Homology cycles in manifolds with locally standard torus actions

Abstract

Let $X$ be a $2n$-manifold with a locally standard action of a compact torus $T^n$. If the free part of action is trivial and proper faces of the orbit space $Q$ are acyclic, then there are three types of homology classes in $X$: (1) classes of face submanifolds; (2) $k$-dimensional classes of $Q$ swept by actions of subtori of dimensions $<k$; (3) relative $k$-classes of $Q$ modulo $\partial Q$ swept by actions of subtori of dimensions $\geqslant k$. The submodule of $H_*(X)$ spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. It is isomorphic to $(\mathbb{Z}[S_Q]/\Theta)/W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset $S_Q$ dual to $Q$; $\Theta$ is the linear system of parameters determined by the characteristic function; and $W$ is a certain submodule, lying in the socle of $\mathbb{Z}[S_Q]/\Theta$. Intersections of homology classes different from face submanifolds are described in terms of intersections on $Q$ and $T^n$.Comment: 25 pages, 3 figures. Minor correction in Lemma 3.3 and a calculations of Subsection 7.