Let \(A\) be the union of the unit sphere in \(\RR^3\) and the interval \(\theset {(t, 0, 0) : -1 \leq t \leq 1} \subset \RR^3\).
Compute \(\pi_1 (A)\) and give an explicit description of the universal cover of \(A\).
Solution
Let \[ p=(-1,0,0), \qquad q=(1,0,0), \] and let \(I\) be the diameter joining \(p\) to \(q\). Thus \[ A=S^2\cup I, \qquad S^2\cap I=\{p,q\}. \]
1. The space \(A\) is homotopy equivalent to \(S^2\vee S^1\).
Give \(A\) a CW structure in which the diameter \(I\) is a contractible subcomplex. Collapsing a contractible CW subcomplex to a point is a homotopy equivalence, so \[ A\simeq A/I. \] The quotient \(A/I\) is the sphere with the two points \(p\) and \(q\) identified. This space is homotopy equivalent to \[ S^2\vee S^1. \] Equivalently, choose an arc in the sphere from \(p\) to \(q\); together with the identified point it supplies the circle factor, while the remaining sphere supplies the \(S^2\) factor.
2. Therefore\[ \boxed{\pi_1(A)\cong\ZZ.} \]
By 1 and homotopy invariance of the fundamental group, \[ \pi_1(A) \cong \pi_1(S^2\vee S^1). \] Van Kampen’s theorem gives \[ \pi_1(S^2\vee S^1) \cong \pi_1(S^2)*\pi_1(S^1) \cong 0*\ZZ \cong \ZZ. \] A generator may be represented by traversing the diameter from \(p\) to \(q\) and then returning from \(q\) to \(p\) along any path on the sphere.
3. Construct a space \(\widetilde A\) from copies\[ S_m^2\cong S^2, \qquad m\in\ZZ, \]and intervals\[ I_m\cong[0,1], \qquad m\in\ZZ, \]as follows.Choose points \(p_m,q_m\in S_m^2\) corresponding to \(p,q\) and attach\[ I_m(0)=p_m, \qquad I_m(1)=q_{m+1}. \]Thus \(\widetilde A\) is a bi-infinite chain of spheres joined by intervals.
Schematically the space is \[ \cdots S_{-1}^2 \mathbin{-}I_{-1}\mathbin{-} S_0^2 \mathbin{-}I_0\mathbin{-} S_1^2 \mathbin{-}I_1\mathbin{-} S_2^2 \cdots, \] where the right end of \(I_m\) is attached to \(q_{m+1}\) and its left end to \(p_m\). This is a connected CW complex.
4. There is a covering map\[ \pi:\widetilde A\longrightarrow A \]that maps every \(S_m^2\) homeomorphically onto \(S^2\) and every \(I_m\) homeomorphically onto the diameter \(I\), taking its initial endpoint to \(p\) and terminal endpoint to \(q\).
On the interiors of the spheres and intervals this is plainly a local homeomorphism with one copy for each \(m\in\ZZ\).
Near \(p\), choose a small neighborhood in \(A\) consisting of a disk neighborhood of \(p\) in \(S^2\) together with a short initial segment of \(I\). Its inverse image is the disjoint union of the corresponding neighborhoods of the points \(p_m\), each using the initial segment of \(I_m\). Each maps homeomorphically onto the chosen neighborhood of \(p\).
Near \(q\), the analogous lifted neighborhood at \(q_m\) uses the terminal segment of \(I_{m-1}\). Again the inverse image is a disjoint union of copies mapped homeomorphically onto the base neighborhood. Thus every point of \(A\) has an evenly covered neighborhood, so \(\pi\) is a covering map.
5. The space \(\widetilde A\) is simply connected.
The incidence graph whose vertices are the spheres \(S_m^2\) and whose edges are the intervals \(I_m\) is the bi-infinite line, hence a tree. Each sphere and each interval is simply connected, and the attachments occur at single points. Van Kampen’s theorem therefore shows that every finite subchain is simply connected.
The CW complex \(\widetilde A\) is locally finite, so the compact image of any loop meets only finitely many cells and is contained in a finite subchain. That finite subchain is simply connected, hence the loop is nullhomotopic. Therefore \[ \pi_1(\widetilde A)=0. \]
6. Hence the covering in 4 is the universal cover of \(A\).
By 3 the total space is connected, and by 5 it is simply connected. Thus \[ \boxed{\widetilde A\to A} \] is the universal covering space. The deck transformation corresponding to the generator of \(\pi_1(A)\cong\ZZ\) shifts every index by one: \[ S_m^2\longmapsto S_{m+1}^2, \qquad I_m\longmapsto I_{m+1}. \]