As an algebraic application of the preceding problem, show that in a free group \(F\) with basis \(x_1, \ldots, x_{2k}\), the product of commutators \([x_1, x_2] \cdots [x_{2k-1}, x_{2k}]\) is not equal to a product of fewer than \(k\) commutators \([\nu_i, w_i]\) of elements \(\nu_i, w_i \in F\).
Solution
Let \[ r_k=[x_1,x_2]\cdots[x_{2k-1},x_{2k}]\in F(x_1,\ldots,x_{2k}). \] Suppose, toward a contradiction, that \[ r_k=[v_1,w_1]\cdots[v_j,w_j] \] with \(j<k\).
Use the standard CW structure on the genus-\(j\) surface \(M_j\): its \(1\)-skeleton is a wedge of \(2j\) circles with generators \(a_i,b_i\), and its \(2\)-cell is attached by \[ [a_1,b_1]\cdots[a_j,b_j]. \] Likewise \(M_k\) has \(1\)-skeleton a wedge of \(2k\) circles with generators \(x_1,\ldots,x_{2k}\) and \(2\)-cell attaching word \(r_k\).
Define a map of \(1\)-skeleta \[ g:(M_j)^1\to(M_k)^1 \] by sending the loop \(a_i\) to a loop representing \(v_i\) and \(b_i\) to a loop representing \(w_i\). The boundary of the \(2\)-cell of \(M_j\) then maps to \[ [v_1,w_1]\cdots[v_j,w_j]=r_k, \] which is exactly the attaching loop of the \(2\)-cell of \(M_k\). Hence \(g\) extends over the \(2\)-cell to a map \[ f:M_j\to M_k \] that maps the source \(2\)-cell once over the target \(2\)-cell. Therefore \(f\) has degree \(1\).
By Exercise 11, a degree-\(1\) map \(M_j\to M_k\) can exist only if \(j\ge k\). This contradicts \(j<k\). Thus \(r_k\) cannot be expressed as a product of fewer than \(k\) commutators.