The space of $r$-immersions of a union of discs in $\mathbb R^n$
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915466704846848 |
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| author | Arone, Gregory Sarcevic, Franjo |
| author_facet | Arone, Gregory Sarcevic, Franjo |
| contents | For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that the preimage of every point in $\mathbb R^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb R^n$ and the $k^{\text{th}}$ power of the space of injective linear maps from $\mathbb R^m$ to $\mathbb R^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_09809 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The space of $r$-immersions of a union of discs in $\mathbb R^n$ Arone, Gregory Sarcevic, Franjo Algebraic Topology Primary: 57R42, Secondary: 55R80, 57R40 For a manifold $M$ and an integer $r>1$, the space of $r$-immersions of $M$ in $\mathbb R^n$ is defined to be the space of immersions of $M$ in $\mathbb R^n$ such that the preimage of every point in $\mathbb R^n$ contains fewer than $r$ points. We consider the space of $r$-immersions when $M$ is a disjoint union of $k$ $m$-dimensional discs, and prove that it is equivalent to the product of the $r$-configuration space of $k$ points in $\mathbb R^n$ and the $k^{\text{th}}$ power of the space of injective linear maps from $\mathbb R^m$ to $\mathbb R^n$. This result is needed in order to apply Michael Weiss's manifold calculus to the study of $r$-immersions. The analogous statement for spaces of embeddings is ``well-known'', but a detailed proof is hard to find in the literature, and the existing proofs seem to use the isotopy extension theorem, if only as a matter of convenience. Isotopy extension does not hold for $r$-immersions, so we spell out the details of a proof that avoids using it, and applies to spaces of $r$-immersions. |
| title | The space of $r$-immersions of a union of discs in $\mathbb R^n$ |
| topic | Algebraic Topology Primary: 57R42, Secondary: 55R80, 57R40 |
| url | https://arxiv.org/abs/2212.09809 |