Manifolds associated with \((\mathbb {Z}_2)^{n+1}\)-colored regular graphs (Q2882749)

The authors study a general construction of combinatorial manifolds from graphs, which goes as follows. Let \(\Gamma\) be a connected \((n+1)\)-regular finite graph whose edges are colored with elements in the group \((\mathbb Z_2)^{n+1}\) in a way such that a certain compatibility condition is met. Then, inductively, for \(k=2,3,\dots,n\), each suitably colored \(k\)-regular subgraph of \(\Gamma\) gives rise to a cell to be attached. This yields a combinatorial \(n\)-manifold without boundary which comes with a regular cell-decomposition. From looking at the barycentric subdivision of any finite regular cell decomposition of some manifold one can see that each combinatorial manifold arises this way. The construction is motivated by work of \textit{D. Bliss, V. W. Guillemin} and \textit{T. S. Holm} [Adv. Math. 185, No. 2, 370--399 (2004; Zbl 1069.53058)] on a mod 2 version of \textit{M. Goresky, R. Kottwitz} and \textit{R. MacPherson} [Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)] as well as work by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)] on small covers.