Given a space \(X\) with basepoint \(x_0 \in X\), we may construct a CW complex \(L(X)\) having a single 0 cell, a 1 cell \(e^1_\gamma\) for each loop \(\gamma\) in \(X\) based at \(x_0\), and a 2 cell \(e^2_\tau\) for each map \(\tau\) of a standard triangle \(PQR\) into \(X\) taking the three vertices \(P\), \(Q\), and \(R\) of the triangle to \(x_0\). The 2 cell \(e^2_\tau\) is attached to the three 1 cells that are the loops obtained by restricting \(\tau\) to the three oriented edges \(PQ\), \(PR\), and \(QR\). Show that the natural map \(L(X) \longrightarrow X\) induces an isomorphism \(\pi_1\big(L(X)\big) \approx \pi_1(X, x_0)\).
Solution
Let \[ F:L(X)\longrightarrow X \] be the natural map: the \(1\)-cell \(e^1_\gamma\) is mapped around the loop \(\gamma\), and each \(2\)-cell \(e^2_\tau\) is mapped by the corresponding triangle map \(\tau\).
1. The induced homomorphism\[ F_*:\pi_1(L(X))\to\pi_1(X,x_0) \]is surjective.
Every element of \(\pi_1(X,x_0)\) is represented by some based loop \(\gamma\). The corresponding \(1\)-cell \(e^1_\gamma\) is a loop in \(L(X)\), and by construction \[ F_*([e^1_\gamma])=[\gamma]. \] Thus every fundamental-group class of \(X\) lies in the image.
2. Let \(c\) denote the constant loop at \(x_0\).Then\[ [e^1_c]=1\in\pi_1(L(X)). \]
Take the constant map \[ \tau:\Delta^2\to\{x_0\}\subseteq X. \] All three oriented edge restrictions are the constant loop \(c\). The corresponding \(2\)-cell is therefore attached along the word \[ e_c e_c e_c^{-1}=e_c. \] Attaching this \(2\)-cell makes \([e_c]\) trivial in the fundamental group.
3. If two based loops \(\gamma_0\) and \(\gamma_1\) are homotopic relative to the basepoint, then\[ [e^1_{\gamma_0}]=[e^1_{\gamma_1}] \quad\text{in }\pi_1(L(X)). \]
Let \[ H:I\times I\to X \] be a based homotopy from \(\gamma_0\) to \(\gamma_1\). Its two vertical edges are constantly \(x_0\). Collapse these two vertical edges of the square to vertices and identify the resulting disk with a triangle. The induced triangle map has oriented edge loops \(\gamma_0\), \(\gamma_1\), and \(c\), up to reversing the orientation of one edge. Hence the associated \(2\)-cell imposes a relation of the form \[ [e_{\gamma_0}][e_c]=[e_{\gamma_1}]. \] By 2, \([e_c]=1\), so \[ [e_{\gamma_0}]=[e_{\gamma_1}]. \]
4. Define\[ G:\pi_1(X,x_0)\longrightarrow\pi_1(L(X)) \]by\[ G([\gamma])=[e^1_\gamma]. \]Then \(G\) is well defined.
If \([\gamma_0]=[\gamma_1]\) in \(\pi_1(X,x_0)\), then the loops are based-homotopic. By 3 their corresponding \(1\)-cells represent the same element of \(\pi_1(L(X))\).
5. The map \(G\) is a homomorphism.
Let \(\alpha\) and \(\beta\) be based loops. The boundary loop \[ \alpha\cdot\beta\cdot\overline{\alpha\cdot\beta} \] is nullhomotopic. Equivalently, there is a triangle map whose oriented edge restrictions are \(\alpha\), \(\beta\), and \(\alpha\cdot\beta\). The corresponding \(2\)-cell in \(L(X)\) gives the relation \[ [e_\alpha][e_\beta]=[e_{\alpha\cdot\beta}]. \] Therefore \[ G([\alpha][\beta]) =G([\alpha\cdot\beta]) =[e_{\alpha\cdot\beta}] =[e_\alpha][e_\beta] =G([\alpha])G([\beta]). \]
6. The composite\[ F_*\circ G \]is the identity on \(\pi_1(X,x_0)\).
For every based loop \(\gamma\), \[ (F_*G)([\gamma]) =F_*([e_\gamma]) =[\gamma] \] by the definition of \(F\).
7. The composite\[ G\circ F_* \]is the identity on \(\pi_1(L(X))\).
The fundamental group of the CW complex \(L(X)\) is generated by the loops given by its \(1\)-cells \(e_\gamma\). For each such generator, \[ (GF_*)([e_\gamma]) =G([\gamma]) =[e_\gamma]. \] Since a homomorphism that fixes a generating set is the identity, the composite is the identity on all of \(\pi_1(L(X))\).
8. Hence\[ \boxed{\pi_1(L(X))\cong\pi_1(X,x_0)}. \]
By <1>6–<1>7, \(F_*\) and \(G\) are mutually inverse homomorphisms.