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A Structural Characterization of the Hit Image in the Motivic Steenrod Algebra

Dang Vo Phuc

TL;DR

This work delivers a complete structural description of the top layer of the motivic hit problem in Kameko’s decomposition, revealing that the obstruction to hitting is governed by an odd-versus-even parity dichotomy in a precisely defined M_1-summand. By introducing a parity functional ε and analyzing the M_1-component of Q_0-images, the authors prove that the hit image equals the even-parity hyperplane, making the non-hit quotient one-dimensional and generated by any odd-parity translate. They derive an infinite arithmetic family with n=2^r+1 and k=n-4 for which β(d)>n, yielding new motivic Peterson-type counterexamples distinct from Kameko’s original family; these results persist under base change to any algebraically closed field of characteristic 0. Collectively, the paper identifies the odd-parity gap as the precise obstruction in this top layer and provides explicit non-hit elements and constructions with broad applicability across base fields, significantly advancing understanding of motivic hit theory and Peterson-type conjectures.

Abstract

The motivic hit problem seeks a minimal generating set for the motivic cohomology of classifying spaces as a module over the motivic Steenrod algebra. In this paper, we provide a complete structural classification of the hit image within the top layer of Kameko's decomposition, revealing a rigidity phenomenon governed by parity. Specifically, for degrees $d=k+2d_1$ with $d_1=(n-1)(2^k-1)$, let $V$ denote the span of the monotone translates of Kameko's monomial $z_k$. We prove that the intersection of $V$ with the hit subspace is exactly the even-parity hyperplane. Consequently, the quotient $V/(\text{hits})$ is one-dimensional and is generated by any odd-parity linear combination of these translates. This establishes the "odd-parity gap" as the precise obstruction to the hit property in this layer. As an arithmetic consequence of this structural result, we identify a new infinite numerical family with $n=2^r+1$ and $k=n-4$ ($r\ge 5$) where $β(d)>n$. This yields a new class of counterexamples to the motivic Peterson-type conjecture, distinct from Kameko's original family. Furthermore, we show that these structural results and counterexamples persist over any algebraically closed field of characteristic $0$.

A Structural Characterization of the Hit Image in the Motivic Steenrod Algebra

TL;DR

This work delivers a complete structural description of the top layer of the motivic hit problem in Kameko’s decomposition, revealing that the obstruction to hitting is governed by an odd-versus-even parity dichotomy in a precisely defined M_1-summand. By introducing a parity functional ε and analyzing the M_1-component of Q_0-images, the authors prove that the hit image equals the even-parity hyperplane, making the non-hit quotient one-dimensional and generated by any odd-parity translate. They derive an infinite arithmetic family with n=2^r+1 and k=n-4 for which β(d)>n, yielding new motivic Peterson-type counterexamples distinct from Kameko’s original family; these results persist under base change to any algebraically closed field of characteristic 0. Collectively, the paper identifies the odd-parity gap as the precise obstruction in this top layer and provides explicit non-hit elements and constructions with broad applicability across base fields, significantly advancing understanding of motivic hit theory and Peterson-type conjectures.

Abstract

The motivic hit problem seeks a minimal generating set for the motivic cohomology of classifying spaces as a module over the motivic Steenrod algebra. In this paper, we provide a complete structural classification of the hit image within the top layer of Kameko's decomposition, revealing a rigidity phenomenon governed by parity. Specifically, for degrees with , let denote the span of the monotone translates of Kameko's monomial . We prove that the intersection of with the hit subspace is exactly the even-parity hyperplane. Consequently, the quotient is one-dimensional and is generated by any odd-parity linear combination of these translates. This establishes the "odd-parity gap" as the precise obstruction to the hit property in this layer. As an arithmetic consequence of this structural result, we identify a new infinite numerical family with and () where . This yields a new class of counterexamples to the motivic Peterson-type conjecture, distinct from Kameko's original family. Furthermore, we show that these structural results and counterexamples persist over any algebraically closed field of characteristic .
Paper Structure (18 sections, 22 theorems, 67 equations)

This paper contains 18 sections, 22 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The classical hit problem and the Peterson conjecture
  4. Motivic cohomology of $BV_n$ and the motivic Steenrod algebra
  5. The motivic hit problem and the monomials $z_k$
  6. Weight vectors and the subspaces $F_n$, $G_n$, and $H_n$
  7. A parity functional on Kameko's top layer
  8. The $M_1$--summand and its parity functional
  9. A linear algebra lemma
  10. The $M_1$--part of the $Q_0$--image
  11. The $A^{>0,*}$--image and the odd-parity gap
  12. Arithmetic of the Kameko degrees
  13. Kameko's original estimates
  14. A new family with $k=n-4$
  15. Base change and motivic counterexamples over general fields
  16. ...and 3 more sections

Key Result

Theorem 1.1

If $\beta(d)>n$, then $QP_n^d = 0$.

Theorems & Definitions (42)

  • Theorem 1.1: Wood
  • Definition 1.2: Motivic Peterson-type conjecture KamekoMotivic
  • Theorem 1.3: Kameko KamekoMotivic
  • Theorem 1.4: Structure of the Top Hit Layer
  • Remark 1.5: Minimality of the Obstruction
  • Theorem 1.6: New numerical family with $\beta(d)>n$
  • Theorem 1.7
  • Proposition 2.1: Kameko KamekoMotivic
  • Definition 2.2: Motivic hit quotients KamekoMotivic
  • Proposition 2.3: Kameko KamekoMotivic
  • ...and 32 more