An application of the hit problem to the algebraic transfer
Nguyen Sum
TL;DR
The paper addresses Singer's cohomological algebraic transfer in the fourth variable case, leveraging results from the Peterson hit problem to compute GL_4-invariant classes in specific degrees. The authors provide explicit bases and invariant generators for $QP_4$ in degrees $d_{s,t}=2^{s+t}+2^s-3$ and $n_{s,t}=2^{s+t}+2^s-2$, proving Singer's conjecture in these families and correcting prior claims in the literature. The main technique combines detailed weight-vector analysis, explicit admissible monomials, and Kameko-type reductions to identify GL_4- and $\ ilde{S}^0$-invariant subspaces and to determine the image and kernel of the algebraic transfer. The results yield precise dimensions and generators for the invariant subspaces, clarifying the structure of Ext groups in these degrees and contributing to a more accurate map between homological and invariant-theoretic data. Overall, the work strengthens the link between the hit problem and the algebraic transfer for k=4 and provides a rigorous foundation for future investigations into higher ranks.
Abstract
Let $P_k$ be the polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_k]$ over the field $\mathbb F_2$ with two elements, in $k$ variables $x_1, x_2, \ldots , x_k$, each variable of degree 1. Denote by $GL_k$ the general linear group over $\mathbb F_2$ which regularly acts on $P_k$. The algebra $P_k$ is a module over the mod-2 Steenrod algebra $\mathcal A$. In 1989, Singer [22] defined the $k$-th homological algebraic transfer, which is a homomorphism $$\varphi_k=(\varphi_k)_m :{\rm Tor}^{\mathcal A}_{k,k+m} (\mathbb F_2,\mathbb F_2) \to (\mathbb F_2\otimes_{\mathcal A}P_k)_m^{GL_k}$$ from the homological group of the mod-2 Steenrod algebra $\mbox{Tor}^{\mathcal A}_{k,k+m} (\mathbb F_2,\mathbb F_2)$ to the subspace $(\mathbb F_2\otimes_{\mathcal A}P_k)_m^{GL_k}$ of $\mathbb F_2{\otimes}_{\mathcal A}P_k$ consisting of all the $GL_k$-invariant classes of degree $m$. In general, the transfer $\varphi_k$ is not a monomorphism and Singer made a conjecture that $\varphi_k$ is an epimorphism for any $k \geqslant 0$. The conjecture is studied by many authors. It is true for $k \leqslant 3$ but unknown for $k \geqslant 4$. In this paper, by using the results of the Peterson hit problem for the polynomial algebra in four variables, we prove that Singer's conjecture for the fourth algebraic transfer is true in the families of generic degrees $d_{s,t} = 2^{s+t}+2^s-3$ and $n_{s,t}=2^{s+t}+2^s-2$ with $s,\, t$ positive integers. Our results also show that many of the results in Phúc [16,17,18] are seriously false. The proofs of the results in Phúc's works are only provided for a few special cases but they are false and incomplete.