15 problems

Let $X$ be a space, and let $d$ be the least integer such that $R_d$ is a nonconstant functor of finite degree. Assume that $d>1$. Degree-growth conjecture. At least one of the fun…

Let $X$ be a space. For each $s$, let $R_s(H^*X)$ denote the analytic functor associated with the reduced unstable module $R_s(H^*X)$. Kuhn's dichotomy conjecture. Either at least…

Let $X$ be a $2d$-connected space, with $d\geq 1$, and let $M=\widetilde H^*(X;\mathbf F_2)$ be its reduced mod-$2$ cohomology. Assume that all cup products in $M$ are trivial, tha…

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…

The Frobenius-extension conjecture.

Let $K^{red}(\mathcal U)$ be the Grothendieck ring of finite direct sums of indecomposable reduced injective unstable modules, and let $v$ be a virtual module in…

Let $\mathcal U$ be the category of unstable modules, and consider the full subcategory whose objects are unstable modules whose injective hull is a finite direct sum of indecompos…

Let $K^{red}(\mathcal U)$ be the Grothendieck ring of finite direct sums of indecomposable reduced injective unstable modules, and let $K^{red}_n(\mathcal U)$ be the subgroup gener…

Let $L_\lambda$ be the indecomposable injective unstable module indexed by a partition $\lambda$, and let $P_\lambda(q)$ denote its Poincaré series. Linear independence conjecture.…

Let $X$ be a space such that the reduced cohomology $\tilde H^*X$ is nilpotent. Let $q_X=w_{QH^*X}$ be the profile function of the unstable module of indecomposable elements of…

Let $X$ be a space such that the reduced cohomology $\tilde H^*X$ is nilpotent. Let $q_X=w_{QH^*X}$ be the profile function of the unstable module of indecomposable elements of…

Locally finite form of the unbounded strong realization conjecture. If $\mathrm{R}_sM$ is of finite weight for each $s$ and $M$ is topologically realizable, then $M$ is locally fin…

Let $M$ be an $\mathcal A$-module concentrated in finitely many degrees, where $\mathcal A$ is the mod 2 Steenrod algebra. Local Realization Conjecture. For all sufficiently large…