Classification of real Bott manifolds and acyclic digraphs

Abstract

We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\mathbb{Z}/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.

arXiv comment27 pages, 5 figures. It is a combination of arXiv:0809.2178 and arXiv:1002.4704, including some new results

References in corpus (5)

• Graphical description of the action of local Clifford transformations on graph states
• Semifree circle actions, Bott towers, and quasitoric manifolds
• Directed Graph Representation of Half-Rate Additive Codes over GF(4)
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• Topology of iterated $S^1$-bundles
• Some results on the topology of real Bott towers
• Spin-structures on real Bott manifolds
• Cohomological rigidity and the number of homeomorphism types for small covers over prisms
• Weakly Equivariant Classification of Small Covers over a Product of Simplicies
• Uniqueness of the direct decomposition of toric manifolds
• On the enumeration of Fano Bott manifolds
• Spin-structures on real Bott manifolds with Kähler structures
• Cubes and Generalized Real Bott Manifolds
• Flat manifolds with holonomy group Z_2^k of diagonal type