Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918534669402112 |
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| author | Irikura, Daiki |
| author_facet | Irikura, Daiki |
| contents | In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $π_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of $\mathrm{Emb}_c(\mathbb{R}^{n-2}, \mathbb{R}^n)$ are infinite-dimensional in infinitely many degrees when $n \ge 5$ is odd. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01903 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2 Irikura, Daiki Algebraic Topology Geometric Topology 58D10 (Primary) 57R40 (Secondary) In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $π_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elements are nontrivial. Consequently, we show that the rational homotopy groups of $\mathrm{Emb}_c(\mathbb{R}^{n-2}, \mathbb{R}^n)$ are infinite-dimensional in infinitely many degrees when $n \ge 5$ is odd. |
| title | Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2 |
| topic | Algebraic Topology Geometric Topology 58D10 (Primary) 57R40 (Secondary) |
| url | https://arxiv.org/abs/2606.01903 |