
Mathematics 1999
Torus actions, equivariant momentangle complexes, and coordinate subspace arrangementsAbstract: We show that the cohomology algebra of the complement of a coordinate subspace arrangement in mdimensional complex space is isomorphic to the cohomology algebra of StanleyReisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and the momentangle complex defined by the simplicial complex. The momentangle complex is a certain subset of a unit polydisk in mdimensional complex space invariant with respect to the action of an mdimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere, but otherwise has more complicated structure. Then we investigate the equivariant topology of the momentangle complex and apply the EilenbergMoore spectral sequence. We also relate our results with well known facts in the theory of toric varieties and symplectic geometry.
