Reduction of Simplicial Complex by Relation and Dowker Complex

Abstract

We show a new reduction method on a simplicial complex. This reduction works well with relations and Dowker complexes. The idea is to add a dummy vertex $ z $ to the simplicial complex $K$. We add the simplicial cone $ z * L $ to $K$ where $ L$ is the union of stars from a set of vertices. If $ L $ is contractible, then we can apply the Gluing theorem to glue $ z * L $ to $K$ to obtain $K'$. Finally, we strong collapse each vertex of $L$ in $K'$ to obtain $K''$. If the conditions are satisfied, then $K$, $K'$ and $K''$ are homotopically equivalent. This trick can be adapted to relation with the associated Dowker complex $K_R$. This notation help to simplify various computations. Relations are simple data structures, and they are represented by binary matrices. This method of reduction with relation is versatile and it can be used on different structures such as simplicial complexes, convex polytopal complexes and covers of topological spaces that satisfy the Nerve Theorem. We develop an algorithm based on the reduction step. Let $n$ be the number of vertices of $K$. We have $ O(n^2) $ subcomplexes $ L$ to verify contractibility. This verification of $ L $ is costly with $ O(d ε(n^2 + m^2)) $ where $d$ is the dimension of $L$, $m$ the number of toplexes in $L$, $n$ the number of vertices in $L$ and $ ε$ the maximal number of toplexes adjacent to a vertex in $L$. But, $L$ is often a small simplicial complex. If $L$ is contractible, then we apply a clean-up method on some columns that takes $ O(d m^2) $. Finally, we show the efficiency of the reduction algorithm on several experimental results.

Figures (2)

• Figure 1: This is an example of the reduction step on simplicial $K$ with $L = \overline{St(x_3)} \cup \overline{St(x_4)}$. Between the Figure (a) and the Figure (c), we glue $a * L$ to $K$ and we strong collapse $x_3$ and $x_4$.
• Figure 2: This is an example of the reduction step on simplicial $K$ where we can't apply any edge collapse. Every edge in Figure \ref{['figPriorCmpa']} does not satisfy the Link condition. But we can still apply our reduction with $L = \overline{St(x_1)} \cup \overline{St(x_2)}$ which is contractible. We obtain the reduced simplicial complex in Figure \ref{['figPriorCmpb']}.

Theorems & Definitions (27)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4: Dowker's Theorem
• Definition 3.5
• Theorem 3.6: barmak2011algebraic
• Theorem 3.7: barmak2011algebraic
• Lemma 4.1
• proof
• Lemma 4.2