Logarithmic Tree-Numbers for Acyclic Complexes

Abstract

For a $d$-dimensional cell complex $\Gamma$ with $\tilde{H}_{i}(\Gamma)=0$ for $-1\leq i < d$, an $i$-dimensional tree is a non-empty collection $B$ of $i$-dimensional cells in $\Gamma$ such that $\tilde{H}_{i}(B\cup \Gamma^{(i-1)})=0$ and $w(B):= |\tilde{H}_{i-1}(B\cup \Gamma^{(i-1)})|$ is finite, where $\Gamma^{(i)}$ is the $i$-skeleton of $\Gamma$. The $i$-th tree-number is defined $k_{i}:=\sum_{B}w(B)^{2}$, where the sum is over all $i$-dimensional trees. In this paper, we will show that if $\Gamma$ is acyclic and $k_{i}>0$ for $-1\leq i \leq d$, then $k_{i}$ and the combinatorial Laplace operators $\Delta_{i}$  are related by  $\sum_{i=-1}^{d}\omega_{i}\,x^{i+1}=(1+x)^{2}\sum_{i=0}^{d-1}\kappa_{i} x^{i}$, where $\omega_{i}=\log \det \Delta_{i}$ and $\kappa_{i}=\log k_{i}$.  We will discuss various consequences and applications of this equation.