Small covers of graph-associahedra and realization of cycles
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arXiv
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| Formato: | Preprint |
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2016
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| _version_ | 1866929347189800960 |
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| author | Gaifullin, Alexander A. |
| author_facet | Gaifullin, Alexander A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_01816 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | Small covers of graph-associahedra and realization of cycles Gaifullin, Alexander A. Algebraic Topology Combinatorics Geometric Topology 57N65, 52B20, 52B70, 05E45, 20F55 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. |
| title | Small covers of graph-associahedra and realization of cycles |
| topic | Algebraic Topology Combinatorics Geometric Topology 57N65, 52B20, 52B70, 05E45, 20F55 |
| url | https://arxiv.org/abs/1611.01816 |