Homology cycles in manifolds with locally standard torus actions

TL;DR

This paper classifies and describes the homology classes of manifolds with locally standard torus actions, revealing their algebraic structure and intersection properties in relation to the orbit space and face submanifolds.

Contribution

It introduces a detailed classification of homology classes in such manifolds and establishes their algebraic structure using face rings and Buchsbaum simplicial posets.

Findings

Homology classes are categorized into three types based on the orbit space and torus actions.

The submodule of face classes forms an ideal in the homology ring, isomorphic to a quotient of a face ring.

Explicit descriptions of intersections of homology classes are provided in terms of the orbit space and 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 $T^n$.