37 problems

Singer's conjecture. The homomorphism

Odd-primary transfer conjecture. The transfer homomorphism $Tr_h^{\mathscr A_p}(\mathbb F_p)$ is one-to-one for any odd prime $p$ and $1\leq h\leq 4$. This extends the known low-ra…

Sum's conjecture.

Let $X$ be a simply connected polyGEM of finite type at $2$, and let $\widetilde{\operatorname{H}}^*X$ denote its reduced cohomology with coefficients in $\mathbb{F}_2$. Grodal's c…

Let $G$ be the group acting in the hyperreal setting, let $C_2$ be its indicated subgroup, and let $E_{2+\sigma}$ and $BBU_{\mathbb R}$ denote the spaces appearing in the source. W…

Let $P_h$ be the polynomial algebra on $h$ variables over $\mathbb F_p$, equipped with the action of the Steenrod algebra $\mathscr A_p$. For a positive integer $n$, write its $p$-…

Detection conjecture. The nonzero element $t$ is detected by the sixth algebraic transfer $Tr_6(\mathbb F_2)$.

Filtration submodule and annihilation conjecture. For each $1\leq k<\min(m,n)$, $\mathcal{F}_{n,k}$ is an $\mathcal{A}$-submodule and a ${\mathcal{D}}_n$-submodule of…

Let $P_k=\mathbb F_2[x_1,\ldots,x_k]$ be the polynomial algebra, let $Q P_k$ denote its quotient by the image of the positive-degree Steenrod algebra, and let … be Kameko's squarin…

Let $\mathscr A_2$ be the mod-$2$ Steenrod algebra, let $n_d$ be the internal degree sequence used in the paper, and let $Tr_5^{\mathscr A_2}$ denote the fifth algebraic transfer.…

Let $n\geq 1$, let $\epsilon$ be $0$ or $1$, and let $l\geq 0$. Set $m=2^{n+1}-\epsilon+2l$, and let $k_n(\operatorname{Gr}_d({\mathbb R}^m))$ denote the dimension of…

Coinvariant-dimension conjecture. Its dimension is

Let $k=\mathbb Z_2$, let $P_4=k[x_1,x_2,x_3,x_4]$, and write $Q^{\otimes 4}=k\otimes_A P_4$. For a positive integer $s$, Kameko's squaring operation is the $kGL_4(k)$-module epimor…

Coinvariant-dimension conjecture. This space is trivial if $s=2$ and has dimension $1$ if $s\neq 2$.

Let $\mathcal{A}$ be the mod-$2$ Steenrod algebra, let $\beta$ denote the relevant operation, and consider the quotient left $\mathcal{A}$-module $\mathcal{A}/\mathcal{A}\beta$. Fo…

Mahowald's conjecture. For any natural numbers $n$ and $k$, the class $v_n^{2^{n+1+k}}$ is defined and nonzero in

Let $A$ be the mod $p$ Steenrod algebra, let $M$ be an unstable $A$-module, and let $P_s=H^*(B(\mathbb Z/p)^s)$. Let $\mathscr{R}_sM$ be the $s$-th Singer construction and let…

Let $A$ be the mod $p$ Steenrod algebra, let $M$ be an unstable $A$-module, and let $\Lambda\otimes M^\#$ be the lambda-algebra complex computing…

Let $K^n(\operatorname{\mathcal{U}})$ be the free abelian group generated by isomorphism classes of direct summands of $H^*V_n$, where $V_n=(\operatorname{\mathbb{F}}_p)^n$, and le…

Admissible-monomial prediction. If $\omega$ is a weight vector, then

Let $M_n$ be the mod $2$ motivic cohomology algebra of the classifying space of the elementary abelian $2$-group of rank $n$, and let $QM_n^{d,*}$ denote its degree-$(d,*)$ quotien…

Let $A$ be the mod-$2$ Steenrod algebra, let $D_k$ be the Dickson algebra, and let … be the Lannes–Zarati homomorphism. Let … be Singer's algebraic transfer, where…

Algebraic conjecture on spherical classes. The homomorphism $\varphi_k$ is zero for all $k>2$.

Let $A$ be the mod $p$ Steenrod algebra, let $E_s$ be the elementary abelian group of rank $s$, and let $\mathscr{B}[s]$ be the subobject defined in the paper inside $H^*BE_s$. Wri…

Let $P_k$ be the polynomial algebra on $k$ generators over $\mathbb{F}_2$, let $QP_k$ denote its indecomposable quotient under the Steenrod algebra, and write $(QP_k)_n$ for its de…