Small covers of graph-associahedra and realization of cycles

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 $M^n$ is called a URC-manifold if for any oriented connected closed manifold $N^n$ of the same dimension there exists a nonzero degree mapping of a finite-fold covering $\widehat{M}^n$ of $M^n$ onto $N^n$. This condition is equivalent to the following: For any $n$-dimensional integral homology class of any topological space $X$, a multiple of it can be realized as the image of the fundamental class of a finite-fold covering $\widehat{M}^n$ of $M^n$ under a continuous mapping $f\colon \widehat{M}^n\to X$. 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 graph-associahedra corresponding to connected graphs, we prove that either they or their two-fold orientation coverings are URC-manifolds. In particular, we obtain that the two-fold covering of the small cover of the usual Stasheff associahedron is a URC-manifold. In dimensions 4 and higher, this manifold is simpler than all previously known URC-manifolds.