Find all the connected covering spaces of \(\mathbb{RP}^2 \vee \mathbb{RP}^2\).
Solution
Write \[ X=\mathbb{RP}^2\vee\mathbb{RP}^2 \] and let \(a,b\) be the two order-two generators coming from the two summands. Then \[ \pi_1(X)=\langle a,b\mid a^2=b^2=1\rangle \cong C_2*C_2=:D_\infty. \] Put \[ r=ab. \]
1. Every subgroup of \(D_\infty\) is one of the following types:\[ 1, \qquad \langle r^n\rangle\ (n\ge1), \qquad \langle r^k a\rangle, \qquad \langle r^n,r^k a\rangle\ (n\ge1). \]
The infinite cyclic subgroup \[ \langle r\rangle\cong\mathbb Z \] has index two in \(D_\infty\), and every element outside it is a reflection \(r^k a\) of order two.
Let \(H\le D_\infty\). If \(H\subseteq\langle r\rangle\), then \(H=1\) or \(H=\langle r^n\rangle\) for a unique \(n\ge1\). Otherwise choose a reflection \(r^k a\in H\). The intersection \[ H\cap\langle r\rangle \] is either trivial or \(\langle r^n\rangle\). In the first case \(H=\langle r^k a\rangle\); in the second every element of \(H\) is in either \(\langle r^n\rangle\) or its coset by \(r^k a\), so \[ H=\langle r^n,r^k a\rangle. \]
2. Up to conjugacy, the reflection subgroups give exactly two infinite-sheeted covers:\[ \langle a\rangle \qquad\text{and}\qquad \langle b\rangle. \]
Conjugation by \(r^m\) changes \[ r^k a\longmapsto r^{k+2m}a, \] while conjugation by a reflection changes \(k\) to its negative modulo an even shift. Thus parity of \(k\) is the only conjugacy invariant. Even \(k\) gives the class of \(a\), and odd \(k\) the class of \(b\).
3. For the finite-index dihedral subgroups\[ H_{n,k}=\langle r^n,r^k a\rangle, \]the index is \(n\).For odd \(n\) there is one conjugacy class; for even \(n\) there are two, represented by\[ H_{n,0} \qquad\text{and}\qquad H_{n,1}. \]
The cosets are represented by \[ 1,r,\dots,r^{n-1}, \] so the index is \(n\). Conjugation changes \(k\) by an even integer and by sign, modulo \(n\). If \(n\) is odd, adding even integers modulo \(n\) reaches every residue class, so all \(k\) are conjugate. If \(n\) is even, parity is preserved and gives exactly two classes.
4. The cyclic subgroup \(\langle r^n\rangle\) has index \(2n\) and gives one normal \(2n\)-sheeted cover for every \(n\ge1\).
The quotient \[ D_\infty/\langle r^n\rangle \] has representatives \[ 1,r,\dots,r^{n-1},a,ra,\dots,r^{n-1}a, \] so it has \(2n\) elements. The subgroup is normal because conjugation sends \(r^n\) to \(r^{\pm n}\).
5. These subgroup types have a direct geometric description.For each lift of the wedge point, an \(a\)-orbit of two vertices is joined by a lifted copy of \(S^2\to\mathbb{RP}^2\) from the first summand, while an \(a\)-fixed vertex carries a copy of \(\mathbb{RP}^2\) itself; similarly for \(b\) and the second summand.
Use the standard CW structure \[ \mathbb{RP}^2=e^0\cup e^1\cup e^2, \] where the \(2\)-cell is attached by the degree-two word \(a^2\) (or \(b^2\)). In a covering, if the monodromy involution \(a\) exchanges two sheets, the two lifted \(a\)-edges and their lifted \(2\)-cells form the standard double cover \[ S^2\to\mathbb{RP}^2 \] between those two lifted wedge points. If \(a\) fixes a sheet, the corresponding summand lifts homeomorphically as \(\mathbb{RP}^2\) attached at that vertex. The same statement holds for \(b\).
6. The trivial subgroup gives the universal cover: a bi-infinite line of lifted wedge points with alternating \(a\)- and \(b\)-copies of \(S^2\) between consecutive points.
The Schreier graph of the free action of \(D_\infty\) on itself has vertices indexed by the group elements and alternating \(a\)- and \(b\)-adjacencies. It is a bi-infinite line. No generator fixes a vertex, so 5 replaces every adjacency by an \(S^2\) piece. The incidence graph is a tree, hence the resulting tree of simply connected spheres is simply connected.
7. The two reflection subgroups \(\langle a\rangle\) and \(\langle b\rangle\) give two infinite-sheeted ray covers.
For \(\langle a\rangle\), the Schreier graph is a ray whose endpoint is fixed by \(a\); therefore the endpoint carries an \(a\)-type \(\mathbb{RP}^2\), followed by an infinite alternating chain of \(S^2\) pieces. For \(\langle b\rangle\), the same construction has a \(b\)-type \(\mathbb{RP}^2\) at the endpoint. The two covers are not isomorphic over \(X\) because the two endpoint projective planes map to different wedge summands.
8. The subgroup \(\langle r^n\rangle\) gives a cyclic necklace of \(2n\) lifted wedge points with alternating \(a\)- and \(b\)-copies of \(S^2\) between consecutive points.
Its Schreier graph is a \(2n\)-cycle with alternating \(a,b\) adjacencies and no fixed vertices. Apply 5.
9. For \(n\ge2\), the subgroup \(H_{n,k}\) gives a finite path of \(n\) lifted wedge points, with \(S^2\) pieces along the path and projective-plane pieces at the two ends.If \(n\) is odd, one endpoint is of \(a\)-type and the other of \(b\)-type.If \(n\) is even, both endpoints have the same type, giving the two nonisomorphic covers according as both are \(a\)-type or both are \(b\)-type.
The Schreier graph on the \(n\) cosets of \(H_{n,k}\) is a path. Interior vertices are exchanged by both involutions, while each endpoint is fixed by the involution whose labelled edge is missing there. Along a path the labels alternate. If \(n\) is even, the path has odd edge-length \(n-1\), so its first and last edge labels are the same; consequently the same opposite involution fixes both endpoints. If \(n\) is odd, the path has even edge-length, so the first and last edge labels are different and the two fixed endpoint involutions are different. Equivalently, a direct coset calculation gives one \(a\)- and one \(b\)-fixed endpoint for odd \(n\), and two endpoints of the same type for even \(n\). The two same-type possibilities for even \(n\) are precisely the two conjugacy classes in 3.
10. For \(n=1\), \(H_{1,0}=D_\infty\) and the corresponding one-sheeted cover is \(X\) itself.
There is one lifted wedge point fixed by both \(a\) and \(b\), so both projective-plane summands occur at that point.
11. The list in <1>6–<1>10 contains every connected covering space of \(\mathbb{RP}^2\vee\mathbb{RP}^2\), up to unbased covering isomorphism.
Connected unbased covering spaces correspond to conjugacy classes of subgroups of \(\pi_1(X)\). The subgroup classification is <1>1–<1>4, and <1>6–<1>10 translates each conjugacy class into its covering space.