Classification of Covering Spaces: Examples

For the forward direction, $m\mathbb{Z}\subseteq n\mathbb{Z}$ puts $m\in n\mathbb{Z}$, so $m=nq$ for an integer $q$ and $n\mid m$. For the reverse direction, if $m=nq$, then every $mk=n(qk)$ is in $n\mathbb{Z}$, so $m\mathbb{Z}\subseteq n\mathbb{Z}$. Since $m,n>0$, the quotient $q=m\mathrm{/}n$ is positive.

The quotient circle is path-connected by [F11] and locally path-connected because [F6] gives arbitrarily small open neighbourhoods homeomorphic to convex intervals, which are path-connected by [F7], so [F8] applies. Hence [L1] and step 1.1 give a unique based morphism $E_m\to E_n$ exactly when $n\mid m$.

Suppose $m=nq$. The local path-connectedness established in step 2.1 lifts to $E_m$ by [F9], and connectedness then makes $E_m$ path-connected by [F10]. Functoriality for the morphism $f:E_m\to E_n$ gives $(p_n{)}_{\ast }{f}_{\ast }=(p_m{)}_{\ast }$, and [F4] identifies ${\pi }_1(E_n)$ with $n\mathbb{Z}$ and $f_{\ast }{\pi }_1(E_m)$ with $m\mathbb{Z}$ inside it. Under the isomorphism $n\mathbb{Z}\to \mathbb{Z}$, $nk\mapsto k$, the subgroup $m\mathbb{Z}=nq\mathbb{Z}$ corresponds to $q\mathbb{Z}$. By [F3], the latter has the $q$ cosets represented by $0,1,\dots ,q-1$. Thus $[n\mathbb{Z}:m\mathbb{Z}]=q$, and the path-connected total space licenses [F2], which gives $q$ sheets. At $q=1$ the two subgroups are equal, so the morphism is an isomorphism.

Let $n\ge 1$. The base hypotheses for [L1] hold by [F3], and [L2] makes $E_n$ regular, so [L1] gives the quotient of the circle group by $n\mathbb{Z}$. The degree isomorphism [F2] identifies this with $(\mathbb{Z},+)\mathrm{/}n\mathbb{Z}$, and [F1] identifies that quotient with $(\mathbb{Z}\mathrm{/}n,+)$. At $n=1$ this group has one element.

For $n=0$, [L3] identifies $E_0$ with the real-line universal cover. It is regular by [L2], and [L1], [F1], and [F2] give its deck group as $(\mathbb{Z},+)\mathrm{/}0\mathbb{Z}=(\mathbb{Z},+)$.

By [L1] and [F1], the assignment $a\mapsto [1]$, $b\mapsto [0]$ extends to a homomorphism $\varphi :F(a,b)\to \mathbb{Z}\mathrm{/}3$. It is surjective because $[1]$ generates $\mathbb{Z}\mathrm{/}3$.

Again by [L1] and [F1], $a\mapsto (12)$ and $b\mapsto (123)$ extend to $\psi :F(a,b)\to S_3$. The elements $(12)$ and $(123)$ generate all six permutations, as seen from $1,(123),(132),(12),(12)(123),(12)(132)$, so $\psi$ is surjective.

Put $H_{\mathrm{r}\mathrm{e}\mathrm{g}}=\ker \varphi$. It is normal by [F2], and [F3] identifies $F(a,b)\mathrm{/}{H}_{\mathrm{r}\mathrm{e}\mathrm{g}}$ with the three-element group $\mathbb{Z}\mathrm{/}3$, so [F9] makes $H_{\mathrm{r}\mathrm{e}\mathrm{g}}$ a subgroup of index three.

Let $J={\mathrm{Stab}}_{S_3}(1)$ and $H_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{e}\mathrm{g}}={\psi }^{-1}(J)$. The set $J$ is a subgroup by [F4], and its preimage is a subgroup because $x,y\in {\psi }^{-1}(J)$ gives $\psi (x{y}^{-1})=\psi (x)\psi (y{)}^{-1}\in J$. The natural action of $S_3$ on $\{1,2,3\}$ is transitive, so [F4] gives three cosets of $J$; surjectivity of $\psi$ gives a bijection between cosets of $H_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{e}\mathrm{g}}$ and cosets of $J$, hence [F9] gives $[F(a,b):H_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{e}\mathrm{g}}]=3$. The subgroup is not normal: $ba{b}^{-1}$ maps to $(123)(12)(132)=(23)$, which fixes $1$, while $b(ba{b}^{-1})b^{-1}$ maps to $(123)(23)(132)=(13)$, which does not fix $1$. Thus conjugation by $b$ takes an element of $H_{\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{r}\mathrm{e}\mathrm{g}}$ outside it.

The base $W$ satisfies the hypotheses of [F5]. It is nonempty and path-connected because every point in either circle is joined to the wedge point. Away from the wedge point, [F11] and [F12] give arbitrarily small open simply connected arcs. At the wedge point, the quotient topology in [F10] makes every open neighbourhood contain a smaller wedge of open arcs; the interval coordinates of [F11] join every point of that smaller wedge to the wedge point along its own branch, so it is path-connected. The particular open overlap supplied by [F10] is simply connected, and therefore has trivial inclusion-induced fundamental group. Thus [F13] gives local path-connectedness and semilocal simple connectedness. Now [F5] realizes the two index-three subgroups from steps 2.1 and 2.2 by connected coverings of $W$. Since $W$ is locally path-connected, [F14] and [F15] make their total spaces path-connected. This licenses [F7], which makes both covers three-sheeted, and [F6], which makes the normal kernel cover regular and the nonnormal stabilizer-preimage cover nonregular.