Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929751913922560 |
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| author | Bao, Linus Hazel, Christy Karkos, Tia Kessler, Alice Nicolas, Austin Ormsby, Kyle Park, Jeremie Schleff, Cait Tilton, Scotty |
| author_facet | Bao, Linus Hazel, Christy Karkos, Tia Kessler, Alice Nicolas, Austin Ormsby, Kyle Park, Jeremie Schleff, Cait Tilton, Scotty |
| contents | We prove that Hill's characteristic function $χ$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $χ$ have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_13835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games Bao, Linus Hazel, Christy Karkos, Tia Kessler, Alice Nicolas, Austin Ormsby, Kyle Park, Jeremie Schleff, Cait Tilton, Scotty Algebraic Topology Combinatorics We prove that Hill's characteristic function $χ$ for transfer systems on a lattice $P$ surjects onto interior operators for $P$. Moreover, the fibers of $χ$ have unique maxima which are exactly the saturated transfer systems. In order to apply this theorem in examples relevant to equivariant homotopy theory, we develop the theory of saturated transfer systems on modular lattices, ultimately producing a ``matchstick game'' that puts saturated transfer systems in bijection with certain structured subsets of covering relations. After an interlude developing a recursion for transfer systems on certain combinations of bounded posets, we apply these results to determine the full lattice of transfer systems for rank two elementary Abelian groups. |
| title | Transfer systems for rank two elementary Abelian groups: characteristic functions and matchstick games |
| topic | Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2310.13835 |