Classification of locally standard $T$-pseudomanifolds over topological stratified pseudomanifolds

TL;DR

The paper develops a comprehensive classification of locally standard $T$-pseudomanifolds over topological stratified pseudomanifolds by introducing characteristic data $(Q,\lambda,c)$. It constructs canonical models $X(Q,\lambda,c)$ and a model space $Y(Q,\lambda,c)$, and proves that, under the condition that the top strata are homotopy equivalent to the whole orbit space, these spaces are classified up to (weak) $T$-equivariant homeomorphism by isomorphism of their characteristic data. The framework unifies complete toric varieties and compact locally standard $T$-manifolds, extending the Davis–Januszkiewicz quasitoric classification to singular settings via a robust G-pseudomanifold/stratified-pseudomanifold toolkit. The results provide canonical models and explicit constructions, enabling both theoretical insights and concrete computations for orbit-space topology, Chern classes of free orbits, and isotropy data encoded by a characteristic functor. This broadens toric topology to a wider class of spaces with torus actions and singular orbit spaces, with direct links to quasitoric and toric-geometric objects.

Abstract

We introduce the notion of a locally standard $T$-pseudomanifold, a class that generalizes both complete toric varieties and locally standard $T$-manifolds. The main goal of this paper is to show that locally standard $T$-pseudomanifolds over topological stratified pseudomanifolds satisfying certain conditions are completely classified, up to (weakly) equivariant homeomorphism, by their characteristic data. This result extends the classification of quasitoric manifolds by Davis-Januszkiewicz.

Figures (5)

• Figure 1: $C_{\Sigma}$, $C_{\Sigma}'$ and spherical duals
• Figure 2: Filtration of $\mathring{c} \left( \widetilde{T_p S^{n-1}} \right)$ and $\mathring{c}\left(\widetilde{T_p N_p}\right)$
• Figure 3: Thom space of the complex line bundle over the Hirzebruch surface
• Figure :
• Figure :

Theorems & Definitions (143)

• Theorem 1.1: Theorem \ref{['relations']}
• Theorem 1.2: classification theorem (Theorem \ref{['classification theorem']})
• Definition 2.1: manifold stratified space Fri20
• Remark 2.2
• Remark 2.3
• Definition 3.1: open cone Max19
• Definition 3.2
• Remark 3.3
• Remark 3.4
• Proposition 3.5