Moscow--Petersburg seminar on low-dimensional mathematics
SPb, PDMI, 12.04.2002, 16:00--17:45, room 311

Taras E. Panov
(Moscow State University)

Combinatorial aspects of torus actions
(based on joint works with Victor M. Buchstaber)

We describe some new relationships between torus actions on manifolds or
more general spaces and combinatorial objects such as polytopes, simplicial
complexes, cubical complexes, and subspace arrangements. The case of our
particular concern is simplicial and cubical subdivisions of manifolds and,
especially, spheres. The constructions allowing to study such combinatorial
objects by means of commutative and homological algebra are described. This
approach unifies commutative algebra methods brought into the combinatorics
by Stanley and topological treatment of torus actions by Davis and
Januszkiewicz and gives rise to the theory of moment-angle complexes,
currently being developed by V.M.Buchstaber and the author, see [1]. The
theory centres around a construction that assigns to each simplicial complex
K with m vertices a T^m -space Z_K with a special bigraded cellular
decomposition. In this framework, the algebraic non-singular toric varieties
arise as orbit spaces of maximal free actions of subtori on moment-angle
complexes corresponding to simplicial spheres. Different combinatorial
invariants of simplicial complexes and other related
combinatorial-geometrical objects acquire a nice and surprisingly regular
interpretation in terms of the bigraded cohomology rings of the
corresponding moment-angle complexes.

[1] V.M.Buchstaber and T.E.Panov.
Torus actions, combinatorial topology and homology algebra,
Russian Math. Surveys 55 (2000), no. 5, 825-921.
Available at http://arXiv.org/abs/math.AT/0010073.