Betti numbers and Euler's formula for combinatorial foliations (Q1357577)

The author defines the notion of a combinatorial foliation \({\mathcal C}=(\Omega,{\mathcal F},m)\) of dimension \(m\), the harmonic measure on \(\mathcal C\), and he proves the simplicial version of the Gauss-Bonnet theorem [\textit{A. Connes}, Proc. Symp. Pure Math. 38, Part 1, 521-628 (1982; Zbl 0531.57023)]. If \(\mathcal C\) is equipped with a probability measure \(\lambda\), one can define the foliation Betti numbers \(b^p_\lambda\) of \(\mathcal C\). Let \(c_p\) be the numbers of \(p\)-simplices in the leaf of \(x\), which contains the vertex \(x\). The simplicial curvature at \(x\) is defined by \(K_{\text{simp}}(x)=\sum^n_{p=0} (-1)^p(1/(p+1))c_p\), where \(n\) denotes the dimension of a finite simplicial complex. The author proves the following theorem: Let \({\mathcal C}=(\Omega,{\mathcal F},m)\) be an \(m\)-dimensional combinatorial foliation and let \(\mu\) be an invariant measure on \(\Omega\). Then \(\int_\Omega K_{\text{simp}}d\mu= \sum^m_{p=0} (-1)^pb^p_\mu\). The second main result of this paper shows that if \({\mathcal C}=(\Omega,{\mathcal F},1)\) is a one-dimensional foliation, this equality holds even if \(\mu\) is only a harmonic measure.