- Describe all connected covering spaces of \(\RP^2 \vee \RP^2\).
Solution
1. Let\[X=\mathbb{RP}^2\vee\mathbb{RP}^2.\]Then\[\pi_1(X)\cong C_2*C_2\cong D_\infty.\]
Van Kampen gives the free product of the two fundamental groups \(C_2\). Writing \[D_\infty=\langle r,s\mid s^2=1,\ srs=r^{-1}\rangle,\] one may take \(s=a\) and \(r=ab\) if \(a,b\) are the two order-\(2\) generators.
2. Connected covering spaces of \(X\) correspond to conjugacy classes of subgroups of \(D_\infty\).
The space \(X\) is connected, locally path connected, and semilocally simply connected, so the standard subgroup-covering correspondence applies. Unbased connected covers correspond to conjugacy classes of subgroups.
3. Every subgroup \(H\le D_\infty\) is one of the following:\[\{1\},\qquad \langle r^n\rangle\ (n\ge1),\qquad \langle r^k s\rangle,\qquad \langle r^n,r^k s\rangle\ (n\ge1).\]
The subgroup \(\langle r\rangle\cong\mathbb Z\) has index \(2\) in \(D_\infty\). If \(H\subseteq\langle r\rangle\), then \(H\) is trivial or \(\langle r^n\rangle\). Otherwise \(H\) contains a reflection \(r^k s\). Its intersection with \(\langle r\rangle\) is either trivial, in which case \(H=\langle r^k s\rangle\cong C_2\), or equals \(\langle r^n\rangle\) for some \(n\ge1\), in which case \(H=\langle r^n,r^k s\rangle\cong D_\infty\).
4. Up to conjugacy, the cyclic rotation subgroups \(\langle r^n\rangle\) are all distinct; the reflection subgroups have exactly two conjugacy classes, represented by \(\langle s\rangle\) and \(\langle rs\rangle\).
Each \(\langle r^n\rangle\) is normal and determined by its intersection with \(\langle r\rangle\). Conjugation by \(r^m\) sends \(r^k s\) to \(r^{k+2m}s\), while conjugation by \(s\) sends \(k\) to \(-k\), so parity of \(k\) is the only invariant for reflection subgroups.
5. For \(H_{n,k}=\langle r^n,r^k s\rangle\), the parameter \(k\) is taken modulo \(n\). If \(n\) is odd, all \(H_{n,k}\) are conjugate; if \(n\) is even, there are exactly two conjugacy classes, represented by \(k=0\) and \(k=1\).
Replacing the reflection generator by \(r^{n\ell}(r^k s)\) changes \(k\) by a multiple of \(n\). Ambient conjugation changes \(k\) by \(2m\) or by a sign. Thus the conjugacy orbits on \(\mathbb Z/n\) are generated by \(k\mapsto k+2m\) and \(k\mapsto-k\). When \(n\) is odd, \(2\) is invertible modulo \(n\), giving one orbit; when \(n\) is even, parity is preserved and yields exactly two orbits.
6. Therefore the connected covers of \(X\) are exactly the covers corresponding to the subgroup classes listed in <1>4–<1>5, together with the universal cover corresponding to the trivial subgroup.
This is the subgroup-covering correspondence from 2 combined with the complete subgroup classification in <1>3–<1>5.
7. Their sheet numbers are: infinite for the trivial and reflection subgroups, \(2n\) for \(\langle r^n\rangle\), and \(n\) for \(\langle r^n,r^k s\rangle\).
The number of sheets is the subgroup index. Since \(\langle r\rangle\) has index \(2\), one has \([D_\infty:\langle r^n\rangle]=2n\). The dihedral subgroup \(\langle r^n,r^k s\rangle\) contains one reflection coset and has index \(n\). Reflection subgroups are finite while \(D_\infty\) is infinite, hence have infinite index.