Classification of real Bott manifolds and acyclic digraphs

Details Classification of real Bott manifolds and acyclic digraphs Classification of real Bott manifolds and acyclic digraphs

2010-06-23

arxiv.org/abs/1006.4658v1

Mathematics

Algebraic Topology

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

Scientific paper

We completely characterize real Bott manifolds up to 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 $\Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. This characterization can also be described combinatorially 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.

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