Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homology cycles in manifolds with locally standard torus actions   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 $T^n$.

7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
18
The categories of this paper are: econ
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga