Cocycles of the space of long embeddings and BCR graphs with more than one loop
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866908592276242432 |
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| author | Yoshioka, Leo |
| author_facet | Yoshioka, Leo |
| contents | The purpose of this paper is to construct non-trivial cocycles of the space $Emb(\mathbb{R}^j, \mathbb{R}^{n})$ of long embeddings. We construct the cocycles by integral over configuration spaces, associated with Bott-Cattaneo-Rossi graphs with more than one loop. As an application, we give explicitly a non-trivial family of trivial long embeddings for odd $n,j$ with $n-j \geq 2$ and $j \geq 3$. This family (cycle) is constructed from a chord diagram on directed lines. The non-triviality is shown by cocycle-cycle paring, described by paring between graphs and chord diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_01573 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cocycles of the space of long embeddings and BCR graphs with more than one loop Yoshioka, Leo Geometric Topology 57K16, 57K45, 57M15, 57R40 The purpose of this paper is to construct non-trivial cocycles of the space $Emb(\mathbb{R}^j, \mathbb{R}^{n})$ of long embeddings. We construct the cocycles by integral over configuration spaces, associated with Bott-Cattaneo-Rossi graphs with more than one loop. As an application, we give explicitly a non-trivial family of trivial long embeddings for odd $n,j$ with $n-j \geq 2$ and $j \geq 3$. This family (cycle) is constructed from a chord diagram on directed lines. The non-triviality is shown by cocycle-cycle paring, described by paring between graphs and chord diagrams. |
| title | Cocycles of the space of long embeddings and BCR graphs with more than one loop |
| topic | Geometric Topology 57K16, 57K45, 57M15, 57R40 |
| url | https://arxiv.org/abs/2212.01573 |