On periodic families in the stable stems of height two
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909772276563968 |
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| author | Carrick, Christian Davies, Jack Morgan |
| author_facet | Carrick, Christian Davies, Jack Morgan |
| contents | We discover a host of infinite periodic families in the 2-primary stable homotopy groups of spheres. We also confirm the existence of many families predicted by Hopkins--Mahowald. These families appear in nineteen different congruence classes of degrees modulo 192, seven of them consist of simple 4-torsion elements, and another four of simple 8-torsion. They all vanish in the homotopy groups of the spectrum TMF of topological modular forms, but we show that they are detected in the fixed-points of TMF with respect to an Atkin--Lehner involution. As a consequence, we confirm the existence of exotic spheres in all dimensions congruent to 72, 144, and 168 modulo 192. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20507 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On periodic families in the stable stems of height two Carrick, Christian Davies, Jack Morgan Algebraic Topology K-Theory and Homology 55T05, 55T15, 55T25, 55N20, 55P42, 55N34, 55Q10, 55Q51 We discover a host of infinite periodic families in the 2-primary stable homotopy groups of spheres. We also confirm the existence of many families predicted by Hopkins--Mahowald. These families appear in nineteen different congruence classes of degrees modulo 192, seven of them consist of simple 4-torsion elements, and another four of simple 8-torsion. They all vanish in the homotopy groups of the spectrum TMF of topological modular forms, but we show that they are detected in the fixed-points of TMF with respect to an Atkin--Lehner involution. As a consequence, we confirm the existence of exotic spheres in all dimensions congruent to 72, 144, and 168 modulo 192. |
| title | On periodic families in the stable stems of height two |
| topic | Algebraic Topology K-Theory and Homology 55T05, 55T15, 55T25, 55N20, 55P42, 55N34, 55Q10, 55Q51 |
| url | https://arxiv.org/abs/2506.20507 |