Cohomology of open torus manifolds

Abstract

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.

The topology of toric origami manifolds

“…Examples of such topological analogues, from most restrictive to most general, are toric manifolds [8] (referred to by some authors as quasitoric manifolds), topological toric manifolds [20] and torus manifolds [23]. We now show that acyclic toric origami manifolds fit into the framework of torus manifolds, and that Theorem 3.6 also follows from the work of Masuda and Panov on the cohomology of torus manifolds [24]. Their theory is more general and their proofs algebraic.…”

“…We now can derive our Theorem 3.6 from Masuda and Panov's work: they prove that face-acyclic locally standard torus manifolds have no odd-degree cohomology [24,Theorem 9.3]. While our proofs have very different flavors, it is interesting to note that a crucial ingredient in their proof is their [24,Lemma 2.3], which is closely related to our Lemma 3.4, as described in Remark 3.5.…”

“…While our proofs have very different flavors, it is interesting to note that a crucial ingredient in their proof is their [24,Lemma 2.3], which is closely related to our Lemma 3.4, as described in Remark 3.5.…”

Todd genera of complex torus manifolds

“…Then 1 .Q 1 /; : : : ; 1 .Q m / are the characteristic submanifolds of X , denoted X 1 ; : : : ; X m before. If Q is a homology polytope, ie any intersection of facets of Q is connected unless empty (this is equivalent to any intersection of characteristic submanifolds of X being connected unless empty), then the geometric realization j † X j of the simplicial complex † X is a homology sphere of dimension n 1 (see [11,Lemma 8.2]), in particular, connected when n 2. Unless Q is a homology polytope, intersections of facets are not necessarily connected.…”

Equivariant classification of 2-torus manifolds

“…Although many interesting examples of torus manifolds are locally standard (e.g. this is the case for compact non-singular toric varieties with restricted action of the compact torus, more generally for torus manifolds with vanishing odd degree cohomology, [11]), the local standardness is not assumed in [8] because a combinatorial object called a multi-fan can be defined without assuming it (see also [10]). As for a 2-torus manifold, we do not require the existence of a fixed point but require that the action be non-free.…”