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**Pseudograph and its associated real toric manifold.**
*(English)*
Zbl 1376.57022

A toric manifold is a compact smooth complex toric variety. The subset of real points in a toric manifold is called real toric manifold. To a Delzant polytope \(P\) one can associate a toric manifold and therefore a real toric manifold.

In the paper under review the situation in which \(P\) is a pseudograph associahedron is studied. In this case a formula for the Betti numbers of the real toric manifold associated to \(P\) is given. It is given in terms of combinatorial data of the associated pseudograph. The proof of this result is based on a result of A. Trevisan and A. Suciu [“Real toric varieties and abelian covers of generalized Davis-Januszkiewicz spaces”, Preprint, 2012] who computed the rational cohomology of real toric manifolds in terms of the combinatorics of the associated Delzant polytope.

The main result generalizes a similar formula of S. Choi and H. Park [J. Math. Soc. Japan, 67, 699–720 (2015, Zbl 1326.57044)] who gave a formula for the Betti numbers in the case that \(P\) is a graph associahedron of a simple graph.

In the paper under review the situation in which \(P\) is a pseudograph associahedron is studied. In this case a formula for the Betti numbers of the real toric manifold associated to \(P\) is given. It is given in terms of combinatorial data of the associated pseudograph. The proof of this result is based on a result of A. Trevisan and A. Suciu [“Real toric varieties and abelian covers of generalized Davis-Januszkiewicz spaces”, Preprint, 2012] who computed the rational cohomology of real toric manifolds in terms of the combinatorics of the associated Delzant polytope.

The main result generalizes a similar formula of S. Choi and H. Park [J. Math. Soc. Japan, 67, 699–720 (2015, Zbl 1326.57044)] who gave a formula for the Betti numbers in the case that \(P\) is a graph associahedron of a simple graph.

Reviewer: Michael Wiemeler (Augsburg)

### MSC:

57N65 | Algebraic topology of manifolds |

52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

05C10 | Planar graphs; geometric and topological aspects of graph theory |

55U10 | Simplicial sets and complexes in algebraic topology |

05C30 | Enumeration in graph theory |

14P05 | Real algebraic sets |