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Moment-angle complexes and combinatorics of simplicial manifolds. (English. Russian original) Zbl 0974.52013

Russ. Math. Surv. 55, No. 3, 567-569 (2000); translation from Usp. Mat. Nauk 55, No. 3, 171-172 (2000).
This paper announces results, some of which have appeared elsewhere, and some of which will appear.
By utilizing a cellular subdivision of the product of \(m\) 2-discs invariant under the action of an \(m\)-torus, the authors can analyze cubical decompositions determined by a simplicial structure. Their analyses for example lead to a description of the multiplicative structure of the cohomology of the complement of the coordinate subspaces of complex \(m\)-space.
They are also able to recover the classical Dehn-Sommerville relations.

MSC:

52B70 Polyhedral manifolds
57Q15 Triangulating manifolds
57R19 Algebraic topology on manifolds and differential topology