Small covers of graph-associahedra and realization of cycles

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2 Citations (Scopus)

Abstract

An oriented connected closed manifold Mn is called a URCmanifold if for any oriented connected closed manifold Nn of the same dimension there exists a nonzero-degree mapping of a finite-fold covering Mn of Mn onto Nn. 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 cMn of Mn under a continuous mapping f : Mn → X. In 2007 the author gave a constructive proof of Thom's classical result 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 the previously known URC-manifolds. Bibliography: 39 titles.

Original languageEnglish
Pages (from-to)1537-1561
Number of pages25
JournalSbornik Mathematics
Volume207
Issue number11
DOIs
Publication statusPublished - 2016

Keywords

  • Domination relation
  • Graph-associahedron
  • Realization of cycles
  • Small cover
  • URC-manifold

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