5 problems

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…

Characteristic-rank conjecture. The characteristic rank is

Strong-form conjecture. As unstable algebras, one has

Weak-form conjecture. As graded algebras, one has

Let $E_8$ be the exceptional Lie group, and let $BE_8$ be its classifying space. Write … for the adjoint representation, and let $w_{128}(\rho_8)$ denote its 128-th Stiefel–Whitney…