Small covers of graph-associahedra and realization of cycles
Alexander A. Gaifullin

TL;DR
This paper proves that certain small covers of graph-associahedra, including those related to the Stasheff associahedron, are URC-manifolds, which can realize any homology class via finite coverings and mappings.
Contribution
It establishes that small covers of graph-associahedra, or their two-fold coverings, are URC-manifolds, providing simpler examples in higher dimensions than previously known.
Findings
Small covers of graph-associahedra are URC-manifolds or their two-fold coverings are.
The two-fold cover of the small cover of the Stasheff associahedron is a URC-manifold.
These manifolds are simpler than previously known URC-manifolds in dimensions 4 and higher.
Abstract
An oriented connected closed manifold is called a URC-manifold if for any oriented connected closed manifold of the same dimension there exists a nonzero degree mapping of a finite-fold covering of onto . This condition is equivalent to the following: For any -dimensional integral homology class of any topological space , a multiple of it can be realized as the image of the fundamental class of a finite-fold covering of under a continuous mapping . In 2007 the author gave a constructive proof of the classical result by Thom that a multiple of any integral homology class can be realized as an image of the fundamental class of an oriented smooth manifold. This construction yields the existence of URC-manifolds of all dimensions. For an important class of manifolds, the so-called small covers of…
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