ as the collection of all covering spaces $p_2:(X_2,x_2)\to (X,x_0)$ where there is a homeomorphism $f:X_1\to X_2$ where $p_1=p_2\circ f$. Then for each subgroup of $\pi_1(X,x_0)$, there is a corresponding $Q(X_1,x_1)$. But when he talks about ignoring the base point, does he mean now that if we keeps $X_1$ the same, but change $x_1$ to $x_1'$, then $Q(X_1,x_1)$ and $Q(X_1,x_1')$ correspond to conjugate subgroups of $\pi_1(X,x_0)$? Is it not possible that $(X_1,x_1')\in Q(X_1,x_1)$?

Best Answer

To each path-connected covering space $p:(\tilde{X},\tilde{x}_0)\to (X,x_0)$ assign the subgroup $G(p) = p_*(\pi_1(\tilde{X},\tilde{x}_0))$ of $\pi_1(X,x_0)$. Then the assertion is

  1. Each subgroup $G \subset \pi_1(X,x_0)$ has the form $G = G(p)$ for some path-connected covering space $p$.

  2. $p,p'$ are basepoint-preserving isomorphic covering spaces if and only if $G(p) = G(p')$.

  3. $p,p'$ are isomorphic covering spaces if and only if $G(p),G(p')$ are conjugate.

In 3. we only require that there exists a homeomorphism $f : \tilde{X} \to \tilde{X}'$ such that $f \circ p = p'$; we do not require $f(\tilde{x}_0) = \tilde{x}'_0$.

You consider a fixed covering space $p_1:(X_1,x_1)\to (X,x_0)$ and define $Q(X_1,x_1)$ as the collection of all covering spaces $p_2:(X_2,x_2)\to (X,x_0)$ where there is a basepoint-preserving homeomorphism $f:X_1\to X_2$ with $p_1=p_2\circ f$. If you drop the reqirement "basepoint-preserving", you get a larger class $Q'(X_1,x_1)$. If you replace $x_1$ by $x'_1$ in the same fiber, you trivially have $(X_1,x'_1) \in Q'(X_1,x_1)$, but in general not $(X_1,x'_1) \in Q(X_1,x_1)$. If $p_1$ is a normal covering space, then it is true for all $x'_1$. Thus it is possible. But if $G(p_1)$ is a non-normal subgroup of $\pi_1(X,x_0)$, you will find $x'_1$ such that $(X_1,x'_1) \notin Q(X_1,x_1)$.

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