Steenrod Lengths and a Problem of Vakil
Khanh Nguyen Duc
TL;DR
This work delivers an explicit combinatorial solution to computing the Steenrod length function $f(n)$ for real projective spaces by reframing the problem on binary classes and Vakil numbers within the directed graphs $T_n$. The authors prove concrete formulas: for Vakil numbers $f(n)=k+\frac{a(a+1)}{2}$, for numbers in a $k$-dimensional binary class $f(n)=S(\overline{n})+\Delta_k$, and in general $f(n)=S(\overline{n})+\Delta^{\widehat{n}}$ where $\widehat{n}$ is the closest Vakil number to $\overline{n}$ with $4|k$, with an efficient reduction algorithm via canonical paths. They also provide a complete proof of these statements (including detailed lemmas and a constructive reduction procedure) and illustrate the method with explicit examples, thereby resolving Vakil’s open question on an explicit combinatorial description of $f(n)$. The results connect homotopy-theoretic invariants to precise combinatorial data, enabling practical computation and offering new insights into the interaction between Steenrod operations, binary expansions, and graph-theoretic encodings of degree constraints.
Abstract
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.