Grasper families of spheres in $S^2 \times D^2$ and barbell diffeomorphisms of $S^1\times S^2 \times I$
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916716313837568 |
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| author | Fernández, Eduardo Gay, David T. Hartman, Daniel Kosanović, Danica |
| author_facet | Fernández, Eduardo Gay, David T. Hartman, Daniel Kosanović, Danica |
| contents | We show that the fundamental group of framed circles in $S^1 \times D^3$ injects into the fundamental group of framed spheres in $S^2\times D^2$, so that the cokernel is the fundamental group of framed neat disks in $D^4$. In particular, grasper families of circles give rise to countably many nontrivial families of spheres. Ambient extensions of either of these two types of families induce the same barbell diffeomorphisms of $S^1\times S^2\times I$. We give two proofs that these diffeomorphisms are nontrivial and pairwise distinct. This implies infinite generation of the abelian group of isotopy classes of diffeomorphisms of $S^1\times S^2\times I$ that are pseudo-isotopic to the identity, recovering a result of Singh. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Grasper families of spheres in $S^2 \times D^2$ and barbell diffeomorphisms of $S^1\times S^2 \times I$ Fernández, Eduardo Gay, David T. Hartman, Daniel Kosanović, Danica Geometric Topology 57R50 57R40 57K40 57S05 58D10 We show that the fundamental group of framed circles in $S^1 \times D^3$ injects into the fundamental group of framed spheres in $S^2\times D^2$, so that the cokernel is the fundamental group of framed neat disks in $D^4$. In particular, grasper families of circles give rise to countably many nontrivial families of spheres. Ambient extensions of either of these two types of families induce the same barbell diffeomorphisms of $S^1\times S^2\times I$. We give two proofs that these diffeomorphisms are nontrivial and pairwise distinct. This implies infinite generation of the abelian group of isotopy classes of diffeomorphisms of $S^1\times S^2\times I$ that are pseudo-isotopic to the identity, recovering a result of Singh. |
| title | Grasper families of spheres in $S^2 \times D^2$ and barbell diffeomorphisms of $S^1\times S^2 \times I$ |
| topic | Geometric Topology 57R50 57R40 57K40 57S05 58D10 |
| url | https://arxiv.org/abs/2412.07467 |