Steenrod Lengths and a Problem of Vakil
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911377293049856 |
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| author | Duc, Khanh Nguyen |
| author_facet | Duc, Khanh Nguyen |
| contents | We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_01672 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Steenrod Lengths and a Problem of Vakil Duc, Khanh Nguyen Representation Theory Algebraic Topology Combinatorics Primary 55P42, Secondary 05C20, 55S10, 68W30 We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers. |
| title | Steenrod Lengths and a Problem of Vakil |
| topic | Representation Theory Algebraic Topology Combinatorics Primary 55P42, Secondary 05C20, 55S10, 68W30 |
| url | https://arxiv.org/abs/2110.01672 |