a topological property of a poset refers to that property of its order complex. For further background on order complexes and poset topology in general see [@Wac07].

We can use the order complex of $[\sigma,\pi]$ to calculate $\mu(\sigma,\pi)$ due to the following formula, which is an application of the Philip Hall Theorem and the Euler-Poincaré formula for the reduced Euler characteristic, see [@Wac07 Section 1.2]: $$\label{eq:Euler}
\mu(\sigma,\pi)=\tilde{\chi}(\Delta(\sigma,\pi))=\sum_{i=-1}^{|\pi|-|\sigma|}(-1)^i \tilde{\beta}_i(\Delta(\sigma,\pi)),$$ where $\tilde{\chi}$ is the reduced Euler characteristic and $\tilde{\beta}_i$ is the $i$’th reduced Betti number, that is, the rank of the $i$’th reduced homology group. Therefore, by calculating the homology of $[\sigma,\pi]$ we can compute the Möbius function. For example, if we can show that $\Delta(\sigma,\pi)$ is contractible this implies $\mu(\sigma,\pi)=0$, and if $\Delta(\sigma,\pi)$ and $\Delta(\alpha,\beta)$ are homotopically equivalent then $\mu(\sigma,\pi)=\mu(\alpha,\beta)$.

The first explicit results on the topology of intervals of permutations appear in [@McSt13] and [@Smith13].

(-2,0) node\[fill,circle,scale=0.5pt\]; (-2,0) node\[left\] [4123]{}; (0,0) node\[fill,circle,scale=0.5pt\]; (0,0) node\[left\] [2341]{}; (2,0) node\[fill,circle,scale=0.5pt\]; (2,0) node\[right\] [1234]{}; (-2,2) node\[fill,circle,scale=0.5pt\]; (-2,2) node\[left\] [45123]{}; (0,2) node\[fill,circle,scale=0.5pt\]; (0,2) node\[left\] [34512]{}; (2,2) node\[fill,circle,scale=0.5pt\]; (2,2) node\[right\] [23451]{}; (-1,4) node\[fill,circle,scale=0.5pt\]; (-1,4) node\[left\] [456123]{}; (1,4) node\[fill,circle,scale=0.5pt\]; (1,4) node\[right\] [345612]{}; (-2,0) – (-2,2) – (-1,4) – (0,2) – (0,0) – (2,2) – (1,4) – (0,2); 