How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.

One way I can think of is to give a map to $Y_{n-1}=\{(y_1,y_2,\ldots , y_{n-1})\in {(\mathbb C^*)}^{n-1}\ |\ y_i\neq y_j\ for\ i\neq j\}$ which is a locally trivial fibration. After replacing $\mathbb C$ by $\mathbb R$ and for $n=2$, I drew a picture and it seems a suitable map should be $(x_1,x_2,\ldots , x_n)\mapsto (\frac{x_i-x_n}{2},\ldots , \frac{x_{n-1}-x_n}{2})$.

But then how does one check that this map is a locally fibration? $Y_n$ is aspherical by taking projection and applying long exact homotopy sequence and induction. Using a result of Fadell-Neuwirth the projections in the case of $Y_n$ are fibrations..

How does one check that the following space is aspherical? $X_n=\{(x_1,x_2,\ldots , x_n)\in {(\mathbb C^*)}^n\ |\ x_i\neq x_j\ and\ x_ix_j\neq 1\ for\ i\neq j\}$.

One way I can think of is to give a map to $Y_{n-1}=\{(y_1,y_2,\ldots , y_{n-1})\in {(\mathbb C^*)}^{n-1}\ |\ y_i\neq y_j\ for\ i\neq j\}$ which is a locally trivial fibration. After replacing $\mathbb C$ by $\mathbb R$ and for $n=2$, I drew a picture and it seems a suitable map should be $(x_1,x_2,\ldots , x_n)\mapsto (\frac{x_i-x_n}{2},\ldots , \frac{x_{n-1}-x_n}{2})$.

But then how does one check that this map is a locally trivial fibration? $Y_n$ is aspherical by taking projection and applying long exact homotopy sequence and induction. Using a result of Fadell-Neuwirth the projections in the case of $Y_n$ are locally trivial fibrations.