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The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

Dang Vo Phuc

Abstract

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra $\mathscr A$ in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko's conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as $\mathscr A$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$ for all $n\ge 3$. For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated $GL(5,\mathbb F_2)$-module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1,\ldots,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.

The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

Abstract

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko's conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that and are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as -modules, and we further determine the homotopy type of the quotient for all . For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated -module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables relative to parameter vectors, and prove that this conjecture holds for all in certain degrees.
Paper Structure (17 sections, 24 theorems, 122 equations, 1 figure)

This paper contains 17 sections, 24 theorems, 122 equations, 1 figure.

Table of Contents

  1. Context and Motivation
  2. Outline of Our Contributions
  3. Preliminaries
  4. The 2-Primary Steenrod Algebra and Related Applications
  5. Some Definitions and Related Results
  6. A Review of Several Known Homomorphisms
  7. Proofs of Theorems \ref{['dlc1']}, \ref{['dlc2']}, and \ref{['dlc3']}
  8. Proof of Theorem \ref{['dlc1']}
  9. Proof of Theorem \ref{['dlc2']}
  10. Proof of Theorem \ref{['dlc3']}
  11. Conclusion and further directions
  12. Appendix
  13. Explicit forms of the invariant representatives $\widetilde{\xi}_{n_0}$ and $\widetilde{\xi}_{n_1}$
  14. An explicit basis for the space $(Q_{n_0 = 18}^{\otimes 5})^{\overline{\mathsf{Param}}_{[4]}}$
  15. The nonadmissible monomials $T_i$, $1\leq i\leq 1685$, with $\mathsf{Param}(T_i) = (3,3,2,1,1)$
  16. ...and 2 more sections

Key Result

Theorem 2.1

Given the arithmetic function where the $\alpha$ function counts the number of ones in the binary expansion of its argument.

Figures (1)

  • Figure 1: A schematic CW-picture of the quotient $X_n=\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$. Collapsing the subcomplex $\mathbb{C}P^{\,n-2}$ to a point leaves a two-cell complex of the form $\mathbb S^{2n-2}\cup_f e^{2n}$. If $n$ is odd, then the attaching map is null-homotopic and $X_n\simeq \mathbb S^{2n-2}\vee \mathbb S^{2n}$. If $n$ is even, then the attaching map is nontrivial and $X_n\simeq \Sigma^{2n-4}\mathbb{C}P^2$.

Theorems & Definitions (35)

  • Theorem 2.1: see R.W, M.K
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Corollary 2.8
  • Theorem 2.9
  • Example 3.1.1
  • ...and 25 more