Torus actions, combinatorial topology and homological algebra
Abstract
The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of momentangle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the wellknown nonsingular toric varieties arise as orbit spaces of maximally free actions of subtori on momentangle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorialgeometrical objects in terms of the bigraded cohomology rings of the corresponding momentangle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some wellknown topological problems.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2000
 arXiv:
 arXiv:math/0010073
 Bibcode:
 2000math.....10073B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Rings and Algebras;
 52B70;
 57Q15;
 57R19;
 14M25;
 52B05;
 13F55;
 05B35
 EPrint:
 87 pages, 11 figures, LaTeX2e, to appear in Russian Math. Surveys 55 (2000), no.5