Using Lemma 1.15, show that if a space \(X\) is obtained from a path-connected subspace \(A\) by attaching a cell \(e^n\) with \(n \geq 2\), then the inclusion \(A \hookrightarrow X\) induces a surjection on \(\pi_1\). Apply this to show:
The wedge sum \(S^1 \lor S^2\) has fundamental group \(\mathbb{Z}\).
For a path-connected CW complex \(X\) the inclusion map \(X^1 \hookrightarrow X\) of its 1 skeleton induces a surjection \(\pi_1(X^1) \longrightarrow \pi_1(X)\). [For the case that \(X\) has infinitely many cells, see Proposition A.1 in the Appendix.]
Solution
Let \[ X=A\cup_\varphi D^n, \qquad n\ge2, \] where the boundary sphere \(S^{n-1}=\partial D^n\) is attached to the path-connected space \(A\) by \(\varphi\).
1. There is an open cover \(X=U\cup V\) such that \(U\) deformation retracts onto \(A\), \(V\) is contractible, and \(U\cap V\) is path connected.
Choose a smaller closed ball \(B\subset\operatorname{int}D^n\) about the center. The image in \(X\) of \[ D^n\setminus B \] deformation retracts radially onto the attaching sphere, hence together with \(A\) gives a neighborhood that deformation retracts onto \(A\). After slightly enlarging this neighborhood in the quotient, call it \(U\).
Take \(V\) to be the image of a slightly larger open ball about the center, joined by a thin radial tube to a chosen point of the attaching sphere and then by a thin tube in \(A\) to the basepoint. This \(V\) is contractible. Its overlap with \(U\) deformation retracts onto a sphere with a tube attached to the basepoint. Since \[ n-1\ge1, \] the sphere \(S^{n-1}\) is path connected, so \(U\cap V\) is path connected. Both \(U\) and \(V\) may be chosen path connected and to contain the basepoint.
2. Every loop in \(X\) based at \(x_0\in A\) is homotopic to a product of loops, each lying entirely in \(U\) or entirely in \(V\).
1.Its hypotheses hold because \(U\) and \(V\) are path connected, both contain \(x_0\), and \(U\cap V\) is path connected.
3. Every factor lying in \(V\) is null-homotopic in \(X\), while every factor lying in \(U\) is homotopic in \(X\) to a loop in \(A\).
The first assertion follows from contractibility of \(V\). For the second, use the deformation retraction \[ U\simeq A \] from 1, chosen to fix the basepoint.
4. The inclusion-induced homomorphism\[ i_*:\pi_1(A,x_0)\longrightarrow\pi_1(X,x_0) \]is surjective.
Let \([\gamma]\in\pi_1(X,x_0)\). By 2, \(\gamma\) is homotopic to a product of loops in \(U\) and \(V\). By 3, discard the null-homotopic \(V\) factors and replace every \(U\) factor by a loop in \(A\). Thus \([\gamma]\) is represented by a product of loops in \(A\), hence lies in the image of \(i_*\).
5. For \(S^1\vee S^2\), the inclusion\[ S^1\hookrightarrow S^1\vee S^2 \]induces a surjection on fundamental groups.
The wedge \(S^1\vee S^2\) is obtained from \(S^1\) by attaching a \(2\)-cell by the constant attaching map. Apply 4.
6. The same inclusion also induces an injection, hence\[ \pi_1(S^1\vee S^2)\cong\mathbb Z. \]
Collapse the \(S^2\) summand to the wedge point. This gives a retraction \[ r:S^1\vee S^2\to S^1. \] If \(i:S^1\hookrightarrow S^1\vee S^2\) is the inclusion, then \[ r\circ i=\operatorname{id}_{S^1}, \] so \[ r_*\circ i_*=\operatorname{id}_{\pi_1(S^1)}. \] Thus \(i_*\) is injective. Together with 5 it is an isomorphism, and \[ \pi_1(S^1)\cong\mathbb Z. \]
7. If \(X\) is a finite path-connected CW complex, then\[ \pi_1(X^1)\longrightarrow\pi_1(X) \]is surjective.
Starting with \(X^1\), form \(X^2\) by attaching the \(2\)-cells one at a time. Each attachment induces a surjection on \(\pi_1\) by 4, so the composite \[ \pi_1(X^1)\twoheadrightarrow\pi_1(X^2) \] is surjective. Then attach the \(3\)-cells one at a time, and continue through all higher dimensions. At every step the cell dimension is at least \(2\), so 4 applies. The composite map \[ \pi_1(X^1)\longrightarrow\pi_1(X) \] is therefore surjective.
8. The same conclusion holds for an arbitrary path-connected CW complex.
Every loop has compact image. By Proposition A.1, a compact subset of a CW complex is contained in a finite subcomplex. Hence any loop in \(X\) lies in a finite subcomplex \(K\subseteq X\). Applying 7 to \(K\) shows that the loop class comes from \(\pi_1(K^1)\), and \[ K^1\subseteq X^1. \] Thus every element of \(\pi_1(X)\) lies in the image of \(\pi_1(X^1)\).