Infinite families of very exotic spheres with free $S^1$- and $S^3$-actions
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| Format: | Preprint |
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2026
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| _version_ | 1866911541731786752 |
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| author | Bauer, Tilman Quigley, J. D. |
| author_facet | Bauer, Tilman Quigley, J. D. |
| contents | There are two kinds of exotic spheres: bp spheres, which bound parallelizable manifolds, and non-bp spheres, or very exotic spheres, which do not. In the 1960s, W.-C. Hsiang showed that in each dimension where bp spheres exist, there is at least one which admits infinitely many inequivalent smooth free $S^1$-actions, and in each dimension congruent to $3$ modulo $4$, there is at least one bp sphere which admits infinitely many inequivalent smooth free $S^3$-actions. On the other hand, for each fixed prime $p$, smooth free $S^1$- and $S^3$- actions are only known to exist on finitely many very exotic spheres with nontrivial $p$-local Kervaire--Milnor invariant, all in dimension less than approximately $p^3$. In this paper, we use topological modular forms to detect smooth free $S^1$- and $S^3$-actions on infinite families of very exotic spheres with nontrivial $2$- and $3$-local Kervaire--Milnor invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_23241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite families of very exotic spheres with free $S^1$- and $S^3$-actions Bauer, Tilman Quigley, J. D. Algebraic Topology Geometric Topology 55M99, 55P42, 55Q55, 57R60, 57S15, 57S25 There are two kinds of exotic spheres: bp spheres, which bound parallelizable manifolds, and non-bp spheres, or very exotic spheres, which do not. In the 1960s, W.-C. Hsiang showed that in each dimension where bp spheres exist, there is at least one which admits infinitely many inequivalent smooth free $S^1$-actions, and in each dimension congruent to $3$ modulo $4$, there is at least one bp sphere which admits infinitely many inequivalent smooth free $S^3$-actions. On the other hand, for each fixed prime $p$, smooth free $S^1$- and $S^3$- actions are only known to exist on finitely many very exotic spheres with nontrivial $p$-local Kervaire--Milnor invariant, all in dimension less than approximately $p^3$. In this paper, we use topological modular forms to detect smooth free $S^1$- and $S^3$-actions on infinite families of very exotic spheres with nontrivial $2$- and $3$-local Kervaire--Milnor invariants. |
| title | Infinite families of very exotic spheres with free $S^1$- and $S^3$-actions |
| topic | Algebraic Topology Geometric Topology 55M99, 55P42, 55Q55, 57R60, 57S15, 57S25 |
| url | https://arxiv.org/abs/2603.23241 |