# Pachner Moves

## Decomposing homeomoprphisms between simplicial complexes.

Supplementary Interactive Figure used in the talk:

Suppose we have a simplex in an $n$-dimensional simplicial complex denoted by $v_0v_1...v_n$ where each $v_i$ is a vertex of this simplex. We may change the combinatorial structure of the simplicial complex by creating a new vertex $u$ in the interior of the simplex (I referred to it as the barycenter in the talk but it does not matter which interior point is chosen) then form $n+1$ new simplices by taking groups of $n$ vertices from the original simplex and joining them with $u$; for example, $v_0v_1...v_{n-1}u$, $v_1v_2...v_nu$, etc.
Suppose we have two adjacent simplices in the simplicial complex given by $v_0v_1...v_n$ and $v_1v_2...v_{n+1}$. These simplices share the vertices $v_1,v_2,...,v_n$, but the vertices $v_0$ and $v_{n+1}$ belong to only one simplex or the other. We may replace this configuration of two simplices with a configuration of $n$ simplices whose vertices are $v_0v_{n+1}$ joined with sets of $n-1$ vertices from the intersection of the original two simplices; for example, $v_0,v_{n+1}v_1v_2...v_{n-1}$, $v_0v_{n+1}v_2v_3...v_n$, etc.