Mathematics – Algebraic Topology
Scientific paper
2003-06-05
Osaka J. Math. 43 (2006), 711-746
Mathematics
Algebraic Topology
30 pages, LaTeX2e; minor changes
Scientific paper
A torus manifold is an even-dimensional manifold acted on by a half-dimensional torus with non-empty fixed point set and some additional orientation data. It may be considered as a far-reaching generalisation of toric manifolds from algebraic geometry. The orbit space of a torus manifold has a rich combinatorial structure, e.g., it is a manifold with corners provided that the action is locally standard. Here we investigate relationships between the cohomological properties of torus manifolds and the combinatorics of their orbit quotients. We show that the cohomology ring of a torus manifold is generated by two-dimensional classes if and only if the quotient is a homology polytope. In this case we retrieve the familiar picture from toric geometry: the equivariant cohomology is the face ring of the nerve simplicial complex and the ordinary cohomology is obtained by factoring out certain linear forms. In a more general situation, we show that the odd-degree cohomology of a torus manifold vanishes if and only if the orbit space is face-acyclic. Although the cohomology is no longer generated in degree two under these circumstances, the equivariant cohomology is still isomorphic to the face ring of an appropriate simplicial poset.
Masuda Mikiya
Panov Taras
No associations
LandOfFree
On the cohomology of torus manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the cohomology of torus manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the cohomology of torus manifolds will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-24200