4 problems

Proposed cohomological lower bound. If

Let $X$ be an $F_0$-space, meaning an elliptic space with evenly graded rational cohomology, and let $p \colon E \to B$ be any fibration of simply connected spaces with fibre $X$.…

Let $f\colon X\rightarrow Y$ be a map whose rationalisation $f_0$ admits a homotopy retraction. Let ${\rm secat}(f_0)$ denote the sectional category of the rationalisation and…

Let $f:Y\rightarrow X$ be a map. A homotopy retraction of $f$ is a map $r:X\rightarrow Y$ such that $rf\simeq \operatorname{id}_Y$. The sectional category of $f$, denoted by…