Homology cycles in manifolds with locally standard torus actions

We describe the homology of a closed manifold $X$ with a locally standard action of a half-dimensional torus under the assumption that proper faces of its orbit space $Q$ are acyclic and the free part of action is trivial. There are three types of homology classes in $X$: (1) classes of face submanifolds, (2) $k$-dimensional classes of $Q$ lifted to $X$ and swept by actions of subtori of dimensions $\gt k$, and (3) relative $k$-classes of $Q$ modulo $\partial Q$ lifted, in appropriate way, to $X$ and swept by actions of subtori of dimensions $\geqslant k$. The submodule spanned by face classes is an ideal in $H_*(X)$ with respect to the intersection product. As a ring it is isomorphic to $(\mathbb{Z}[S_Q] / \Theta) / W$, where $\mathbb{Z}[S_Q]$ is the face ring of the Buchsbaum simplicial poset dual to $Q$, $\Theta$ is an ideal generated by a linear system of parameters, and $W$ is a submodule lying in the socle of $\mathbb{Z}[S_Q] / \Theta$. The intersection product in homology is described in terms of the product in the face ring and intersection products on the orbit space and on the torus. Manifolds with torus actions provide a topological interpretation for the results of Novik and Swartz concerning socles of Buchsbaum face rings.

Keywords

locally standard torus action, orbit type filtration, face submanifold, characteristic submanifold, intersection product, face ring, Buchsbaum simplicial poset, socle of a module

2010 Mathematics Subject Classification

05E45, 06A07, 13F50, 13F55, 13H10, 16W50, 55N45, 55R91, 57N65

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Published 31 May 2016