Construct a simply-connected covering space of the space \(X \subset \mathbb{R}^3\) that is the union of a sphere and a diameter. Do the same when \(X\) is the union of a sphere and a circle intersecting it in two points.
Solution
Let \(p,q\in S^2\) be the two endpoints of the added diameter in the first space.
1. For the sphere-with-diameter space, construct \(\widetilde X\) from copies\[ S_n^2\cong S^2, \qquad n\in\mathbb Z, \]by adjoining an interval \(I_n\) from the point \(q_n\in S_n^2\) to the point \(p_{n+1}\in S_{n+1}^2\) for every \(n\).
Here \(p_n,q_n\) denote the copies of \(p,q\) in \(S_n^2\). Thus the incidence pattern is the infinite line \[ \cdots-S_{-1}^2-I_{-1}-S_0^2-I_0-S_1^2-I_1-\cdots . \]
2. Define\[ P:\widetilde X\to X \]by mapping every \(S_n^2\) homeomorphically onto the sphere and every \(I_n\) homeomorphically onto the diameter, with its endpoints sent to \(q\) and \(p\) respectively.Then \(P\) is a covering map.
Away from \(p\) and \(q\), the assertion is immediate: a small neighborhood lies either in the sphere away from the attachment points or in the interior of the diameter, and its inverse image is a disjoint union of copies.
Choose a small disk neighborhood \(D_p\) of \(p\) in the sphere and a short initial subinterval \(J_p\) of the diameter meeting the sphere only at \(p\). The union \[ U_p=D_p\cup J_p \] is a neighborhood of \(p\) in \(X\). For each \(n\), its lift at \(p_n\) is the union of the copy of \(D_p\) in \(S_n^2\) and the terminal short segment of \(I_{n-1}\) ending at \(p_n\). These lifted neighborhoods are pairwise disjoint and each maps homeomorphically to \(U_p\).
The same construction at \(q\) uses the initial segment of \(I_n\) issuing from \(q_n\). Thus \(p\) and \(q\) are evenly covered as well.
3. The covering space \(\widetilde X\) is simply connected.
Each sphere \(S_n^2\) and each interval \(I_n\) is simply connected, adjacent pieces meet in a single point, and the incidence graph is the infinite line, hence a tree. Any loop has compact image, so it meets only finitely many of these pieces and lies in a finite subtree of the graph of spaces. Repeated application of van Kampen along point intersections shows that every such finite union is simply connected. Hence \[ \pi_1(\widetilde X)=0. \]
4. Now let \(X'\) be the union of \(S^2\) with a circle meeting the sphere exactly at \(p\) and \(q\).Write the two arcs of the added circle from \(p\) to \(q\) as \(A\) and \(B\).Let\[ F_2=\langle a,b\rangle \]be the free group on two generators.
The two arcs \(A\) and \(B\) are disjoint except at their endpoints \(p,q\). The symbols \(a,b\) will record which of these two arc types is crossed in the covering.
5. Construct \(\widetilde X'\) from a copy \(S_g^2\) of the sphere for every \(g\in F_2\) as follows:
- for each \(g\), attach an interval \(A_g\) from \(p_g\in S_g^2\) to \(q_{ga}\in S_{ga}^2\);
- for each \(g\), attach an interval \(B_g\) from \(p_g\in S_g^2\) to \(q_{gb}\in S_{gb}^2\).
If each sphere copy is collapsed to a vertex, the resulting incidence graph has vertex set \(F_2\) and edges \[ g\longleftrightarrow ga, \qquad g\longleftrightarrow gb. \] This is the Cayley graph of \(F_2\) with respect to \(a,b\), hence a tree.
6. Map each \(S_g^2\) homeomorphically to the sphere, each \(A_g\) homeomorphically to \(A\), and each \(B_g\) homeomorphically to \(B\).This defines a covering map\[ P':\widetilde X'\to X'. \]
Away from \(p,q\), the local covering property is immediate.
At \(p_g\), exactly one lift of a short initial segment of \(A\) and one lift of a short initial segment of \(B\) issue from the sphere sheet \(S_g^2\), namely the segments in \(A_g\) and \(B_g\). Together with a small disk neighborhood of \(p_g\) in \(S_g^2\), they map homeomorphically to the corresponding neighborhood of \(p\) in \(X'\).
At \(q_g\), the incident lifted terminal segments are those of \[ A_{ga^{-1}}\quad\text{and}\quad B_{gb^{-1}}, \] again giving exactly one lift of each local branch at \(q\). Thus neighborhoods of both intersection points are evenly covered.
7. The space \(\widetilde X'\) is simply connected.
5.As in 3, every loop lies in a finite subtree of sphere and interval pieces, and van Kampen along the point intersections gives trivial fundamental group.
8. Therefore the constructions in 1 and 5 are simply connected covering spaces of the two requested spaces.
The covering properties are 2 and 6, and simple connectivity is 3 and 7.