Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915157741928448 |
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| author | Yoshioka, Leo |
| author_facet | Yoshioka, Leo |
| contents | In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2-loop or 3-loop part of modified configuration space integrals using a modified Bott-Cattaneo-Rossi graph complex. We show the non-triviality of the cycles by pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2-loop part, we provide an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long embeddings of codimension two, which Budney--Gabai and Watanabe first established. We also show the non-finite generation of the 2(j-1)-th homotopy group by using the 3-loop part. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs Yoshioka, Leo Algebraic Topology Geometric Topology 57K45, 57K16, 57R40, 57R22, 57M15 In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2-loop or 3-loop part of modified configuration space integrals using a modified Bott-Cattaneo-Rossi graph complex. We show the non-triviality of the cycles by pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2-loop part, we provide an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long embeddings of codimension two, which Budney--Gabai and Watanabe first established. We also show the non-finite generation of the 2(j-1)-th homotopy group by using the 3-loop part. |
| title | Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs |
| topic | Algebraic Topology Geometric Topology 57K45, 57K16, 57R40, 57R22, 57M15 |
| url | https://arxiv.org/abs/2502.12547 |