Choi, Suyoung; Masuda, Mikiya; Oum, Sang-Il Classification of real Bott manifolds and acyclic digraphs. (English) Zbl 1368.37053 Trans. Am. Math. Soc. 369, No. 4, 2987-3011 (2017). Given a sequence of \({\mathbb{R}}P^1\)-bundles \[ M_n \to M_{n-1} \to \cdots \to M_1 \to M_0, \] where \(M_0\) is a point, and each bundle is obtained as the projectivization of the sum of the trivial line bundle with another real line bundle, one calls \(M_n\) a real Bott manifold. Real Bott manifolds are real toric manifolds that admit a flat invariant Riemannian metric. In particular, their universal cover is the flat Euclidean space. Two such manifolds are called affinely diffeomorphic if there exists a diffeomorphism between them whose lift to the universal cover is an affine transformation of the Euclidean space.All real Bott manifolds arise in the following way: One defines a matrix \(A\) to be a Bott matrix if it is conjugate by a permutation matrix to a strictly upper triangular binary matrix. To any \(n\times n\) Bott matrix one can associate a free action of \({\mathbb{Z}}_2^n\) on the torus \(T^n=(S^1)^n\); the orbit space of this action is a real Bott manifold, denoted \(M(A)\). For an upper triangular matrix \(A\), the columns of \(A\) encode the first Stiefel-Whitney classes of the bundles in the sequence above. The authors show that two real Bott manifolds \(M(A)\) and \(M(B)\) are affinely diffeomorphic if and only if the Bott matrix \(B\) is obtained from \(A\) by a sequence of three operations, called (Op1), (Op2) and (Op3). They visualize these three operations by identifying a Bott matrix \(A\) with an acyclic digraph with adjacency matrix \(A\). The three operations correspond to permuting labels of vertices, a known operation called a local complementation, and a new one which they call a slide. Moreover, the authors prove a refinement of the known statement that two real Bott manifolds are diffeomorphic if and only if their graded cohomology rings with \({\mathbb{Z}}_2\)-coefficients are isomorphic: they show that any such graded ring isomorphism is in fact induced by an affine diffeomorphism.There is a number of other results in the paper, such as a proof of the toral rank conjecture for real Bott manifolds, as well as a list containing the number of real Bott manifolds in low dimensions up to diffeomorphism, obtained via the interpretation of Bott matrices as graphs. Reviewer: Oliver Goertsches (Marburg) Cited in 1 ReviewCited in 18 Documents MSC: 37F20 Combinatorics and topology in relation with holomorphic dynamical systems 05C90 Applications of graph theory 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies Keywords:real toric manifold; acyclic digraph; local complementation; flat Riemannian manifold; toral rank conjecture PDFBibTeX XMLCite \textit{S. Choi} et al., Trans. Am. Math. Soc. 369, No. 4, 2987--3011 (2017; Zbl 1368.37053) Full Text: DOI arXiv References: [1] Allday, C.; Puppe, V., Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics 32, xii+470 pp. (1993), Cambridge University Press, Cambridge · Zbl 0799.55001 [2] Bang-Jensen, J{\o }rgen; Gutin, Gregory, Digraphs, Springer Monographs in Mathematics, xxii+754 pp. (2001), Springer-Verlag London, Ltd., London · Zbl 0958.05002 [3] Bouchet, Andr{\'e}, Connectivity of isotropic systems. 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