Logarithmic Tree-Numbers for Acyclic Complexes

For a d-dimensional cell complex Γ with H̃i(Γ) = 0 for −1 6 i < d, an idimensional tree is a non-empty collection B of i-dimensional cells in Γ such that H̃i(B ∪ Γ(i−1)) = 0 and w(B) := |H̃i−1(B ∪ Γ(i−1))| is finite, where Γ(i) is the iskeleton of Γ. The i-th tree-number is defined ki := ∑ B w(B) 2, where the sum is over all i-dimensional trees. In this paper, we will show that if Γ is acyclic and ki > 0 for −1 6 i 6 d, then ki and the combinatorial Laplace operators ∆i are related by ∑d i=−1 ωi x i+1 = (1 + x)2 ∑d−1 i=0 κix i, where ωi = log det ∆i and κi = log ki. We will discuss various consequences and applications of this equation.