Exercises

1. Use the general results of this section to give an efficient and transparent proof that $\pi_1(S^1, 1)=\mathbb{Z}$. First show that for any loop $\gamma$ based at $1$, the map $\pi_1(S^1, 1)\longrightarrow \mathbb{Z}$ given by $[\gamma]\mapsto {\tilde \gamma}(1)$ is well defined by theorem 16.1, is a group homomorphism using uniqueness of lifts. Show that surjectivity follows from uniqueness of lifts and injectivity follows from theorem 16.1.
2. Let $X$ be a topological spaces and $a, b\in X$. A simple chain connecting $a$ and $b$ is a finite sequence $U_1, U_2, \dots, U_n$ of open sets such that $a \in U_1$, $b\in U_n$ and for $1\leq i < j\leq n$, $U_i\cap U_j\neq \emptyset$ implies $j = i+1$.

   Figure: Chain connectedness

   [width=.9]GKSBook/fig14/fig14.eps
   Show that if $X$ is a connected metric space and ${\cal U}$ is an open covering of $X$ then any two points $a, b\in X$ can be connected by a simple chain. This property is referred to as chain connectedness. Is $\mathbb{Q}$ chain connected?
3. Use the above exercise to show that if $X$ is a chain-connected space and $p: {\tilde X}\longrightarrow X$ is a covering projection then for any pair of points $x, y \in X$ the fibers $p^{-1}(x)$ and $p^{-1}(y)$ have the same cardinality. The point here is that $X$ need not be path connected and the idea of using a path joining $x$ and $y$ as was done in the proof of theorem 14.4 is no longer available.
4. A toral knot is a group homomorphism $\kappa:S^1\longrightarrow S^1\times S^1$ given by $z\mapsto (z^m, z^n)$ where $m, n \in \mathbb{N}.$ Regarding the toral knot as a loop on the torus determine its lifts with respect to the covering projection $\mathbb{R}\times \mathbb{R}\longrightarrow S^1\times S^1$.
5. For the group homomorphism $\kappa$ of the previous exercise describe the induced map $\kappa_*$.

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Lecture XVII - Action of $\pi_1(X, x_0)$ on the fibers $p^{-1}(x_0)$

Given a covering projection $p: {\tilde X}\longrightarrow X$, the lifting lemma would imply that the fundamental group of the base space $X$ acts naturally on the fibers $p^{-1}(x_0)$ ($x_0 \in X$). We define this action and examine its basic properties such as its transitivity. The action provides a great deal of information about the fundamental group $\pi_1(X, x_0)$ and this is the primary application of the theory of covering spaces in this course.